Integrand size = 23, antiderivative size = 247 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{2 b d (a+b \text {arcsinh}(c+d x))^2}-\frac {3 e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))}-\frac {2 e^3 (c+d x)^4}{b^2 d (a+b \text {arcsinh}(c+d x))}+\frac {e^3 \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{b^3 d}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^3 d} \]
-3/2*e^3*(d*x+c)^2/b^2/d/(a+b*arcsinh(d*x+c))-2*e^3*(d*x+c)^4/b^2/d/(a+b*a rcsinh(d*x+c))-1/2*e^3*cosh(2*a/b)*Shi(2*(a+b*arcsinh(d*x+c))/b)/b^3/d+e^3 *cosh(4*a/b)*Shi(4*(a+b*arcsinh(d*x+c))/b)/b^3/d+1/2*e^3*Chi(2*(a+b*arcsin h(d*x+c))/b)*sinh(2*a/b)/b^3/d-e^3*Chi(4*(a+b*arcsinh(d*x+c))/b)*sinh(4*a/ b)/b^3/d-1/2*e^3*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^2
Time = 0.63 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.72 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=\frac {e^3 \left (-\frac {b^2 (c+d x)^3 \sqrt {1+(c+d x)^2}}{(a+b \text {arcsinh}(c+d x))^2}+\frac {b \left (-3 (c+d x)^2-4 (c+d x)^4\right )}{a+b \text {arcsinh}(c+d x)}+\text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-2 \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+2 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )}{2 b^3 d} \]
(e^3*(-((b^2*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2 ) + (b*(-3*(c + d*x)^2 - 4*(c + d*x)^4))/(a + b*ArcSinh[c + d*x]) + CoshIn tegral[2*(a/b + ArcSinh[c + d*x])]*Sinh[(2*a)/b] - 2*CoshIntegral[4*(a/b + ArcSinh[c + d*x])]*Sinh[(4*a)/b] - Cosh[(2*a)/b]*SinhIntegral[2*(a/b + Ar cSinh[c + d*x])] + 2*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c + d*x]) ]))/(2*b^3*d)
Result contains complex when optimal does not.
Time = 1.50 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.15, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {6274, 27, 6194, 6233, 6195, 25, 5971, 27, 2009, 3042, 26, 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.
| \(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \text {arcsinh}(c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \text {arcsinh}(c+d x))^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{2 b}+\frac {2 \int \frac {(c+d x)^4}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (\frac {2 \int \frac {c+d x}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}+\frac {2 \left (\frac {4 \int \frac {(c+d x)^3}{a+b \text {arcsinh}(c+d x)}d(c+d x)}{b}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (\frac {4 \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}+\frac {3 \left (\frac {2 \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^3 \left (\frac {2 \left (-\frac {4 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}+\frac {3 \left (-\frac {2 \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {2 \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 (a+b \text {arcsinh}(c+d x))}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}+\frac {2 \left (-\frac {4 \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 (a+b \text {arcsinh}(c+d x))}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 (a+b \text {arcsinh}(c+d x))}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}+\frac {2 \left (-\frac {4 \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 (a+b \text {arcsinh}(c+d x))}-\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 (a+b \text {arcsinh}(c+d x))}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}-\frac {\int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))+\cosh \left (\frac {2 a}{b}\right ) \int -\frac {i \sinh \left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \cosh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \cosh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {e^3 \left (\frac {3 \left (-\frac {(c+d x)^2}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}\right )}{2 b}+\frac {2 \left (\frac {4 \left (\frac {1}{4} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{8} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{8} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{b^2}-\frac {(c+d x)^4}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{d}\) |
(e^3*(-1/2*((c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(b*(a + b*ArcSinh[c + d*x]) ^2) + (3*(-((c + d*x)^2/(b*(a + b*ArcSinh[c + d*x]))) + (I*(I*CoshIntegral [(2*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(2*a)/b] - I*Cosh[(2*a)/b]*SinhInteg ral[(2*(a + b*ArcSinh[c + d*x]))/b]))/b^2))/(2*b) + (2*(-((c + d*x)^4/(b*( a + b*ArcSinh[c + d*x]))) + (4*((CoshIntegral[(2*(a + b*ArcSinh[c + d*x])) /b]*Sinh[(2*a)/b])/4 - (CoshIntegral[(4*(a + b*ArcSinh[c + d*x]))/b]*Sinh[ (4*a)/b])/8 - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b]) /4 + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/8))/b^2) )/b))/d
3.2.69.3.1 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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (- Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n + 1)/ Sqrt[1 + c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
Int[((a_.) + ArcSinh[(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* ArcSinh[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. \(578\) vs. \(2(237)=474\).
