Integrand size = 29, antiderivative size = 731 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {33464 b^2 d^2 \sqrt {d-c^2 d x^2}}{694575 c^4}+\frac {3358 b^2 d^2 x^2 \sqrt {d-c^2 d x^2}}{694575 c^2}+\frac {484 b^2 d^2 x^4 \sqrt {d-c^2 d x^2}}{77175}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {8 b^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}{8505 c^4}+\frac {2 b^2 d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2}}{4725 c^4}-\frac {20 b^2 d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2}}{3969 c^4}+\frac {2 b^2 d^2 (1-c x)^4 (1+c x)^4 \sqrt {d-c^2 d x^2}}{729 c^4}+\frac {4 b d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \] Output:
-33464/694575*b^2*d^2*(-c^2*d*x^2+d)^(1/2)/c^4+3358/694575*b^2*d^2*x^2*(-c ^2*d*x^2+d)^(1/2)/c^2+484/77175*b^2*d^2*x^4*(-c^2*d*x^2+d)^(1/2)-10/3087*b ^2*c^2*d^2*x^6*(-c^2*d*x^2+d)^(1/2)+8/8505*b^2*d^2*(-c*x+1)*(c*x+1)*(-c^2* d*x^2+d)^(1/2)/c^4+2/4725*b^2*d^2*(-c*x+1)^2*(c*x+1)^2*(-c^2*d*x^2+d)^(1/2 )/c^4-20/3969*b^2*d^2*(-c*x+1)^3*(c*x+1)^3*(-c^2*d*x^2+d)^(1/2)/c^4+2/729* b^2*d^2*(-c*x+1)^4*(c*x+1)^4*(-c^2*d*x^2+d)^(1/2)/c^4+4/63*b*d^2*x*(-c^2*d *x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2/189*b*d ^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c/(c*x-1)^(1/2)/(c*x+1)^(1/ 2)-2/21*b*c*d^2*x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/ (c*x+1)^(1/2)+38/441*b*c^3*d^2*x^7*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x)) /(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/81*b*c^5*d^2*x^9*(-c^2*d*x^2+d)^(1/2)*(a+b* arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/63*d^2*(-c^2*d*x^2+d)^(1/2)*(a +b*arccosh(c*x))^2/c^4-1/63*d^2*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x) )^2/c^2+1/21*d^2*x^4*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2+5/63*d*x^4* (-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+1/9*x^4*(-c^2*d*x^2+d)^(5/2)*(a+ b*arccosh(c*x))^2
Time = 0.57 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.39 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (3969 a^2 \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right )-126 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-126-21 c^2 x^2+189 c^4 x^4-171 c^6 x^6+49 c^8 x^8\right )+2 b^2 \left (6140-7039 c^2 x^2-106 c^4 x^4+2152 c^6 x^6-1490 c^8 x^8+343 c^{10} x^{10}\right )+126 b \left (63 a \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right )+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (126+21 c^2 x^2-189 c^4 x^4+171 c^6 x^6-49 c^8 x^8\right )\right ) \text {arccosh}(c x)+3969 b^2 \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right ) \text {arccosh}(c x)^2\right )}{250047 c^4 \left (-1+c^2 x^2\right )} \] Input:
Integrate[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]
Output:
(d^2*Sqrt[d - c^2*d*x^2]*(3969*a^2*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2) - 126* a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-126 - 21*c^2*x^2 + 189*c^4*x^4 - 17 1*c^6*x^6 + 49*c^8*x^8) + 2*b^2*(6140 - 7039*c^2*x^2 - 106*c^4*x^4 + 2152* c^6*x^6 - 1490*c^8*x^8 + 343*c^10*x^10) + 126*b*(63*a*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(126 + 21*c^2*x^2 - 189* c^4*x^4 + 171*c^6*x^6 - 49*c^8*x^8))*ArcCosh[c*x] + 3969*b^2*(-1 + c^2*x^2 )^4*(2 + 7*c^2*x^2)*ArcCosh[c*x]^2))/(250047*c^4*(-1 + c^2*x^2))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int x^4 (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int x^4 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{315 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{315} b c \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c^2 x^2-1}}dx}{315 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {c^2 x^2-1}}dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {35 \left (c^2 x^2-1\right )^{7/2}}{c^4}+\frac {50 \left (c^2 x^2-1\right )^{5/2}}{c^4}+\frac {3 \left (c^2 x^2-1\right )^{3/2}}{c^4}-\frac {4 \sqrt {c^2 x^2-1}}{c^4}+\frac {8}{c^4 \sqrt {c^2 x^2-1}}\right )dx^2}{630 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{9} d \int