Integrand size = 23, antiderivative size = 340 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^4} \, dx=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e^3 (c+d x)^2}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^3 (c+d x) \sqrt {1+(c+d x)^2}}{b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {8 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d}+\frac {4 e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d}+\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d}-\frac {4 e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{3 b^4 d} \]
-1/2*e^3*(d*x+c)^2/b^2/d/(a+b*arcsinh(d*x+c))^2-2/3*e^3*(d*x+c)^4/b^2/d/(a +b*arcsinh(d*x+c))^2-1/3*e^3*Chi(2*(a+b*arcsinh(d*x+c))/b)*cosh(2*a/b)/b^4 /d+4/3*e^3*Chi(4*(a+b*arcsinh(d*x+c))/b)*cosh(4*a/b)/b^4/d+1/3*e^3*Shi(2*( a+b*arcsinh(d*x+c))/b)*sinh(2*a/b)/b^4/d-4/3*e^3*Shi(4*(a+b*arcsinh(d*x+c) )/b)*sinh(4*a/b)/b^4/d-1/3*e^3*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcs inh(d*x+c))^3-e^3*(d*x+c)*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))-8 /3*e^3*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))
Time = 1.17 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.94 \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\frac {e^3 \left (-\frac {2 b^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{(a+b \text {arcsinh}(c+d x))^3}+\frac {b^2 \left (-3 (c+d x)^2-4 (c+d x)^4\right )}{(a+b \text {arcsinh}(c+d x))^2}-\frac {2 b \sqrt {1+(c+d x)^2} \left (3 (c+d x)+8 (c+d x)^3\right )}{a+b \text {arcsinh}(c+d x)}+6 \log (a+b \text {arcsinh}(c+d x))+30 \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-\log (a+b \text {arcsinh}(c+d x))-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )+8 \left (-4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )+3 \log (a+b \text {arcsinh}(c+d x))+4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )\right )}{6 b^4 d} \]
(e^3*((-2*b^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^ 3 + (b^2*(-3*(c + d*x)^2 - 4*(c + d*x)^4))/(a + b*ArcSinh[c + d*x])^2 - (2 *b*Sqrt[1 + (c + d*x)^2]*(3*(c + d*x) + 8*(c + d*x)^3))/(a + b*ArcSinh[c + d*x]) + 6*Log[a + b*ArcSinh[c + d*x]] + 30*(Cosh[(2*a)/b]*CoshIntegral[2* (a/b + ArcSinh[c + d*x])] - Log[a + b*ArcSinh[c + d*x]] - Sinh[(2*a)/b]*Si nhIntegral[2*(a/b + ArcSinh[c + d*x])]) + 8*(-4*Cosh[(2*a)/b]*CoshIntegral [2*(a/b + ArcSinh[c + d*x])] + Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh [c + d*x])] + 3*Log[a + b*ArcSinh[c + d*x]] + 4*Sinh[(2*a)/b]*SinhIntegral [2*(a/b + ArcSinh[c + d*x])] - Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh [c + d*x])])))/(6*b^4*d)
Time = 1.39 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6274, 27, 6194, 6233, 6193, 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.
| \(\displaystyle \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^3 (c+d x)^3}{(a+b \text {arcsinh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int \frac {(c+d x)^3}{(a+b \text {arcsinh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {e^3 \left (\frac {\int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}d(c+d x)}{b}+\frac {4 \int \frac {(c+d x)^4}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\int \frac {c+d x}{(a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \int \frac {(c+d x)^3}{(a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (\frac {\int \left (\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 (a+b \text {arcsinh}(c+d x))}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{2 (a+b \text {arcsinh}(c+d x))}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {\int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e^3 \left (\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {\cosh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \sinh \left (\frac {2 a}{b}\right ) \int -\frac {i \sinh \left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\cosh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-\sinh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-\sinh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^3 \left (\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))+\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {e^3 \left (\frac {-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{b}}{b}+\frac {4 \left (\frac {2 \left (\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {e^3 \left (\frac {\frac {\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x) \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{b}-\frac {(c+d x)^2}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}+\frac {4 \left (\frac {2 \left (\frac {-\frac {1}{2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^3 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{b}-\frac {(c+d x)^4}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^3}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
(e^3*(-1/3*((c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(b*(a + b*ArcSinh[c + d*x]) ^3) + (-1/2*(c + d*x)^2/(b*(a + b*ArcSinh[c + d*x])^2) + (-(((c + d*x)*Sqr t[1 + (c + d*x)^2])/(b*(a + b*ArcSinh[c + d*x]))) + (Cosh[(2*a)/b]*CoshInt egral[(2*(a + b*ArcSinh[c + d*x]))/b] - Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/b^2)/b)/b + (4*(-1/2*(c + d*x)^4/(b*(a + b*ArcSi nh[c + d*x])^2) + (2*(-(((c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(b*(a + b*ArcS inh[c + d*x]))) + (-1/2*(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b]) + (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/ 2 + (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/2 - (Sinh [(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/2)/b^2))/b))/(3*b) ))/d
3.2.75.3.1 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[((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] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
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_)*((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. \(799\) vs. \(2(318)=636\).
