Integrand size = 23, antiderivative size = 331 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {e^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{24 b^4 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 b^4 d} \]
-1/3*e^2*(d*x+c)/b^2/d/(a+b*arcsinh(d*x+c))^2-1/2*e^2*(d*x+c)^3/b^2/d/(a+b *arcsinh(d*x+c))^2-1/24*e^2*cosh(a/b)*Shi((a+b*arcsinh(d*x+c))/b)/b^4/d+9/ 8*e^2*cosh(3*a/b)*Shi(3*(a+b*arcsinh(d*x+c))/b)/b^4/d+1/24*e^2*Chi((a+b*ar csinh(d*x+c))/b)*sinh(a/b)/b^4/d-9/8*e^2*Chi(3*(a+b*arcsinh(d*x+c))/b)*sin h(3*a/b)/b^4/d-1/3*e^2*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+ c))^3-1/3*e^2*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))-3/2*e^2*(d*x+ c)^2*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x+c))
Time = 0.93 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.78 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\frac {e^2 \left (-\frac {8 b^3 (c+d x)^2 \sqrt {1+(c+d x)^2}}{(a+b \text {arcsinh}(c+d x))^3}+\frac {4 b^2 \left (-2 (c+d x)-3 (c+d x)^3\right )}{(a+b \text {arcsinh}(c+d x))^2}-\frac {4 b \sqrt {1+(c+d x)^2} \left (2+9 (c+d x)^2\right )}{a+b \text {arcsinh}(c+d x)}-80 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )+80 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+27 \left (3 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )\right )}{24 b^4 d} \]
(e^2*((-8*b^3*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^ 3 + (4*b^2*(-2*(c + d*x) - 3*(c + d*x)^3))/(a + b*ArcSinh[c + d*x])^2 - (4 *b*Sqrt[1 + (c + d*x)^2]*(2 + 9*(c + d*x)^2))/(a + b*ArcSinh[c + d*x]) - 8 0*CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] + 80*Cosh[a/b]*SinhIntegr al[a/b + ArcSinh[c + d*x]] + 27*(3*CoshIntegral[a/b + ArcSinh[c + d*x]]*Si nh[a/b] - CoshIntegral[3*(a/b + ArcSinh[c + d*x])]*Sinh[(3*a)/b] - 3*Cosh[ a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + Cosh[(3*a)/b]*SinhIntegral[3*( a/b + ArcSinh[c + d*x])])))/(24*b^4*d)
Result contains complex when optimal does not.
Time = 1.78 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.09, 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, 6188, 6193, 2009, 6234, 25, 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.
| \(\displaystyle \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {e^2 (c+d x)^2}{(a+b \text {arcsinh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \int \frac {c+d x}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}d(c+d x)}{3 b}+\frac {\int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}d(c+d x)}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {\int \frac {1}{(a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{2 b}-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6188 |
| \(\displaystyle \frac {e^2 \left (\frac {\frac {3 \int \frac {(c+d x)^2}{(a+b \text {arcsinh}(c+d x))^2}d(c+d x)}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}+\frac {2 \left (\frac {\frac {\int \frac {c+d x}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}d(c+d x)}{b}-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {e^2 \left (\frac {\frac {3 \left (\frac {\int \left (\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 (a+b \text {arcsinh}(c+d x))}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 (a+b \text {arcsinh}(c+d x))}\right )d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}+\frac {2 \left (\frac {\frac {\int \frac {c+d x}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}d(c+d x)}{b}-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {\frac {\int \frac {c+d x}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}d(c+d x)}{b}-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {\frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (\frac {-\frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}-\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))+\cosh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arcsinh}(c+d x)}d(a+b \text {arcsinh}(c+d x))-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {e^2 \left (\frac {2 \left (-\frac {c+d x}{2 b (a+b \text {arcsinh}(c+d x))^2}+\frac {-\frac {\sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}+\frac {i \left (i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )}{b^2}}{2 b}\right )}{3 b}+\frac {\frac {3 \left (\frac {\frac {1}{4} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )-\frac {3}{4} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-\frac {1}{4} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\frac {3}{4} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{b^2}-\frac {(c+d x)^2 \sqrt {(c+d x)^2+1}}{b (a+b \text {arcsinh}(c+d x))}\right )}{2 b}-\frac {(c+d x)^3}{2 b (a+b \text {arcsinh}(c+d x))^2}}{b}-\frac {\sqrt {(c+d x)^2+1} (c+d x)^2}{3 b (a+b \text {arcsinh}(c+d x))^3}\right )}{d}\) |