x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {5}{9} d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int -x^4 (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{9} d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int x^4 (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {5}{9} d \left (-\frac {2 b c d \sqrt {d-c^2 d x^2} \int x^4 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \int \frac {x^5}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\int \frac {4 x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {4 \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {4 \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \int x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \int x^4 (a+b \text {arccosh}(c x))dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\int \frac {4 x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}+\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {5}{9} d \left (\frac {3}{7} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{5 c^2}+\frac {4 \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 x^9 (a+b \text {arccosh}(c x))-\frac {2}{7} c^2 x^7 (a+b \text {arccosh}(c x))+\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \text {arccosh}(c x)) x^9-\frac {2}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \int x^2 (a+b \text {arccosh}(c x))dx}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \text {arccosh}(c x)) x^9-\frac {2}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \text {arccosh}(c x)) x^9-\frac {2}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )\right )}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{9} c^4 (a+b \text {arccosh}(c x)) x^9-\frac {2}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {70 \left (c^2 x^2-1\right )^{9/2}}{9 c^6}+\frac {100 \left (c^2 x^2-1\right )^{7/2}}{7 c^6}+\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^6}+\frac {16 \sqrt {c^2 x^2-1}}{c^6}\right )}{630 \sqrt {c x-1} \sqrt {c x+1}}\right )}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{9} d \left (\frac {1}{7} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^2 (a+b \text {arccosh}(c x)) x^7+\frac {1}{5} (a+b \text {arccosh}(c x)) x^5-\frac {1}{35} b c \left (\frac {19}{7} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )-\frac {5}{7} x^6 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3}{7} d \left (\frac {1}{5} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^4-\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {1}{5} x^5 (a+b \text {arccosh}(c x))-\frac {1}{5} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^4}{5 c^2}+\frac {4 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}\right )}{5 c^2}\right )\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^2}{3 c^2}+\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )\right )}{3 c}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}\right )}{5 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
Input:
Int[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]
Output:
$Aborted
Time = 0.71 (sec) , antiderivative size = 648, normalized size of antiderivative = 0.89
| method | result | size |
| orering | \(\frac {\left (74431 x^{12} c^{12}-287825 c^{10} x^{10}+383719 c^{8} x^{8}-112267 c^{6} x^{6}-485758 c^{4} x^{4}+324980 c^{2} x^{2}-73680\right ) \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{250047 c^{6} \left (c x -1\right ) \left (c x +1\right ) x^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {4 \left (686 c^{10} x^{10}-2503 c^{8} x^{8}+2822 c^{6} x^{6}+511 c^{4} x^{4}-8786 c^{2} x^{2}+3070\right ) \left (3 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}-5 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d +\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{83349 c^{6} \left (c x -1\right ) \left (c x +1\right ) x^{4} \left (c^{2} x^{2}-1\right )}+\frac {\left (343 c^{8} x^{8}-1147 c^{6} x^{6}+1005 c^{4} x^{4}+899 c^{2} x^{2}-6140\right ) \left (6 x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}-35 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d +\frac {12 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}+15 x^{5} \sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{4} d^{2}-\frac {20 x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{3} d b}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{250047 c^{6} \left (c x -1\right ) \left (c x +1\right ) x^{3}}\) | \(648\) |
