Integrand size = 29, antiderivative size = 653 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {359 b^2 d^2 x \sqrt {d-c^2 d x^2}}{36864 c^2}+\frac {1079 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{55296}-\frac {209 b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}}{13824}+\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d-c^2 d x^2}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{9216 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{12288 c^3 (1-c x) (1+c x)} \] Output:
359/36864*b^2*d^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+1079/55296*b^2*d^2*x^3*(-c^2* d*x^2+d)^(1/2)-209/13824*b^2*c^2*d^2*x^5*(-c^2*d*x^2+d)^(1/2)+1/256*b^2*c^ 4*d^2*x^7*(-c^2*d*x^2+d)^(1/2)+35/9216*b^2*d^2*(-c^2*d*x^2+d)^(1/2)*arccos h(c*x)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/128*b*d^2*x^2*(-c^2*d*x^2+d)^(1/2 )*(a+b*arccosh(c*x))/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-59/384*b*c*d^2*x^4*(-c^ 2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^(1/2)+17/144*b*c ^3*d^2*x^6*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1/2)/(c*x+1)^( 1/2)-1/32*b*c^5*d^2*x^8*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/(c*x-1)^(1 /2)/(c*x+1)^(1/2)-5/128*d^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/c^ 2+5/64*d^2*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2+5/48*d*x^3*(-c^2* d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2+1/8*x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arcc osh(c*x))^2-5/384*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^3/b/c^3/(c*x -1)^(1/2)/(c*x+1)^(1/2)-73/12288*b^2*d^2*(c^2*x^2-1)^(1/2)*(-c^2*d*x^2+d)^ (1/2)*arctanh(c*x/(c^2*x^2-1)^(1/2))/c^3/(-c*x+1)/(c*x+1)
Time = 5.76 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.39 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=-\frac {d^2 \left (34560 a^2 c x \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+34560 a^2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-271872 a^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-271872 a^2 c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+313344 a^2 c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+313344 a^2 c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-110592 a^2 c^7 x^7 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-110592 a^2 c^8 x^8 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+11520 b^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^3+34560 a^2 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+34560 a^2 c \sqrt {d} x \sqrt {\frac {-1+c x}{1+c x}} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+13824 a b \sqrt {d-c^2 d x^2} \cosh (2 \text {arccosh}(c x))+3456 a b \sqrt {d-c^2 d x^2} \cosh (4 \text {arccosh}(c x))-1536 a b \sqrt {d-c^2 d x^2} \cosh (6 \text {arccosh}(c x))+216 a b \sqrt {d-c^2 d x^2} \cosh (8 \text {arccosh}(c x))-6912 b^2 \sqrt {d-c^2 d x^2} \sinh (2 \text {arccosh}(c x))-864 b^2 \sqrt {d-c^2 d x^2} \sinh (4 \text {arccosh}(c x))+256 b^2 \sqrt {d-c^2 d x^2} \sinh (6 \text {arccosh}(c x))-27 b^2 \sqrt {d-c^2 d x^2} \sinh (8 \text {arccosh}(c x))+24 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x) (576 b \cosh (2 \text {arccosh}(c x))+144 b \cosh (4 \text {arccosh}(c x))-64 b \cosh (6 \text {arccosh}(c x))+9 b \cosh (8 \text {arccosh}(c x))-1152 a \sinh (2 \text {arccosh}(c x))-576 a \sinh (4 \text {arccosh}(c x))+384 a \sinh (6 \text {arccosh}(c x))-72 a \sinh (8 \text {arccosh}(c x)))-288 b \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2 (-120 a+48 b \sinh (2 \text {arccosh}(c x))+24 b \sinh (4 \text {arccosh}(c x))-16 b \sinh (6 \text {arccosh}(c x))+3 b \sinh (8 \text {arccosh}(c x)))\right )}{884736 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \] Input:
Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]
Output:
-1/884736*(d^2*(34560*a^2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^ 2] + 34560*a^2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 27 1872*a^2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 271872*a ^2*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 313344*a^2*c^5 *x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 313344*a^2*c^6*x^6*S qrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 110592*a^2*c^7*x^7*Sqrt[(- 1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 110592*a^2*c^8*x^8*Sqrt[(-1 + c* x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 11520*b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[ c*x]^3 + 34560*a^2*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 34560*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 13824*a*b*Sqrt[d - c^2*d*x^2]*Cosh[2*ArcCosh[c*x]] + 3456*a*b*Sqrt[d - c^2*d*x^2]*Cosh[4*ArcCosh[c*x]] - 1536*a*b*Sqrt[d - c^2*d*x^2]*Cosh[6*ArcC osh[c*x]] + 216*a*b*Sqrt[d - c^2*d*x^2]*Cosh[8*ArcCosh[c*x]] - 6912*b^2*Sq rt[d - c^2*d*x^2]*Sinh[2*ArcCosh[c*x]] - 864*b^2*Sqrt[d - c^2*d*x^2]*Sinh[ 4*ArcCosh[c*x]] + 256*b^2*Sqrt[d - c^2*d*x^2]*Sinh[6*ArcCosh[c*x]] - 27*b^ 2*Sqrt[d - c^2*d*x^2]*Sinh[8*ArcCosh[c*x]] + 24*b*Sqrt[d - c^2*d*x^2]*ArcC osh[c*x]*(576*b*Cosh[2*ArcCosh[c*x]] + 144*b*Cosh[4*ArcCosh[c*x]] - 64*b*C osh[6*ArcCosh[c*x]] + 9*b*Cosh[8*ArcCosh[c*x]] - 1152*a*Sinh[2*ArcCosh[c*x ]] - 576*a*Sinh[4*ArcCosh[c*x]] + 384*a*Sinh[6*ArcCosh[c*x]] - 72*a*Sin...
