Integrand size = 24, antiderivative size = 386 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx=-\frac {12632 b^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}{3375 c}+\frac {8}{135} b^3 c d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}+\frac {8 b^3 d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{375 c}-\frac {6 b^3 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{625 c}+\frac {298}{75} b^2 d^2 x (a+b \text {arccosh}(c x))-\frac {76}{225} b^2 c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {6}{125} b^2 c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {8 b d^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{5 c}+\frac {4 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))^2}{15 c}-\frac {3 b d^2 (-1+c x)^{5/2} (1+c x)^{5/2} (a+b \text {arccosh}(c x))^2}{25 c}+\frac {8}{15} d^2 x (a+b \text {arccosh}(c x))^3+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3 \] Output:
-12632/3375*b^3*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+8/135*b^3*c*d^2*x^2*(c*x -1)^(1/2)*(c*x+1)^(1/2)+8/375*b^3*d^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c-6/625* b^3*d^2*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c+298/75*b^2*d^2*x*(a+b*arccosh(c*x))- 76/225*b^2*c^2*d^2*x^3*(a+b*arccosh(c*x))+6/125*b^2*c^4*d^2*x^5*(a+b*arcco sh(c*x))-8/5*b*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^2/c+4/15 *b*d^2*(c*x-1)^(3/2)*(c*x+1)^(3/2)*(a+b*arccosh(c*x))^2/c-3/25*b*d^2*(c*x- 1)^(5/2)*(c*x+1)^(5/2)*(a+b*arccosh(c*x))^2/c+8/15*d^2*x*(a+b*arccosh(c*x) )^3+4/15*d^2*x*(-c^2*x^2+1)*(a+b*arccosh(c*x))^3+1/5*d^2*x*(-c^2*x^2+1)^2* (a+b*arccosh(c*x))^3
Time = 1.37 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.90 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx=\frac {d^2 \left (1125 a^3 c x \left (15-10 c^2 x^2+3 c^4 x^4\right )-225 a^2 b \sqrt {-1+c x} \sqrt {1+c x} \left (149-38 c^2 x^2+9 c^4 x^4\right )+30 a b^2 c x \left (2235-190 c^2 x^2+27 c^4 x^4\right )-2 b^3 \sqrt {-1+c x} \sqrt {1+c x} \left (31841-842 c^2 x^2+81 c^4 x^4\right )+15 b \left (225 a^2 c x \left (15-10 c^2 x^2+3 c^4 x^4\right )-30 a b \sqrt {-1+c x} \sqrt {1+c x} \left (149-38 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (2235-190 c^2 x^2+27 c^4 x^4\right )\right ) \text {arccosh}(c x)-225 b^2 \left (-15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (149-38 c^2 x^2+9 c^4 x^4\right )\right ) \text {arccosh}(c x)^2+1125 b^3 c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)^3\right )}{16875 c} \] Input:
Integrate[(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^3,x]
Output:
(d^2*(1125*a^3*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) - 225*a^2*b*Sqrt[-1 + c*x ]*Sqrt[1 + c*x]*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 30*a*b^2*c*x*(2235 - 190* c^2*x^2 + 27*c^4*x^4) - 2*b^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(31841 - 842*c^ 2*x^2 + 81*c^4*x^4) + 15*b*(225*a^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) - 30 *a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 2*b^2*c *x*(2235 - 190*c^2*x^2 + 27*c^4*x^4))*ArcCosh[c*x] - 225*b^2*(-15*a*c*x*(1 5 - 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(149 - 38*c^2 *x^2 + 9*c^4*x^4))*ArcCosh[c*x]^2 + 1125*b^3*c*x*(15 - 10*c^2*x^2 + 3*c^4* x^4)*ArcCosh[c*x]^3))/(16875*c)
Time = 3.02 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.26, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6312, 27, 6312, 6294, 6330, 25, 2009, 6304, 6309, 27, 960, 83, 1905, 1576, 1140, 2009}
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.
