Integrand size = 23, antiderivative size = 252 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^3} \, dx=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {e^2 (c+d x)}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {3 e^2 (c+d x)^3}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {e^2 \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^3 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^3 d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{8 b^3 d} \]
e^2*(d*x+c)/b^2/d/(a+b*arccosh(d*x+c))-3/2*e^2*(d*x+c)^3/b^2/d/(a+b*arccos h(d*x+c))+1/8*e^2*cosh(a/b)*Shi((a+b*arccosh(d*x+c))/b)/b^3/d+9/8*e^2*cosh (3*a/b)*Shi(3*(a+b*arccosh(d*x+c))/b)/b^3/d-1/8*e^2*Chi((a+b*arccosh(d*x+c ))/b)*sinh(a/b)/b^3/d-9/8*e^2*Chi(3*(a+b*arccosh(d*x+c))/b)*sinh(3*a/b)/b^ 3/d-1/2*e^2*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x +c))^2
Time = 0.52 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.88 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^3} \, dx=\frac {e^2 \left (-\frac {4 b^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^2}+\frac {4 b \left (2 (c+d x)-3 (c+d x)^3\right )}{a+b \text {arccosh}(c+d x)}+8 \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-8 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+9 \left (-\text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )\right )}{8 b^3 d} \]
(e^2*((-4*b^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(a + b*Arc Cosh[c + d*x])^2 + (4*b*(2*(c + d*x) - 3*(c + d*x)^3))/(a + b*ArcCosh[c + d*x]) + 8*CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - 8*Cosh[a/b]*Sin hIntegral[a/b + ArcCosh[c + d*x]] + 9*(-(CoshIntegral[a/b + ArcCosh[c + d* x]]*Sinh[a/b]) - CoshIntegral[3*(a/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] + Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] + Cosh[(3*a)/b]*SinhIntegra l[3*(a/b + ArcCosh[c + d*x])])))/(8*b^3*d)
Result contains complex when optimal does not.
Time = 1.80 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.12, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {6411, 27, 6301, 6366, 6296, 25, 3042, 26, 3784, 26, 3042, 26, 3779, 3782, 6302, 25, 5971, 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.
| \(\displaystyle \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{(a+b \text {arccosh}(c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{(a+b \text {arccosh}(c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6301 |
| \(\displaystyle \frac {e^2 \left (-\frac {\int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}d(c+d x)}{b}+\frac {3 \int \frac {(c+d x)^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^2}d(c+d x)}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 6366 |
| \(\displaystyle \frac {e^2 \left (-\frac {\frac {\int \frac {1}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 6296 |
| \(\displaystyle \frac {e^2 \left (-\frac {\frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}-\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+\cosh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))\right )}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b^2}}{b}+\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {e^2 \left (\frac {3 \left (\frac {3 \int \frac {(c+d x)^2}{a+b \text {arccosh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 6302 |
| \(\displaystyle \frac {e^2 \left (\frac {3 \left (\frac {3 \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^2 \left (\frac {3 \left (-\frac {3 \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {e^2 \left (\frac {3 \left (-\frac {3 \int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )}{4 (a+b \text {arccosh}(c+d x))}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{4 (a+b \text {arccosh}(c+d x))}\right )d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b^2}}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (-\frac {-\frac {c+d x}{b (a+b \text {arccosh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )}{b^2}}{b}+\frac {3 \left (\frac {3 \left (-\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\frac {1}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )+\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\frac {1}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^3}{b (a+b \text {arccosh}(c+d x))}\right )}{2 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b (a+b \text {arccosh}(c+d x))^2}\right )}{d}\) |
(e^2*(-1/2*(Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*(a + b*Ar cCosh[c + d*x])^2) - (-((c + d*x)/(b*(a + b*ArcCosh[c + d*x]))) + (I*(I*Co shIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b] - I*Cosh[a/b]*SinhIntegra l[(a + b*ArcCosh[c + d*x])/b]))/b^2)/b + (3*(-((c + d*x)^3/(b*(a + b*ArcCo sh[c + d*x]))) + (3*(-1/4*(CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a /b]) - (CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/4 + (C osh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/4 + (Cosh[(3*a)/b]*Sinh Integral[(3*(a + b*ArcCosh[c + d*x]))/b])/4))/b^2))/(2*b)))/d
3.2.44.3.1 Defintions of rubi rules used
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[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
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_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
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. \(556\) vs. \(2(236)=472\).