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.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle -\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{24} b c \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (3 c^4 x^4-8 c^2 x^2+6\right )}{\sqrt {c^2 x^2-1}}dx}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {c^2 x^4 \left (48-43 c^2 x^2\right )}{\sqrt {c^2 x^2-1}}dx}{8 c^2}+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \int \frac {x^4 \left (48-43 c^2 x^2\right )}{\sqrt {c^2 x^2-1}}dx+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \int \frac {x^4}{\sqrt {c^2 x^2-1}}dx-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{8} d \int x^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6345 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {b c d \sqrt {d-c^2 d x^2} \int -x^3 (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^3 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6336 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6341 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^3 (a+b \text {arccosh}(c x))dx}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {\int \frac {3 x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (-\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {3 \left (\frac {\text {arccosh}(c x)}{2 c^3}+\frac {x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c x-1} \sqrt {c x+1}}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 x^8 (a+b \text {arccosh}(c x))-\frac {1}{3} c^2 x^6 (a+b \text {arccosh}(c x))+\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2-1}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{8} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 (a+b \text {arccosh}(c x)) x^8-\frac {1}{3} c^2 (a+b \text {arccosh}(c x)) x^6+\frac {1}{4} (a+b \text {arccosh}(c x)) x^4-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {3}{8} c^2 \sqrt {c^2 x^2-1} x^7+\frac {1}{8} \left (\frac {73}{6} \left (\frac {\sqrt {c^2 x^2-1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c^2 x^2-1} x}{2 c^2}+\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}\right )}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{8} d \left (\frac {1}{6} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 (a+b \text {arccosh}(c x)) x^6+\frac {1}{4} (a+b \text {arccosh}(c x)) x^4-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x}{2 c^2}+\frac {\text {arccosh}(c x)}{2 c^3}\right )}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} d \left (\frac {1}{4} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x}{2 c^2}+\frac {\text {arccosh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}-\frac {b \int x (a+b \text {arccosh}(c x))dx}{c}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {1}{8} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {1}{8} c^4 (a+b \text {arccosh}(c x)) x^8-\frac {1}{3} c^2 (a+b \text {arccosh}(c x)) x^6+\frac {1}{4} (a+b \text {arccosh}(c x)) x^4-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {3}{8} c^2 \sqrt {c^2 x^2-1} x^7+\frac {1}{8} \left (\frac {73}{6} \left (\frac {\sqrt {c^2 x^2-1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c^2 x^2-1} x}{2 c^2}+\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}\right )}{4 c^2}\right )-\frac {43}{6} x^5 \sqrt {c^2 x^2-1}\right )\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{8} d \left (\frac {1}{6} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} c^2 (a+b \text {arccosh}(c x)) x^6+\frac {1}{4} (a+b \text {arccosh}(c x)) x^4-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x}{2 c^2}+\frac {\text {arccosh}(c x)}{2 c^3}\right )}{4 c^2}\right )-\frac {1}{3} x^5 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{2} d \left (\frac {1}{4} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 x^3-\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{4} x^4 (a+b \text {arccosh}(c x))-\frac {1}{4} b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x^3}{4 c^2}+\frac {3 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} x}{2 c^2}+\frac {\text {arccosh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {d-c^2 d x^2} \left (\frac {x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arccosh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{c}+\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}\right )}{4 \sqrt {c x-1} \sqrt {c x+1}}\right )\right )\) |
Input:
Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(2527\) vs. \(2(565)=1130\).
