Integrand size = 23, antiderivative size = 360 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}+\frac {e^3 (c+d x)^2}{2 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d (a+b \text {arccosh}(c+d x))^2}+\frac {e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b^3 d (a+b \text {arccosh}(c+d x))}-\frac {8 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b^3 d (a+b \text {arccosh}(c+d x))}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{3 b^4 d}+\frac {4 e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{3 b^4 d}-\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{3 b^4 d}-\frac {4 e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{3 b^4 d} \] Output:
-1/3*e^3*(d*x+c-1)^(1/2)*(d*x+c)^3*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c) )^3+1/2*e^3*(d*x+c)^2/b^2/d/(a+b*arccosh(d*x+c))^2-2/3*e^3*(d*x+c)^4/b^2/d /(a+b*arccosh(d*x+c))^2+e^3*(d*x+c-1)^(1/2)*(d*x+c)*(d*x+c+1)^(1/2)/b^3/d/ (a+b*arccosh(d*x+c))-8/3*e^3*(d*x+c-1)^(1/2)*(d*x+c)^3*(d*x+c+1)^(1/2)/b^3 /d/(a+b*arccosh(d*x+c))+1/3*e^3*cosh(2*a/b)*Chi(2*(a+b*arccosh(d*x+c))/b)/ b^4/d+4/3*e^3*cosh(4*a/b)*Chi(4*(a+b*arccosh(d*x+c))/b)/b^4/d-1/3*e^3*sinh (2*a/b)*Shi(2*(a+b*arccosh(d*x+c))/b)/b^4/d-4/3*e^3*sinh(4*a/b)*Shi(4*(a+b *arccosh(d*x+c))/b)/b^4/d
Time = 1.24 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.92 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx=\frac {e^3 \left (-\frac {2 b^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^3}+\frac {b^2 \left (3 (c+d x)^2-4 (c+d x)^4\right )}{(a+b \text {arccosh}(c+d x))^2}-\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-3 (c+d x)+8 (c+d x)^3\right )}{a+b \text {arccosh}(c+d x)}+6 \log (a+b \text {arccosh}(c+d x))-30 \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\log (a+b \text {arccosh}(c+d x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )+8 \left (4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )+3 \log (a+b \text {arccosh}(c+d x))-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{6 b^4 d} \] Input:
Integrate[(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^4,x]
Output:
(e^3*((-2*b^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(a + b*Arc Cosh[c + d*x])^3 + (b^2*(3*(c + d*x)^2 - 4*(c + d*x)^4))/(a + b*ArcCosh[c + d*x])^2 - (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(-3*(c + d*x) + 8*(c + d*x)^3))/(a + b*ArcCosh[c + d*x]) + 6*Log[a + b*ArcCosh[c + d*x]] - 30* (Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c + d*x])] + Log[a + b*ArcCos h[c + d*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c + d*x])]) + 8* (4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c + d*x])] + Cosh[(4*a)/b]* CoshIntegral[4*(a/b + ArcCosh[c + d*x])] + 3*Log[a + b*ArcCosh[c + d*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c + d*x])] - Sinh[(4*a)/b]* SinhIntegral[4*(a/b + ArcCosh[c + d*x])])))/(6*b^4*d)
Time = 1.78 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6411, 27, 6301, 6366, 6300, 25, 2009, 3042, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {e^3 \left (-\frac {\int \frac {(c+d x)^2}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{b}+\frac {4 \int \frac {(c+d x)^4}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {e^3 \left (-\frac {\frac {\int \frac {c+d x}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \int \frac {(c+d x)^3}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6300 |
| \(\displaystyle \frac {e^3 \left (-\frac {\frac {-\frac {\int -\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (-\frac {\int \left (-\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{2 (a+b \text {arccosh}(c+d x))}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{2 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^3 \left (-\frac {\frac {\frac {\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (-\frac {\int \left (-\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{2 (a+b \text {arccosh}(c+d x))}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{2 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (-\frac {\frac {\frac {\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (-\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (-\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (-\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e^3 \left (-\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+i \sinh \left (\frac {2 a}{b}\right ) \int -\frac {i \sinh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (-\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^3 \left (-\frac {\frac {-\frac {\sinh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (-\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (-\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {\sinh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (-\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^3 \left (-\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {-i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (-\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {e^3 \left (-\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (-\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {e^3 \left (-\frac {\frac {-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b (a+b \text {arccosh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (-\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )-\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b (a+b \text {arccosh}(c+d x))^3}\right )}{d}\) |
Input:
Int[(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^4,x]
Output:
(e^3*(-1/3*(Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(b*(a + b*Ar cCosh[c + d*x])^3) - (-1/2*(c + d*x)^2/(b*(a + b*ArcCosh[c + d*x])^2) + (- ((Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d* x]))) - (-(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b]) + S inh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/b^2)/b)/b + (4* (-1/2*(c + d*x)^4/(b*(a + b*ArcCosh[c + d*x])^2) + (2*(-((Sqrt[-1 + c + d* x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x]))) - (-1/2*(C osh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b]) - (Cosh[(4*a)/b ]*CoshIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/2 + (Sinh[(2*a)/b]*SinhInt egral[(2*(a + b*ArcCosh[c + d*x]))/b])/2 + (Sinh[(4*a)/b]*SinhIntegral[(4* (a + b*ArcCosh[c + d*x]))/b])/2)/b^2))/b))/(3*b)))/d
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + Simp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) )), x] + (-Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcCosh[c*x ])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) ), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 _) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x ]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp [Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Int[ (f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(859\) vs. \(2(332)=664\).