Time = 1.02 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.34
| method | result | size |
| derivativedivides | \(\frac {-\frac {\left (-8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{4}-4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (4 b \,\operatorname {arcsinh}\left (d x +c \right )+4 a -b \right )}{32 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {4 a}{b}\right )}{2 b^{3}}+\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e^{3} \left (2 b \,\operatorname {arcsinh}\left (d x +c \right )+2 a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{3}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{32 b \left (a +b \,\operatorname {arcsinh}\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 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{8 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {4 a}{b}\right )}{2 b^{3}}}{d}\) | \(579\) |
| default | \(\frac {-\frac {\left (-8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{4}-4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (4 b \,\operatorname {arcsinh}\left (d x +c \right )+4 a -b \right )}{32 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \operatorname {Ei}_{1}\left (4 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {4 a}{b}\right )}{2 b^{3}}+\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e^{3} \left (2 b \,\operatorname {arcsinh}\left (d x +c \right )+2 a -b \right )}{16 b^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )+a^{2}\right )}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{3}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{32 b \left (a +b \,\operatorname {arcsinh}\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 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{8 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \operatorname {Ei}_{1}\left (-4 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {4 a}{b}\right )}{2 b^{3}}}{d}\) | \(579\) |
1/d*(-1/32*(-8*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+8*(d*x+c)^4-4*(d*x+c)*(1+(d*x +c)^2)^(1/2)+8*(d*x+c)^2+1)*e^3*(4*b*arcsinh(d*x+c)+4*a-b)/b^2/(b^2*arcsin h(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2)+1/2*e^3/b^3*exp(4*a/b)*Ei(1,4*arcsinh (d*x+c)+4*a/b)+1/16*(-2*(d*x+c)*(1+(d*x+c)^2)^(1/2)+2*(d*x+c)^2+1)*e^3*(2* b*arcsinh(d*x+c)+2*a-b)/b^2/(b^2*arcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2 )-1/4*e^3/b^3*exp(2*a/b)*Ei(1,2*arcsinh(d*x+c)+2*a/b)+1/16/b*e^3*(2*(d*x+c )^2+1+2*(d*x+c)*(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2+1/8/b^2*e^3*(2 *(d*x+c)^2+1+2*(d*x+c)*(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))+1/4/b^3*e ^3*exp(-2*a/b)*Ei(1,-2*arcsinh(d*x+c)-2*a/b)-1/32/b*e^3*(8*(d*x+c)^4+8*(d* x+c)^2+8*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+1)/(a +b*arcsinh(d*x+c))^2-1/8/b^2*e^3*(8*(d*x+c)^4+8*(d*x+c)^2+8*(d*x+c)^3*(1+( d*x+c)^2)^(1/2)+4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+1)/(a+b*arcsinh(d*x+c))-1/2/ b^3*e^3*exp(-4*a/b)*Ei(1,-4*arcsinh(d*x+c)-4*a/b))