Time = 0.59 (sec) , antiderivative size = 2528, normalized size of antiderivative = 3.87
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2528\) |
| parts | \(\text {Expression too large to display}\) | \(2528\) |
Input:
int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)
Output:
-1/8*a^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+1/48*a^2/c^2*x*(-c^2*d*x^2+d)^(5/2)+ 5/192*a^2/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a^2/c^2*d^2*x*(-c^2*d*x^2+d)^ (1/2)+5/128*a^2/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d )^(1/2))+b^2*(-5/384*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^ 3*arccosh(c*x)^3*d^2+1/65536*(-d*(c^2*x^2-1))^(1/2)*(128*c^9*x^9-320*c^7*x ^7+128*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^8*c^8+272*c^5*x^5-256*c^6*x^6*(c*x-1) ^(1/2)*(c*x+1)^(1/2)-88*c^3*x^3+160*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+8* c*x-32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(3 2*arccosh(c*x)^2-8*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)-1/6912*(-d*(c^2 *x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*c^6*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/ 2)+38*c^3*x^3-48*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c*x+18*(c*x-1)^(1/2 )*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(18*arccosh(c*x)^2-6* arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)+1/2048*(-d*(c^2*x^2-1))^(1/2)*(8*c ^5*x^5-12*c^3*x^3+8*c^4*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x-8*(c*x-1)^(1 /2)*(c*x+1)^(1/2)*c^2*x^2+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(8*arccosh(c*x)^2-4 *arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)+1/256*(-d*(c^2*x^2-1))^(1/2)*(2*c ^3*x^3-2*c*x+2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2-(c*x-1)^(1/2)*(c*x+1)^( 1/2))*(2*arccosh(c*x)^2-2*arccosh(c*x)+1)*d^2/(c*x+1)/c^3/(c*x-1)+1/256*(- d*(c^2*x^2-1))^(1/2)*(-2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+2*c^3*x^3+(c* x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*(2*arccosh(c*x)^2+2*arccosh(c*x)+1)*d^2...
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="fric as")
Output:
integral((a^2*c^4*d^2*x^6 - 2*a^2*c^2*d^2*x^4 + a^2*d^2*x^2 + (b^2*c^4*d^2 *x^6 - 2*b^2*c^2*d^2*x^4 + b^2*d^2*x^2)*arccosh(c*x)^2 + 2*(a*b*c^4*d^2*x^ 6 - 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d), x )
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**2,x)
Output:
Timed out
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="maxi ma")
Output:
1/384*(8*(-c^2*d*x^2 + d)^(5/2)*x/c^2 - 48*(-c^2*d*x^2 + d)^(7/2)*x/(c^2*d ) + 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^2 + 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^2 + 15*d^(5/2)*arcsin(c*x)/c^3)*a^2 + integrate((-c^2*d*x^2 + d)^(5/2)*b^2*x ^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2*(-c^2*d*x^2 + d)^(5/2)*a*b *x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \] Input:
integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2,x, algorithm="giac ")
Output:
integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)^2*x^2, x)
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int(x^2*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2),x)
Output:
int(x^2*(a + b*acosh(c*x))^2*(d - c^2*d*x^2)^(5/2), x)
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {asin} \left (c x \right ) a^{2}+48 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{7} x^{7}-136 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{5} x^{5}+118 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}-15 \sqrt {-c^{2} x^{2}+1}\, a^{2} c x +768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{6}d x \right ) a b \,c^{7}-1536 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}+768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}-768 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+384 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}\right )}{384 c^{3}} \] Input:
int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))^2,x)
Output:
(sqrt(d)*d**2*(15*asin(c*x)*a**2 + 48*sqrt( - c**2*x**2 + 1)*a**2*c**7*x** 7 - 136*sqrt( - c**2*x**2 + 1)*a**2*c**5*x**5 + 118*sqrt( - c**2*x**2 + 1) *a**2*c**3*x**3 - 15*sqrt( - c**2*x**2 + 1)*a**2*c*x + 768*int(sqrt( - c** 2*x**2 + 1)*acosh(c*x)*x**6,x)*a*b*c**7 - 1536*int(sqrt( - c**2*x**2 + 1)* acosh(c*x)*x**4,x)*a*b*c**5 + 768*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)*x* *2,x)*a*b*c**3 + 384*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x**6,x)*b**2 *c**7 - 768*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x**4,x)*b**2*c**5 + 3 84*int(sqrt( - c**2*x**2 + 1)*acosh(c*x)**2*x**2,x)*b**2*c**3))/(384*c**3)