| \(\displaystyle \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {4}{5} d \int d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3dx-\frac {3}{5} b c d^2 \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} d^2 \int \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3dx-\frac {3}{5} b c d^2 \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {4}{5} d^2 \left (b c \int x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2dx+\frac {2}{3} \int (a+b \text {arccosh}(c x))^3dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3\right )-\frac {3}{5} b c d^2 \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 6294 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )+b c \int x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2dx+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3\right )-\frac {3}{5} b c d^2 \int x (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2dx+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \int (a+b \text {arccosh}(c x))dx}{c}\right )\right )+b c \left (\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}-\frac {2 b \int -((1-c x) (c x+1) (a+b \text {arccosh}(c x)))dx}{3 c}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3\right )-\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \int (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \int (a+b \text {arccosh}(c x))dx}{c}\right )\right )+b c \left (\frac {2 b \int (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3\right )-\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \int (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{5} d^2 \left (b c \left (\frac {2 b \int (1-c x) (c x+1) (a+b \text {arccosh}(c x))dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \int (1-c x)^2 (c x+1)^2 (a+b \text {arccosh}(c x))dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle \frac {4}{5} d^2 \left (b c \left (\frac {2 b \int \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))dx}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \int \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))dx}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 6309 |
| \(\displaystyle \frac {4}{5} d^2 \left (b c \left (\frac {2 b \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \left (-b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{15 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} d^2 \left (b c \left (\frac {2 b \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle \frac {4}{5} d^2 \left (b c \left (\frac {2 b \left (-\frac {1}{3} b c \left (\frac {7}{3} \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle -\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3+\frac {4}{5} d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+b c \left (\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )\) |
| \(\Big \downarrow \) 1905 |
| \(\displaystyle -\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {c^2 x^2-1}}dx}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3+\frac {4}{5} d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+b c \left (\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle -\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {b c \sqrt {c^2 x^2-1} \int \frac {3 c^4 x^4-10 c^2 x^2+15}{\sqrt {c^2 x^2-1}}dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3+\frac {4}{5} d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+b c \left (\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle -\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {b c \sqrt {c^2 x^2-1} \int \left (3 \left (c^2 x^2-1\right )^{3/2}-4 \sqrt {c^2 x^2-1}+\frac {8}{\sqrt {c^2 x^2-1}}\right )dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3+\frac {4}{5} d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+b c \left (\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {arccosh}(c x))^3+\frac {4}{5} d^2 \left (\frac {1}{3} x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^3+b c \left (\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {7 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}-\frac {1}{3} x^2 \sqrt {c x-1} \sqrt {c x+1}\right )\right )}{3 c}+\frac {(c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))^2}{3 c^2}\right )+\frac {2}{3} \left (x (a+b \text {arccosh}(c x))^3-3 b c \left (\frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))^2}{5 c^2}-\frac {2 b \left (\frac {1}{5} c^4 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 x^3 (a+b \text {arccosh}(c x))+x (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\right )}{5 c}\right )\) |
Input:
Int[(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^3,x]
Output:
(d^2*x*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x])^3)/5 - (3*b*c*d^2*(((-1 + c*x) ^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x])^2)/(5*c^2) - (2*b*(-1/30*(b*c* Sqrt[-1 + c^2*x^2]*((16*Sqrt[-1 + c^2*x^2])/c^2 - (8*(-1 + c^2*x^2)^(3/2)) /(3*c^2) + (6*(-1 + c^2*x^2)^(5/2))/(5*c^2)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x ]) + x*(a + b*ArcCosh[c*x]) - (2*c^2*x^3*(a + b*ArcCosh[c*x]))/3 + (c^4*x^ 5*(a + b*ArcCosh[c*x]))/5))/(5*c)))/5 + (4*d^2*((x*(1 - c^2*x^2)*(a + b*Ar cCosh[c*x])^3)/3 + b*c*(((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c *x])^2)/(3*c^2) + (2*b*(-1/3*(b*c*((7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2 ) - (x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/3)) + x*(a + b*ArcCosh[c*x]) - (c^2 *x^3*(a + b*ArcCosh[c*x]))/3))/(3*c)) + (2*(x*(a + b*ArcCosh[c*x])^3 - 3*b *c*((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/c^2 - (2*b*(a*x - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + b*x*ArcCosh[c*x]))/c)))/3))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) *(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n *(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ (b1*b2*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n /2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] , x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p )] Int[x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.23 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {d^{2} a^{3} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{3} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{3} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {8 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{5}+\frac {4144 c x \,\operatorname {arccosh}\left (c x \right )}{1125}-\frac {4144 \sqrt {c x -1}\, \sqrt {c x +1}}{1125}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {6 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right ) \left (c x +1\right )}{1125}-\frac {6 \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{625}+\frac {272 \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{3375}+\frac {4 \operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{15}\right )+3 d^{2} a \,b^{2} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 c x \left (c x -1\right ) \left (c x +1\right )}{3375}+\frac {8 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{45}\right )+3 d^{2} a^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(492\) |
