Integrand size = 24, antiderivative size = 239 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {b c^3}{3 d^2 \sqrt {1+c^2 x^2}}-\frac {b c}{6 d^2 x^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (1+c^2 x^2\right )}+\frac {5 c^2 (a+b \text {arcsinh}(c x))}{3 d^2 x \left (1+c^2 x^2\right )}+\frac {5 c^4 x (a+b \text {arcsinh}(c x))}{2 d^2 \left (1+c^2 x^2\right )}+\frac {5 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d^2}+\frac {13 b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{6 d^2}-\frac {5 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{2 d^2}+\frac {5 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 d^2} \]
1/3*(-a-b*arcsinh(c*x))/d^2/x^3/(c^2*x^2+1)+5/3*c^2*(a+b*arcsinh(c*x))/d^2 /x/(c^2*x^2+1)+5/2*c^4*x*(a+b*arcsinh(c*x))/d^2/(c^2*x^2+1)+5*c^3*(a+b*arc sinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/d^2+13/6*b*c^3*arctanh((c^2*x^2+1 )^(1/2))/d^2-5/2*I*b*c^3*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^2+5/2*I*b *c^3*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/d^2+1/3*b*c^3/d^2/(c^2*x^2+1)^(1 /2)-1/6*b*c/d^2/x^2/(c^2*x^2+1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.42 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {-\frac {5 a}{3 x^3}+\frac {5 a c^2}{x}-\frac {5 b c \sqrt {1+c^2 x^2}}{6 x^2}+\frac {a}{x^3+c^2 x^5}-\frac {5 b \text {arcsinh}(c x)}{3 x^3}+\frac {5 b c^2 \text {arcsinh}(c x)}{x}+\frac {b \text {arcsinh}(c x)}{x^3+c^2 x^5}+5 a c^3 \arctan (c x)+\frac {35}{6} b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {b c^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},1+c^2 x^2\right )}{\sqrt {1+c^2 x^2}}-5 b \left (-c^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+5 b \left (-c^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+5 b \left (-c^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-5 b \left (-c^2\right )^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{2 d^2} \]
((-5*a)/(3*x^3) + (5*a*c^2)/x - (5*b*c*Sqrt[1 + c^2*x^2])/(6*x^2) + a/(x^3 + c^2*x^5) - (5*b*ArcSinh[c*x])/(3*x^3) + (5*b*c^2*ArcSinh[c*x])/x + (b*A rcSinh[c*x])/(x^3 + c^2*x^5) + 5*a*c^3*ArcTan[c*x] + (35*b*c^3*ArcTanh[Sqr t[1 + c^2*x^2]])/6 + (b*c^3*Hypergeometric2F1[-1/2, 2, 1/2, 1 + c^2*x^2])/ Sqrt[1 + c^2*x^2] - 5*b*(-c^2)^(3/2)*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x ])/Sqrt[-c^2]] + 5*b*(-c^2)^(3/2)*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSi nh[c*x])/c] + 5*b*(-c^2)^(3/2)*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 5*b*(-c^2)^(3/2)*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(2*d^2)
Time = 1.23 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.17, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {6224, 27, 243, 52, 61, 73, 221, 6224, 243, 61, 73, 221, 6203, 241, 6204, 3042, 4668, 2715, 2838}
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.