| default | \(\text {Expression too large to display}\) | \(2224\) |
| parts | \(\text {Expression too large to display}\) | \(2224\) |
Input:
int(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/250047*(74431*c^12*x^12-287825*c^10*x^10+383719*c^8*x^8-112267*c^6*x^6-4 85758*c^4*x^4+324980*c^2*x^2-73680)/c^6/(c*x-1)/(c*x+1)/x^2/(c^2*x^2-1)^2* (-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2-4/83349*(686*c^10*x^10-2503*c^8* x^8+2822*c^6*x^6+511*c^4*x^4-8786*c^2*x^2+3070)/c^6/(c*x-1)/(c*x+1)/x^4/(c ^2*x^2-1)*(3*x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2-5*x^4*(-c^2*d*x ^2+d)^(3/2)*(a+b*arccosh(c*x))^2*c^2*d+2*x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arc cosh(c*x))*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2))+1/250047*(343*c^8*x^8-1147*c^6 *x^6+1005*c^4*x^4+899*c^2*x^2-6140)/c^6/(c*x-1)/(c*x+1)/x^3*(6*x*(-c^2*d*x ^2+d)^(5/2)*(a+b*arccosh(c*x))^2-35*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh( c*x))^2*c^2*d+12*x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))*b*c/(c*x-1)^( 1/2)/(c*x+1)^(1/2)+15*x^5*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2*c^4*d^ 2-20*x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))*c^3*d*b/(c*x-1)^(1/2)/(c* x+1)^(1/2)+2*x^3*(-c^2*d*x^2+d)^(5/2)*b^2*c^2/(c*x-1)/(c*x+1)-x^3*(-c^2*d* x^2+d)^(5/2)*(a+b*arccosh(c*x))*b*c^2/(c*x-1)^(3/2)/(c*x+1)^(1/2)-x^3*(-c^ 2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))*b*c^2/(c*x-1)^(1/2)/(c*x+1)^(3/2))
Time = 0.11 (sec) , antiderivative size = 558, normalized size of antiderivative = 0.76 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {3969 \, {\left (7 \, b^{2} c^{10} d^{2} x^{10} - 26 \, b^{2} c^{8} d^{2} x^{8} + 34 \, b^{2} c^{6} d^{2} x^{6} - 16 \, b^{2} c^{4} d^{2} x^{4} - b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 126 \, {\left (49 \, a b c^{9} d^{2} x^{9} - 171 \, a b c^{7} d^{2} x^{7} + 189 \, a b c^{5} d^{2} x^{5} - 21 \, a b c^{3} d^{2} x^{3} - 126 \, a b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 126 \, {\left ({\left (49 \, b^{2} c^{9} d^{2} x^{9} - 171 \, b^{2} c^{7} d^{2} x^{7} + 189 \, b^{2} c^{5} d^{2} x^{5} - 21 \, b^{2} c^{3} d^{2} x^{3} - 126 \, b^{2} c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 63 \, {\left (7 \, a b c^{10} d^{2} x^{10} - 26 \, a b c^{8} d^{2} x^{8} + 34 \, a b c^{6} d^{2} x^{6} - 16 \, a b c^{4} d^{2} x^{4} - a b c^{2} d^{2} x^{2} + 2 \, a b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (343 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{10} d^{2} x^{10} - 2 \, {\left (51597 \, a^{2} + 1490 \, b^{2}\right )} c^{8} d^{2} x^{8} + 2 \, {\left (67473 \, a^{2} + 2152 \, b^{2}\right )} c^{6} d^{2} x^{6} - 4 \, {\left (15876 \, a^{2} + 53 \, b^{2}\right )} c^{4} d^{2} x^{4} - {\left (3969 \, a^{2} + 14078 \, b^{2}\right )} c^{2} d^{2} x^{2} + 2 \, {\left (3969 \, a^{2} + 6140 \, b^{2}\right )} d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{250047 \, {\left (c^{6} x^{2} - c^{4}\right )}} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="fric as")
Output:
1/250047*(3969*(7*b^2*c^10*d^2*x^10 - 26*b^2*c^8*d^2*x^8 + 34*b^2*c^6*d^2* x^6 - 16*b^2*c^4*d^2*x^4 - b^2*c^2*d^2*x^2 + 2*b^2*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 126*(49*a*b*c^9*d^2*x^9 - 171*a*b*c^7* d^2*x^7 + 189*a*b*c^5*d^2*x^5 - 21*a*b*c^3*d^2*x^3 - 126*a*b*c*d^2*x)*sqrt (-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 126*((49*b^2*c^9*d^2*x^9 - 171*b^2*c^ 7*d^2*x^7 + 189*b^2*c^5*d^2*x^5 - 21*b^2*c^3*d^2*x^3 - 126*b^2*c*d^2*x)*sq rt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 63*(7*a*b*c^10*d^2*x^10 - 26*a*b*c^ 8*d^2*x^8 + 34*a*b*c^6*d^2*x^6 - 16*a*b*c^4*d^2*x^4 - a*b*c^2*d^2*x^2 + 2* a*b*d^2)*sqrt(-c^2*d*x^2 + d))*log(c*x + sqrt(c^2*x^2 - 1)) + (343*(81*a^2 + 2*b^2)*c^10*d^2*x^10 - 2*(51597*a^2 + 1490*b^2)*c^8*d^2*x^8 + 2*(67473* a^2 + 2152*b^2)*c^6*d^2*x^6 - 4*(15876*a^2 + 53*b^2)*c^4*d^2*x^4 - (3969*a ^2 + 14078*b^2)*c^2*d^2*x^2 + 2*(3969*a^2 + 6140*b^2)*d^2)*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x**3*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**2,x)