(e^2*(-1/3*((c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(b*(a + b*ArcSinh[c + d*x]) ^3) + (2*(-1/2*(c + d*x)/(b*(a + b*ArcSinh[c + d*x])^2) + (-(Sqrt[1 + (c + d*x)^2]/(b*(a + b*ArcSinh[c + d*x]))) + (I*(I*CoshIntegral[(a + b*ArcSinh [c + d*x])/b]*Sinh[a/b] - I*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x] )/b]))/b^2)/(2*b)))/(3*b) + (-1/2*(c + d*x)^3/(b*(a + b*ArcSinh[c + d*x])^ 2) + (3*(-(((c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(b*(a + b*ArcSinh[c + d*x]) )) + ((CoshIntegral[(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b])/4 - (3*CoshInte gral[(3*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(3*a)/b])/4 - (Cosh[a/b]*SinhInt egral[(a + b*ArcSinh[c + d*x])/b])/4 + (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/4)/b^2))/(2*b))/b))/d
3.2.76.3.1 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_Symbol] :> Simp[Sqrt[1 + c^ 2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1) ) Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && 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] - 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_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
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. \(708\) vs. \(2(307)=614\).
Time = 0.47 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.14
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (9 b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+18 a b \,\operatorname {arcsinh}\left (d x +c \right )-3 b^{2} \operatorname {arcsinh}\left (d x +c \right )+9 a^{2}-3 a b +2 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 {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )-b^{2} \operatorname {arcsinh}\left (d x +c \right )+a^{2}-a b +2 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 {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}+\frac {e^{2} \left (d x +c +\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^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\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^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \,\operatorname {arcsinh}\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 {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\) | \(709\) |
| default | \(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (9 b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+18 a b \,\operatorname {arcsinh}\left (d x +c \right )-3 b^{2} \operatorname {arcsinh}\left (d x +c \right )+9 a^{2}-3 a b +2 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 {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )-b^{2} \operatorname {arcsinh}\left (d x +c \right )+a^{2}-a b +2 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 {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}+\frac {e^{2} \left (d x +c +\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^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\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^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \,\operatorname {arcsinh}\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 {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\) | \(709\) |
1/d*(1/48*(-4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)^3-(1+(d*x+c)^2)^(1/2 )+3*d*x+3*c)*e^2*(9*b^2*arcsinh(d*x+c)^2+18*a*b*arcsinh(d*x+c)-3*b^2*arcsi nh(d*x+c)+9*a^2-3*a*b+2*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)+9/16*e^2/b^4*exp(3*a/b)*Ei(1,3*arcsinh(d *x+c)+3*a/b)-1/48*(-(1+(d*x+c)^2)^(1/2)+d*x+c)*e^2*(b^2*arcsinh(d*x+c)^2+2 *a*b*arcsinh(d*x+c)-b^2*arcsinh(d*x+c)+a^2-a*b+2*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)-1/48*e^2/b^4*ex p(a/b)*Ei(1,arcsinh(d*x+c)+a/b)+1/24/b*e^2*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+ b*arcsinh(d*x+c))^3+1/48/b^2*e^2*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh( d*x+c))^2+1/48/b^3*e^2*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))+1/ 48/b^4*e^2*exp(-a/b)*Ei(1,-arcsinh(d*x+c)-a/b)-1/24/b*e^2*(4*(d*x+c)^3+3*d *x+3*c+4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d *x+c))^3-1/16/b^2*e^2*(4*(d*x+c)^3+3*d*x+3*c+4*(d*x+c)^2*(1+(d*x+c)^2)^(1/ 2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2-3/16/b^3*e^2*(4*(d*x+c)^3+3 *d*x+3*c+4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh (d*x+c))-9/16/b^4*e^2*exp(-3*a/b)*Ei(1,-3*arcsinh(d*x+c)-3*a/b))
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(b^4*arcsinh(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)
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
e**2*(Integral(c**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) + Int egral(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(2*c*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. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\text {Timed out} \]
\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4} \,d x \]