\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (c^2 d x^2+d\right )^2} \, dx\) |
\(\Big \downarrow \) 6224 |
\(\displaystyle -\frac {5}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{d^2 x^2 \left (c^2 x^2+1\right )^2}dx+\frac {b c \int \frac {1}{x^3 \left (c^2 x^2+1\right )^{3/2}}dx}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^2}dx}{3 d^2}+\frac {b c \int \frac {1}{x^3 \left (c^2 x^2+1\right )^{3/2}}dx}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^2}dx}{3 d^2}+\frac {b c \int \frac {1}{x^4 \left (c^2 x^2+1\right )^{3/2}}dx^2}{6 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^2}dx}{3 d^2}+\frac {b c \left (-\frac {3}{2} c^2 \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx^2-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^2}dx}{3 d^2}+\frac {b c \left (-\frac {3}{2} c^2 \left (\int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2+\frac {2}{\sqrt {c^2 x^2+1}}\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^2}dx}{3 d^2}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2 \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c^2}+\frac {2}{\sqrt {c^2 x^2+1}}\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^2}dx}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 6224 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+b c \int \frac {1}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx^2-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+\frac {1}{2} b c \left (\int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2+\frac {2}{\sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+\frac {1}{2} b c \left (\frac {2 \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c^2}+\frac {2}{\sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 6203 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx-\frac {1}{2} b c \int \frac {x}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 241 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 6204 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {5 c^2 \left (-3 c^2 \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}+\frac {1}{2} b c \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )\right )}{3 d^2}-\frac {a+b \text {arcsinh}(c x)}{3 d^2 x^3 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {3}{2} c^2 \left (\frac {2}{\sqrt {c^2 x^2+1}}-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )\right )-\frac {1}{x^2 \sqrt {c^2 x^2+1}}\right )}{6 d^2}\) |
-1/3*(a + b*ArcSinh[c*x])/(d^2*x^3*(1 + c^2*x^2)) + (b*c*(-(1/(x^2*Sqrt[1 + c^2*x^2])) - (3*c^2*(2/Sqrt[1 + c^2*x^2] - 2*ArcTanh[Sqrt[1 + c^2*x^2]]) )/2))/(6*d^2) - (5*c^2*(-((a + b*ArcSinh[c*x])/(x*(1 + c^2*x^2))) + (b*c*( 2/Sqrt[1 + c^2*x^2] - 2*ArcTanh[Sqrt[1 + c^2*x^2]]))/2 - 3*c^2*(b/(2*c*Sqr t[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x]))/(2*(1 + c^2*x^2)) + (2*(a + b*A rcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c))))/(3*d^2)
3.1.45.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x ], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
Time = 0.23 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}+\frac {c x}{2 c^{2} x^{2}+2}+\frac {5 \arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {1}{3 \sqrt {c^{2} x^{2}+1}}-\frac {1}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}+\frac {13 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {5 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {5 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2}}\right )\) | \(268\) |
default | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}+\frac {c x}{2 c^{2} x^{2}+2}+\frac {5 \arctan \left (c x \right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {1}{3 \sqrt {c^{2} x^{2}+1}}-\frac {1}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}+\frac {13 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {5 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {5 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2}}\right )\) | \(268\) |
parts | \(\frac {a \left (c^{4} \left (\frac {x}{2 c^{2} x^{2}+2}+\frac {5 \arctan \left (c x \right )}{2 c}\right )-\frac {1}{3 x^{3}}+\frac {2 c^{2}}{x}\right )}{d^{2}}+\frac {b \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {\operatorname {arcsinh}\left (c x \right ) c x}{2 c^{2} x^{2}+2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{2}+\frac {1}{3 \sqrt {c^{2} x^{2}+1}}-\frac {1}{6 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}+\frac {13 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {5 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}-\frac {5 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}+\frac {5 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2}\right )}{d^{2}}\) | \(271\) |
c^3*(a/d^2*(-1/3/c^3/x^3+2/c/x+1/2*c*x/(c^2*x^2+1)+5/2*arctan(c*x))+b/d^2* (-1/3*arcsinh(c*x)/c^3/x^3+2*arcsinh(c*x)/c/x+1/2*c*x/(c^2*x^2+1)*arcsinh( c*x)+5/2*arcsinh(c*x)*arctan(c*x)+1/3/(c^2*x^2+1)^(1/2)-1/6/c^2/x^2/(c^2*x ^2+1)^(1/2)+13/6*arctanh(1/(c^2*x^2+1)^(1/2))+5/2*arctan(c*x)*ln(1+I*(1+I* c*x)/(c^2*x^2+1)^(1/2))-5/2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2) )-5/2*I*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+5/2*I*dilog(1-I*(1+I*c*x)/( c^2*x^2+1)^(1/2))))
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{4}} \,d x } \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{8} + 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{8} + 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \]
(Integral(a/(c**4*x**8 + 2*c**2*x**6 + x**4), x) + Integral(b*asinh(c*x)/( c**4*x**8 + 2*c**2*x**6 + x**4), x))/d**2
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{4}} \,d x } \]
1/6*(15*c^3*arctan(c*x)/d^2 + (15*c^4*x^4 + 10*c^2*x^2 - 2)/(c^2*d^2*x^5 + d^2*x^3))*a + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/(c^4*d^2*x^8 + 2*c ^2*d^2*x^6 + d^2*x^4), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]