Time = 0.61 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.21
| method | result | size |
| derivativedivides | \(\frac {-\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (3 b \,\operatorname {arccosh}\left (d x +c \right )+3 a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b \,\operatorname {arccosh}\left (d x +c \right )+a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d}\) | \(557\) |
| default | \(\frac {-\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (3 b \,\operatorname {arccosh}\left (d x +c \right )+3 a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b \,\operatorname {arccosh}\left (d x +c \right )+a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d}\) | \(557\) |
1/d*(-1/16*(-4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+(d*x+c-1)^(1/2)*( d*x+c+1)^(1/2)+4*(d*x+c)^3-3*d*x-3*c)*e^2*(3*b*arccosh(d*x+c)+3*a-b)/b^2/( b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+9/16*e^2/b^3*exp(3*a/b)*Ei( 1,3*arccosh(d*x+c)+3*a/b)-1/16*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*e^ 2*(b*arccosh(d*x+c)+a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^ 2)+1/16*e^2/b^3*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/16/b*e^2*(d*x+c+(d*x+c -1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^2-1/16/b^2*e^2*(d*x+c+(d*x +c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/16/b^3*e^2*exp(-a/b)*E i(1,-arccosh(d*x+c)-a/b)-1/16/b*e^2*(4*(d*x+c)^3-3*d*x-3*c+4*(d*x+c)^2*(d* x+c-1)^(1/2)*(d*x+c+1)^(1/2)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh (d*x+c))^2-3/16/b^2*e^2*(4*(d*x+c)^3-3*d*x-3*c+4*(d*x+c)^2*(d*x+c-1)^(1/2) *(d*x+c+1)^(1/2)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-9/1 6/b^3*e^2*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b))
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(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), x)
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^3} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \]
e**2*(Integral(c**2/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d *x)**2 + b**3*acosh(c + d*x)**3), x) + Integral(d**2*x**2/(a**3 + 3*a**2*b *acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*acosh(c + d*x)**3), x) + Integral(2*c*d*x/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d *x)**2 + b**3*acosh(c + d*x)**3), x))
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
-1/2*((3*a*d^9*e^2 + b*d^9*e^2)*x^9 + 9*(3*a*c*d^8*e^2 + b*c*d^8*e^2)*x^8 + 3*(3*(12*c^2*d^7*e^2 - d^7*e^2)*a + (12*c^2*d^7*e^2 - d^7*e^2)*b)*x^7 + 21*(3*(4*c^3*d^6*e^2 - c*d^6*e^2)*a + (4*c^3*d^6*e^2 - c*d^6*e^2)*b)*x^6 + 3*(3*(42*c^4*d^5*e^2 - 21*c^2*d^5*e^2 + d^5*e^2)*a + (42*c^4*d^5*e^2 - 21 *c^2*d^5*e^2 + d^5*e^2)*b)*x^5 + 3*(3*(42*c^5*d^4*e^2 - 35*c^3*d^4*e^2 + 5 *c*d^4*e^2)*a + (42*c^5*d^4*e^2 - 35*c^3*d^4*e^2 + 5*c*d^4*e^2)*b)*x^4 + ( (3*a*d^6*e^2 + b*d^6*e^2)*x^6 + 6*(3*a*c*d^5*e^2 + b*c*d^5*e^2)*x^5 + ((45 *c^2*d^4*e^2 - 4*d^4*e^2)*a + (15*c^2*d^4*e^2 - d^4*e^2)*b)*x^4 + 4*((15*c ^3*d^3*e^2 - 4*c*d^3*e^2)*a + (5*c^3*d^3*e^2 - c*d^3*e^2)*b)*x^3 + ((45*c^ 4*d^2*e^2 - 24*c^2*d^2*e^2 + d^2*e^2)*a + 3*(5*c^4*d^2*e^2 - 2*c^2*d^2*e^2 )*b)*x^2 + (3*c^6*e^2 - 4*c^4*e^2 + c^2*e^2)*a + (c^6*e^2 - c^4*e^2)*b + 2 *((9*c^5*d*e^2 - 8*c^3*d*e^2 + c*d*e^2)*a + (3*c^5*d*e^2 - 2*c^3*d*e^2)*b) *x)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + (3*(84*c^6*d^3*e^2 - 105*c^4 *d^3*e^2 + 30*c^2*d^3*e^2 - d^3*e^2)*a + (84*c^6*d^3*e^2 - 105*c^4*d^3*e^2 + 30*c^2*d^3*e^2 - d^3*e^2)*b)*x^3 + (3*(3*a*d^7*e^2 + b*d^7*e^2)*x^7 + 2 1*(3*a*c*d^6*e^2 + b*c*d^6*e^2)*x^6 + ((189*c^2*d^5*e^2 - 17*d^5*e^2)*a + (63*c^2*d^5*e^2 - 5*d^5*e^2)*b)*x^5 + 5*((63*c^3*d^4*e^2 - 17*c*d^4*e^2)*a + (21*c^3*d^4*e^2 - 5*c*d^4*e^2)*b)*x^4 + (5*(63*c^4*d^3*e^2 - 34*c^2*d^3 *e^2 + 2*d^3*e^2)*a + (105*c^4*d^3*e^2 - 50*c^2*d^3*e^2 + 2*d^3*e^2)*b)*x^ 3 + ((189*c^5*d^2*e^2 - 170*c^3*d^2*e^2 + 30*c*d^2*e^2)*a + (63*c^5*d^2...
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3} \,d x \]