Output:
Timed out
Time = 0.15 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.64 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a b \operatorname {arcosh}\left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a^{2} + \frac {2}{250047} \, b^{2} {\left (\frac {343 \, \sqrt {c^{2} x^{2} - 1} c^{6} \sqrt {-d} d^{2} x^{8} - 1147 \, \sqrt {c^{2} x^{2} - 1} c^{4} \sqrt {-d} d^{2} x^{6} + 1005 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d^{2} x^{4} + 899 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2} x^{2} - \frac {6140 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2}}{c^{2}}}{c^{2}} - \frac {63 \, {\left (49 \, c^{8} \sqrt {-d} d^{2} x^{9} - 171 \, c^{6} \sqrt {-d} d^{2} x^{7} + 189 \, c^{4} \sqrt {-d} d^{2} x^{5} - 21 \, c^{2} \sqrt {-d} d^{2} x^{3} - 126 \, \sqrt {-d} d^{2} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (49 \, c^{8} \sqrt {-d} d^{2} x^{9} - 171 \, c^{6} \sqrt {-d} d^{2} x^{7} + 189 \, c^{4} \sqrt {-d} d^{2} x^{5} - 21 \, c^{2} \sqrt {-d} d^{2} x^{3} - 126 \, \sqrt {-d} d^{2} x\right )} a b}{3969 \, c^{3}} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxi ma")
Output:
-1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^ 4*d))*b^2*arccosh(c*x)^2 - 2/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2* (-c^2*d*x^2 + d)^(7/2)/(c^4*d))*a*b*arccosh(c*x) - 1/63*(7*(-c^2*d*x^2 + d )^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^4*d))*a^2 + 2/250047*b^2 *((343*sqrt(c^2*x^2 - 1)*c^6*sqrt(-d)*d^2*x^8 - 1147*sqrt(c^2*x^2 - 1)*c^4 *sqrt(-d)*d^2*x^6 + 1005*sqrt(c^2*x^2 - 1)*c^2*sqrt(-d)*d^2*x^4 + 899*sqrt (c^2*x^2 - 1)*sqrt(-d)*d^2*x^2 - 6140*sqrt(c^2*x^2 - 1)*sqrt(-d)*d^2/c^2)/ c^2 - 63*(49*c^8*sqrt(-d)*d^2*x^9 - 171*c^6*sqrt(-d)*d^2*x^7 + 189*c^4*sqr t(-d)*d^2*x^5 - 21*c^2*sqrt(-d)*d^2*x^3 - 126*sqrt(-d)*d^2*x)*arccosh(c*x) /c^3) - 2/3969*(49*c^8*sqrt(-d)*d^2*x^9 - 171*c^6*sqrt(-d)*d^2*x^7 + 189*c ^4*sqrt(-d)*d^2*x^5 - 21*c^2*sqrt(-d)*d^2*x^3 - 126*sqrt(-d)*d^2*x)*a*b/c^ 3
Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="giac ")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int(x^3*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2),x)
Output:
int(x^3*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2), x)
\[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (7 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{8} x^{8}-19 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{6} x^{6}+15 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}-\sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2}+126 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{7}d x \right ) a b \,c^{8}-252 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{5}d x \right ) a b \,c^{6}+126 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}+63 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{7}d x \right ) b^{2} c^{8}-126 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+63 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}\right )}{63 c^{4}} \] Input:
int(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))^2,x)
Output:
(sqrt(d)*d**2*(7*sqrt( - c**2*x**2 + 1)*a**2*c**8*x**8 - 19*sqrt( - c**2*x **2 + 1)*a**2*c**6*x**6 + 15*sqrt( - c**2*x**2 + 1)*a**2*c**4*x**4 - sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - 2*sqrt( - c**2*x**2 + 1)*a**2 + 126*int (sqrt( - c**2*x**2 + 1)*acosh(c*x)*x**7,x)*a*b*c**8 - 252*int(sqrt( - c**2 *x**2 + 1)*acosh(c*x)*x**5,x)*a*b*c**6 + 126*int(sqrt( - c**2*x**2 + 1)*ac osh(c*x)*x**3,x)*a*b*c**4 + 63*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x* *7,x)*b**2*c**8 - 126*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x**5,x)*b** 2*c**6 + 63*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x**3,x)*b**2*c**4))/( 63*c**4)