Time = 0.22 (sec) , antiderivative size = 860, normalized size of antiderivative = 2.39
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}+4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (8 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+16 a b \,\operatorname {arccosh}\left (d x +c \right )-2 b^{2} \operatorname {arccosh}\left (d x +c \right )+8 a^{2}-2 a b +b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {2 e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{3 b^{4}}+\frac {\left (-2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e^{3} \left (2 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+4 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{24 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{6 b^{4}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )\right )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )\right )}{24 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )\right )}{12 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{6 b^{4}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+1\right )}{48 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+1\right )}{24 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+1\right )}{6 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {2 e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{3 b^{4}}}{d}\) | \(860\) |
| default | \(\frac {\frac {\left (-8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}+4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (8 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+16 a b \,\operatorname {arccosh}\left (d x +c \right )-2 b^{2} \operatorname {arccosh}\left (d x +c \right )+8 a^{2}-2 a b +b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {2 e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (4 \,\operatorname {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{3 b^{4}}+\frac {\left (-2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e^{3} \left (2 b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+4 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{24 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{6 b^{4}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )\right )}{24 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )\right )}{24 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )\right )}{12 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{6 b^{4}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+1\right )}{48 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+1\right )}{24 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+1\right )}{6 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {2 e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {expIntegral}_{1}\left (-4 \,\operatorname {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{3 b^{4}}}{d}\) | \(860\) |
Input:
int((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(1/48*(-8*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^3+4*(d*x+c-1)^(1/2)* (d*x+c+1)^(1/2)*(d*x+c)+8*(d*x+c)^4-8*(d*x+c)^2+1)*e^3*(8*b^2*arccosh(d*x+ c)^2+16*a*b*arccosh(d*x+c)-2*b^2*arccosh(d*x+c)+8*a^2-2*a*b+b^2)/b^3/(b^3* arccosh(d*x+c)^3+3*a*b^2*arccosh(d*x+c)^2+3*a^2*b*arccosh(d*x+c)+a^3)-2/3* e^3/b^4*exp(4*a/b)*Ei(1,4*arccosh(d*x+c)+4*a/b)+1/24*(-2*(d*x+c-1)^(1/2)*( d*x+c+1)^(1/2)*(d*x+c)+2*(d*x+c)^2-1)*e^3*(2*b^2*arccosh(d*x+c)^2+4*a*b*ar ccosh(d*x+c)-b^2*arccosh(d*x+c)+2*a^2-a*b+b^2)/b^3/(b^3*arccosh(d*x+c)^3+3 *a*b^2*arccosh(d*x+c)^2+3*a^2*b*arccosh(d*x+c)+a^3)-1/6*e^3/b^4*exp(2*a/b) *Ei(1,2*arccosh(d*x+c)+2*a/b)-1/24/b*e^3*(2*(d*x+c)^2-1+2*(d*x+c-1)^(1/2)* (d*x+c+1)^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))^3-1/24/b^2*e^3*(2*(d*x+c)^2- 1+2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))^2-1/12/b ^3*e^3*(2*(d*x+c)^2-1+2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c))/(a+b*arcc osh(d*x+c))-1/6/b^4*e^3*exp(-2*a/b)*Ei(1,-2*arccosh(d*x+c)-2*a/b)-1/48/b*e ^3*(8*(d*x+c)^4-8*(d*x+c)^2+8*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^3-4* (d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)+1)/(a+b*arccosh(d*x+c))^3-1/24/b^2 *e^3*(8*(d*x+c)^4-8*(d*x+c)^2+8*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^3- 4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)+1)/(a+b*arccosh(d*x+c))^2-1/6/b^ 3*e^3*(8*(d*x+c)^4-8*(d*x+c)^2+8*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^3 -4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)+1)/(a+b*arccosh(d*x+c))-2/3/b^4 *e^3*exp(-4*a/b)*Ei(1,-4*arccosh(d*x+c)-4*a/b))
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:
integrate((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")
Output:
integral((d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3)/(b^4*ar ccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2*arccosh(d*x + c) ^2 + 4*a^3*b*arccosh(d*x + c) + a^4), x)
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx\right ) \] Input:
integrate((d*e*x+c*e)**3/(a+b*acosh(d*x+c))**4,x)
Output:
e**3*(Integral(c**3/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d*x)**2 + 4*a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x) + Int egral(d**3*x**3/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d* x)**2 + 4*a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x) + Integra l(3*c*d**2*x**2/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d* x)**2 + 4*a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x) + Integra l(3*c**2*d*x/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d*x)* *2 + 4*a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), x))
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:
integrate((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^4,x, algorithm="giac")
Output:
integrate((d*e*x + c*e)^3/(b*arccosh(d*x + c) + a)^4, x)
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \] Input:
int((c*e + d*e*x)^3/(a + b*acosh(c + d*x))^4,x)
Output:
int((c*e + d*e*x)^3/(a + b*acosh(c + d*x))^4, x)
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arccosh}(c+d x))^4} \, dx=e^{3} \left (\left (\int \frac {x^{3}}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) d^{3}+3 \left (\int \frac {x^{2}}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c \,d^{2}+3 \left (\int \frac {x}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c^{2} d +\left (\int \frac {1}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \right ) c^{3}\right ) \] Input:
int((d*e*x+c*e)^3/(a+b*acosh(d*x+c))^4,x)
Output:
e**3*(int(x**3/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*ac osh(c + d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*d**3 + 3*in t(x**2/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*c*d**2 + 3*int(x/(a cosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)**2*a **2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*c**2*d + int(1/(acosh(c + d* x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)*c**3)