| default | \(\frac {d^{2} a^{3} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{3} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{3} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {8 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{5}+\frac {4144 c x \,\operatorname {arccosh}\left (c x \right )}{1125}-\frac {4144 \sqrt {c x -1}\, \sqrt {c x +1}}{1125}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {6 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right ) \left (c x +1\right )}{1125}-\frac {6 \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{625}+\frac {272 \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{3375}+\frac {4 \operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{15}\right )+3 d^{2} a \,b^{2} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 c x \left (c x -1\right ) \left (c x +1\right )}{3375}+\frac {8 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{45}\right )+3 d^{2} a^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(492\) |
| parts | \(d^{2} a^{3} \left (\frac {1}{5} c^{4} x^{5}-\frac {2}{3} c^{2} x^{3}+x \right )+\frac {d^{2} b^{3} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{3} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{3} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {8 \operatorname {arccosh}\left (c x \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{5}+\frac {4144 c x \,\operatorname {arccosh}\left (c x \right )}{1125}-\frac {4144 \sqrt {c x -1}\, \sqrt {c x +1}}{1125}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {6 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 \,\operatorname {arccosh}\left (c x \right ) c x \left (c x -1\right ) \left (c x +1\right )}{1125}-\frac {6 \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{625}+\frac {272 \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{3375}+\frac {4 \operatorname {arccosh}\left (c x \right )^{2} \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{15}\right )}{c}+\frac {3 d^{2} a \,b^{2} \left (\frac {8 \operatorname {arccosh}\left (c x \right )^{2} c x}{15}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{5}-\frac {4 \operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{15}-\frac {16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {5}{2}} \left (c x +1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c x -1\right )^{2} \left (c x +1\right )^{2}}{125}-\frac {272 c x \left (c x -1\right ) \left (c x +1\right )}{3375}+\frac {8 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{45}\right )}{c}+\frac {3 d^{2} a^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\) | \(495\) |
| orering | \(\text {Expression too large to display}\) | \(933\) |
Input:
int((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c*(d^2*a^3*(1/5*c^5*x^5-2/3*c^3*x^3+c*x)+d^2*b^3*(8/15*arccosh(c*x)^3*c* x+1/5*arccosh(c*x)^3*c*x*(c*x-1)^2*(c*x+1)^2-4/15*arccosh(c*x)^3*c*x*(c*x- 1)*(c*x+1)-8/5*arccosh(c*x)^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)+4144/1125*c*x*ar ccosh(c*x)-4144/1125*(c*x-1)^(1/2)*(c*x+1)^(1/2)-3/25*arccosh(c*x)^2*(c*x- 1)^(5/2)*(c*x+1)^(5/2)+6/125*arccosh(c*x)*c*x*(c*x-1)^2*(c*x+1)^2-272/1125 *arccosh(c*x)*c*x*(c*x-1)*(c*x+1)-6/625*(c*x-1)^(5/2)*(c*x+1)^(5/2)+272/33 75*(c*x-1)^(3/2)*(c*x+1)^(3/2)+4/15*arccosh(c*x)^2*(c*x-1)^(3/2)*(c*x+1)^( 3/2))+3*d^2*a*b^2*(8/15*arccosh(c*x)^2*c*x+1/5*arccosh(c*x)^2*c*x*(c*x-1)^ 2*(c*x+1)^2-4/15*arccosh(c*x)^2*c*x*(c*x-1)*(c*x+1)-16/15*arccosh(c*x)*(c* x-1)^(1/2)*(c*x+1)^(1/2)+4144/3375*c*x-2/25*arccosh(c*x)*(c*x-1)^(5/2)*(c* x+1)^(5/2)+2/125*c*x*(c*x-1)^2*(c*x+1)^2-272/3375*c*x*(c*x-1)*(c*x+1)+8/45 *arccosh(c*x)*(c*x-1)^(3/2)*(c*x+1)^(3/2))+3*d^2*a^2*b*(1/5*arccosh(c*x)*c ^5*x^5-2/3*c^3*x^3*arccosh(c*x)+c*x*arccosh(c*x)-1/225*(c*x-1)^(1/2)*(c*x+ 1)^(1/2)*(9*c^4*x^4-38*c^2*x^2+149)))