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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^3*ar csinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*arcsinh(d*x + c) + a^3), x)
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx\right ) \]
e**3*(Integral(c**3/(a**3 + 3*a**2*b*asinh(c + d*x) + 3*a*b**2*asinh(c + d *x)**2 + b**3*asinh(c + d*x)**3), x) + Integral(d**3*x**3/(a**3 + 3*a**2*b *asinh(c + d*x) + 3*a*b**2*asinh(c + d*x)**2 + b**3*asinh(c + d*x)**3), x) + Integral(3*c*d**2*x**2/(a**3 + 3*a**2*b*asinh(c + d*x) + 3*a*b**2*asinh (c + d*x)**2 + b**3*asinh(c + d*x)**3), x) + Integral(3*c**2*d*x/(a**3 + 3 *a**2*b*asinh(c + d*x) + 3*a*b**2*asinh(c + d*x)**2 + b**3*asinh(c + d*x)* *3), x))
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
-1/2*((4*a*d^10*e^3 + b*d^10*e^3)*x^10 + 10*(4*a*c*d^9*e^3 + b*c*d^9*e^3)* x^9 + 3*(4*(15*c^2*d^8*e^3 + d^8*e^3)*a + (15*c^2*d^8*e^3 + d^8*e^3)*b)*x^ 8 + 24*(4*(5*c^3*d^7*e^3 + c*d^7*e^3)*a + (5*c^3*d^7*e^3 + c*d^7*e^3)*b)*x ^7 + 3*(4*(70*c^4*d^6*e^3 + 28*c^2*d^6*e^3 + d^6*e^3)*a + (70*c^4*d^6*e^3 + 28*c^2*d^6*e^3 + d^6*e^3)*b)*x^6 + 6*(4*(42*c^5*d^5*e^3 + 28*c^3*d^5*e^3 + 3*c*d^5*e^3)*a + (42*c^5*d^5*e^3 + 28*c^3*d^5*e^3 + 3*c*d^5*e^3)*b)*x^5 + (4*(210*c^6*d^4*e^3 + 210*c^4*d^4*e^3 + 45*c^2*d^4*e^3 + d^4*e^3)*a + ( 210*c^6*d^4*e^3 + 210*c^4*d^4*e^3 + 45*c^2*d^4*e^3 + d^4*e^3)*b)*x^4 + 4*( 4*(30*c^7*d^3*e^3 + 42*c^5*d^3*e^3 + 15*c^3*d^3*e^3 + c*d^3*e^3)*a + (30*c ^7*d^3*e^3 + 42*c^5*d^3*e^3 + 15*c^3*d^3*e^3 + c*d^3*e^3)*b)*x^3 + 3*(4*(1 5*c^8*d^2*e^3 + 28*c^6*d^2*e^3 + 15*c^4*d^2*e^3 + 2*c^2*d^2*e^3)*a + (15*c ^8*d^2*e^3 + 28*c^6*d^2*e^3 + 15*c^4*d^2*e^3 + 2*c^2*d^2*e^3)*b)*x^2 + ((4 *a*d^7*e^3 + b*d^7*e^3)*x^7 + 7*(4*a*c*d^6*e^3 + b*c*d^6*e^3)*x^6 + (6*(14 *c^2*d^5*e^3 + d^5*e^3)*a + (21*c^2*d^5*e^3 + d^5*e^3)*b)*x^5 + 5*(2*(14*c ^3*d^4*e^3 + 3*c*d^4*e^3)*a + (7*c^3*d^4*e^3 + c*d^4*e^3)*b)*x^4 + (2*(70* c^4*d^3*e^3 + 30*c^2*d^3*e^3 + d^3*e^3)*a + 5*(7*c^4*d^3*e^3 + 2*c^2*d^3*e ^3)*b)*x^3 + (6*(14*c^5*d^2*e^3 + 10*c^3*d^2*e^3 + c*d^2*e^3)*a + (21*c^5* d^2*e^3 + 10*c^3*d^2*e^3)*b)*x^2 + 2*(2*c^7*e^3 + 3*c^5*e^3 + c^3*e^3)*a + (c^7*e^3 + c^5*e^3)*b + (2*(14*c^6*d*e^3 + 15*c^4*d*e^3 + 3*c^2*d*e^3)*a + (7*c^6*d*e^3 + 5*c^4*d*e^3)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2)...
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^3} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3} \,d x \]