Time = 0.89 (sec) , antiderivative size = 800, normalized size of antiderivative = 2.35
| 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 (8 b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+16 a b \,\operatorname {arcsinh}\left (d x +c \right )-2 b^{2} \operatorname {arcsinh}\left (d x +c \right )+8 a^{2}-2 a b +b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\right )}-\frac {2 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 )}{3 b^{4}}-\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^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+4 a b \,\operatorname {arcsinh}\left (d x +c \right )-b^{2} \operatorname {arcsinh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{24 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\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 )}{6 b^{4}}+\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 )}{24 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{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 )}{24 b^{2} \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 )}{12 b^{3} \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 )}{6 b^{4}}-\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 )}{48 b \left (a +b \,\operatorname {arcsinh}\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 \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 )}{24 b^{2} \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 )}{6 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {2 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 )}{3 b^{4}}}{d}\) | \(800\) |
| 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 (8 b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+16 a b \,\operatorname {arcsinh}\left (d x +c \right )-2 b^{2} \operatorname {arcsinh}\left (d x +c \right )+8 a^{2}-2 a b +b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\right )}-\frac {2 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 )}{3 b^{4}}-\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^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+4 a b \,\operatorname {arcsinh}\left (d x +c \right )-b^{2} \operatorname {arcsinh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{24 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\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 )}{6 b^{4}}+\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 )}{24 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{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 )}{24 b^{2} \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 )}{12 b^{3} \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 )}{6 b^{4}}-\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 )}{48 b \left (a +b \,\operatorname {arcsinh}\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 \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 )}{24 b^{2} \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 )}{6 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {2 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 )}{3 b^{4}}}{d}\) | \(800\) |
1/d*(1/48*(-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*(8*b^2*arcsinh(d*x+c)^2+16*a*b*arcsinh(d*x+ c)-2*b^2*arcsinh(d*x+c)+8*a^2-2*a*b+b^2)/b^3/(b^3*arcsinh(d*x+c)^3+3*a*b^2 *arcsinh(d*x+c)^2+3*a^2*b*arcsinh(d*x+c)+a^3)-2/3*e^3/b^4*exp(4*a/b)*Ei(1, 4*arcsinh(d*x+c)+4*a/b)-1/24*(-2*(d*x+c)*(1+(d*x+c)^2)^(1/2)+2*(d*x+c)^2+1 )*e^3*(2*b^2*arcsinh(d*x+c)^2+4*a*b*arcsinh(d*x+c)-b^2*arcsinh(d*x+c)+2*a^ 2-a*b+b^2)/b^3/(b^3*arcsinh(d*x+c)^3+3*a*b^2*arcsinh(d*x+c)^2+3*a^2*b*arcs inh(d*x+c)+a^3)+1/6*e^3/b^4*exp(2*a/b)*Ei(1,2*arcsinh(d*x+c)+2*a/b)+1/24/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))^3+ 1/24/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))^2+1/12/b^3*e^3*(2*(d*x+c)^2+1+2*(d*x+c)*(1+(d*x+c)^2)^(1/2))/(a+b*ar csinh(d*x+c))+1/6/b^4*e^3*exp(-2*a/b)*Ei(1,-2*arcsinh(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)^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))^3-1/24/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))^2-1/6/b^3*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))-2/3 /b^4*e^3*exp(-4*a/b)*Ei(1,-4*arcsinh(d*x+c)-4*a/b))
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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^4*ar csinh(d*x + c)^4 + 4*a*b^3*arcsinh(d*x + c)^3 + 6*a^2*b^2*arcsinh(d*x + c) ^2 + 4*a^3*b*arcsinh(d*x + c) + a^4), x)
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^4} \, dx=e^{3} \left (\int \frac {c^{3}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
e**3*(Integral(c**3/(a**4 + 4*a**3*b*asinh(c + d*x) + 6*a**2*b**2*asinh(c + d*x)**2 + 4*a*b**3*asinh(c + d*x)**3 + b**4*asinh(c + d*x)**4), x) + Int egral(d**3*x**3/(a**4 + 4*a**3*b*asinh(c + d*x) + 6*a**2*b**2*asinh(c + d* x)**2 + 4*a*b**3*asinh(c + d*x)**3 + b**4*asinh(c + d*x)**4), x) + Integra l(3*c*d**2*x**2/(a**4 + 4*a**3*b*asinh(c + d*x) + 6*a**2*b**2*asinh(c + d* x)**2 + 4*a*b**3*asinh(c + d*x)**3 + b**4*asinh(c + d*x)**4), x) + Integra l(3*c**2*d*x/(a**4 + 4*a**3*b*asinh(c + d*x) + 6*a**2*b**2*asinh(c + d*x)* *2 + 4*a*b**3*asinh(c + d*x)**3 + b**4*asinh(c + d*x)**4), x))
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\text {Timed out} \]
\[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^3}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4} \,d x \]