Time = 0.19 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.18 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx=\frac {135 \, {\left (25 \, a^{3} + 6 \, a b^{2}\right )} c^{5} d^{2} x^{5} - 150 \, {\left (75 \, a^{3} + 38 \, a b^{2}\right )} c^{3} d^{2} x^{3} + 225 \, {\left (75 \, a^{3} + 298 \, a b^{2}\right )} c d^{2} x + 1125 \, {\left (3 \, b^{3} c^{5} d^{2} x^{5} - 10 \, b^{3} c^{3} d^{2} x^{3} + 15 \, b^{3} c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{3} + 225 \, {\left (45 \, a b^{2} c^{5} d^{2} x^{5} - 150 \, a b^{2} c^{3} d^{2} x^{3} + 225 \, a b^{2} c d^{2} x - {\left (9 \, b^{3} c^{4} d^{2} x^{4} - 38 \, b^{3} c^{2} d^{2} x^{2} + 149 \, b^{3} d^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 15 \, {\left (27 \, {\left (25 \, a^{2} b + 2 \, b^{3}\right )} c^{5} d^{2} x^{5} - 10 \, {\left (225 \, a^{2} b + 38 \, b^{3}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} b + 298 \, b^{3}\right )} c d^{2} x - 30 \, {\left (9 \, a b^{2} c^{4} d^{2} x^{4} - 38 \, a b^{2} c^{2} d^{2} x^{2} + 149 \, a b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (81 \, {\left (25 \, a^{2} b + 2 \, b^{3}\right )} c^{4} d^{2} x^{4} - 2 \, {\left (4275 \, a^{2} b + 842 \, b^{3}\right )} c^{2} d^{2} x^{2} + {\left (33525 \, a^{2} b + 63682 \, b^{3}\right )} d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{16875 \, c} \] Input:
integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^3,x, algorithm="fricas")
Output:
1/16875*(135*(25*a^3 + 6*a*b^2)*c^5*d^2*x^5 - 150*(75*a^3 + 38*a*b^2)*c^3* d^2*x^3 + 225*(75*a^3 + 298*a*b^2)*c*d^2*x + 1125*(3*b^3*c^5*d^2*x^5 - 10* b^3*c^3*d^2*x^3 + 15*b^3*c*d^2*x)*log(c*x + sqrt(c^2*x^2 - 1))^3 + 225*(45 *a*b^2*c^5*d^2*x^5 - 150*a*b^2*c^3*d^2*x^3 + 225*a*b^2*c*d^2*x - (9*b^3*c^ 4*d^2*x^4 - 38*b^3*c^2*d^2*x^2 + 149*b^3*d^2)*sqrt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1))^2 + 15*(27*(25*a^2*b + 2*b^3)*c^5*d^2*x^5 - 10*(225*a^ 2*b + 38*b^3)*c^3*d^2*x^3 + 15*(225*a^2*b + 298*b^3)*c*d^2*x - 30*(9*a*b^2 *c^4*d^2*x^4 - 38*a*b^2*c^2*d^2*x^2 + 149*a*b^2*d^2)*sqrt(c^2*x^2 - 1))*lo g(c*x + sqrt(c^2*x^2 - 1)) - (81*(25*a^2*b + 2*b^3)*c^4*d^2*x^4 - 2*(4275* a^2*b + 842*b^3)*c^2*d^2*x^2 + (33525*a^2*b + 63682*b^3)*d^2)*sqrt(c^2*x^2 - 1))/c
\[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx=d^{2} \left (\int a^{3}\, dx + \int b^{3} \operatorname {acosh}^{3}{\left (c x \right )}\, dx + \int 3 a b^{2} \operatorname {acosh}^{2}{\left (c x \right )}\, dx + \int 3 a^{2} b \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 a^{3} c^{2} x^{2}\right )\, dx + \int a^{3} c^{4} x^{4}\, dx + \int \left (- 2 b^{3} c^{2} x^{2} \operatorname {acosh}^{3}{\left (c x \right )}\right )\, dx + \int b^{3} c^{4} x^{4} \operatorname {acosh}^{3}{\left (c x \right )}\, dx + \int \left (- 6 a b^{2} c^{2} x^{2} \operatorname {acosh}^{2}{\left (c x \right )}\right )\, dx + \int 3 a b^{2} c^{4} x^{4} \operatorname {acosh}^{2}{\left (c x \right )}\, dx + \int \left (- 6 a^{2} b c^{2} x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int 3 a^{2} b c^{4} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)**2*(a+b*acosh(c*x))**3,x)
Output:
d**2*(Integral(a**3, x) + Integral(b**3*acosh(c*x)**3, x) + Integral(3*a*b **2*acosh(c*x)**2, x) + Integral(3*a**2*b*acosh(c*x), x) + Integral(-2*a** 3*c**2*x**2, x) + Integral(a**3*c**4*x**4, x) + Integral(-2*b**3*c**2*x**2 *acosh(c*x)**3, x) + Integral(b**3*c**4*x**4*acosh(c*x)**3, x) + Integral( -6*a*b**2*c**2*x**2*acosh(c*x)**2, x) + Integral(3*a*b**2*c**4*x**4*acosh( c*x)**2, x) + Integral(-6*a**2*b*c**2*x**2*acosh(c*x), x) + Integral(3*a** 2*b*c**4*x**4*acosh(c*x), x))
Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (330) = 660\).
Time = 0.07 (sec) , antiderivative size = 861, normalized size of antiderivative = 2.23 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx =\text {Too large to display} \] Input:
integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^3,x, algorithm="maxima")
Output:
1/5*b^3*c^4*d^2*x^5*arccosh(c*x)^3 + 3/5*a*b^2*c^4*d^2*x^5*arccosh(c*x)^2 + 1/5*a^3*c^4*d^2*x^5 - 2/3*b^3*c^2*d^2*x^3*arccosh(c*x)^3 - 2*a*b^2*c^2*d ^2*x^3*arccosh(c*x)^2 + 1/25*(15*x^5*arccosh(c*x) - (3*sqrt(c^2*x^2 - 1)*x ^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*a^2*b*c ^4*d^2 - 2/375*(15*(3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/ c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c*arccosh(c*x) - (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*a*b^2*c^4*d^2 - 1/5625*(225*(3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*s qrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c*arccosh(c*x)^2 + 2*c *((27*sqrt(c^2*x^2 - 1)*c^2*x^4 + 136*sqrt(c^2*x^2 - 1)*x^2 + 2072*sqrt(c^ 2*x^2 - 1)/c^2)/c^4 - 15*(9*c^4*x^5 + 20*c^2*x^3 + 120*x)*arccosh(c*x)/c^5 ))*b^3*c^4*d^2 - 2/3*a^3*c^2*d^2*x^3 + b^3*d^2*x*arccosh(c*x)^3 - 2/3*(3*x ^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4)) *a^2*b*c^2*d^2 + 4/9*(3*c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1) /c^4)*arccosh(c*x) - (c^2*x^3 + 6*x)/c^2)*a*b^2*c^2*d^2 + 2/27*(9*c*(sqrt( c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4)*arccosh(c*x)^2 + 2*c*((sqr t(c^2*x^2 - 1)*x^2 + 20*sqrt(c^2*x^2 - 1)/c^2)/c^2 - 3*(c^2*x^3 + 6*x)*arc cosh(c*x)/c^3))*b^3*c^2*d^2 + 3*a*b^2*d^2*x*arccosh(c*x)^2 - 3*(sqrt(c^2*x ^2 - 1)*arccosh(c*x)^2/c - 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))/c)*b^3 *d^2 + 6*a*b^2*d^2*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^3*d^2*x + 3* (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a^2*b*d^2/c
Exception generated. \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^3,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. \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^3\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \] Input:
int((a + b*acosh(c*x))^3*(d - c^2*d*x^2)^2,x)
Output:
int((a + b*acosh(c*x))^3*(d - c^2*d*x^2)^2, x)
\[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))^3 \, dx=\frac {d^{2} \left (45 \mathit {acosh} \left (c x \right ) a^{2} b \,c^{5} x^{5}-150 \mathit {acosh} \left (c x \right ) a^{2} b \,c^{3} x^{3}+225 \mathit {acosh} \left (c x \right ) a^{2} b c x -9 \sqrt {c^{2} x^{2}-1}\, a^{2} b \,c^{4} x^{4}+38 \sqrt {c^{2} x^{2}-1}\, a^{2} b \,c^{2} x^{2}+76 \sqrt {c^{2} x^{2}-1}\, a^{2} b -225 \sqrt {c x +1}\, \sqrt {c x -1}\, a^{2} b +75 \left (\int \mathit {acosh} \left (c x \right )^{3}d x \right ) b^{3} c +225 \left (\int \mathit {acosh} \left (c x \right )^{2}d x \right ) a \,b^{2} c +75 \left (\int \mathit {acosh} \left (c x \right )^{3} x^{4}d x \right ) b^{3} c^{5}-150 \left (\int \mathit {acosh} \left (c x \right )^{3} x^{2}d x \right ) b^{3} c^{3}+225 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{4}d x \right ) a \,b^{2} c^{5}-450 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2} c^{3}+15 a^{3} c^{5} x^{5}-50 a^{3} c^{3} x^{3}+75 a^{3} c x \right )}{75 c} \] Input:
int((-c^2*d*x^2+d)^2*(a+b*acosh(c*x))^3,x)
Output:
(d**2*(45*acosh(c*x)*a**2*b*c**5*x**5 - 150*acosh(c*x)*a**2*b*c**3*x**3 + 225*acosh(c*x)*a**2*b*c*x - 9*sqrt(c**2*x**2 - 1)*a**2*b*c**4*x**4 + 38*sq rt(c**2*x**2 - 1)*a**2*b*c**2*x**2 + 76*sqrt(c**2*x**2 - 1)*a**2*b - 225*s qrt(c*x + 1)*sqrt(c*x - 1)*a**2*b + 75*int(acosh(c*x)**3,x)*b**3*c + 225*i nt(acosh(c*x)**2,x)*a*b**2*c + 75*int(acosh(c*x)**3*x**4,x)*b**3*c**5 - 15 0*int(acosh(c*x)**3*x**2,x)*b**3*c**3 + 225*int(acosh(c*x)**2*x**4,x)*a*b* *2*c**5 - 450*int(acosh(c*x)**2*x**2,x)*a*b**2*c**3 + 15*a**3*c**5*x**5 - 50*a**3*c**3*x**3 + 75*a**3*c*x))/(75*c)