Integrand size = 21, antiderivative size = 178 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c d^3} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5788, 5789, 4265, 2317, 2438, 267} \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {3 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{4 c d^3}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (c^2 x^2+1\right )}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {3 b}{8 c d^3 \sqrt {c^2 x^2+1}}+\frac {b}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}} \]
[In]
[Out]
Rule 267
Rule 2317
Rule 2438
Rule 4265
Rule 5788
Rule 5789
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {(b c) \int \frac {x}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^2} \, dx}{4 d} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}-\frac {(3 b c) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{8 d^3}+\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx}{8 d^2} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{8 c d^3} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {(3 i b) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{8 c d^3}+\frac {(3 i b) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{8 c d^3} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{8 c d^3} \\ & = \frac {b}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {3 b}{8 c d^3 \sqrt {1+c^2 x^2}}+\frac {x (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {3 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 c d^3}-\frac {3 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 c d^3}+\frac {3 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 c d^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.92 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {15 a c x+9 a c^3 x^3+11 b \sqrt {1+c^2 x^2}+9 b c^2 x^2 \sqrt {1+c^2 x^2}+15 b c x \text {arcsinh}(c x)+9 b c^3 x^3 \text {arcsinh}(c x)+9 a \arctan (c x)+18 a c^2 x^2 \arctan (c x)+9 a c^4 x^4 \arctan (c x)+9 i b \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+18 i b c^2 x^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+9 i b c^4 x^4 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )-9 i b \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-18 i b c^2 x^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-9 i b c^4 x^4 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-9 i b \left (1+c^2 x^2\right )^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+9 i b \left (1+c^2 x^2\right )^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{24 c d^3 \left (1+c^2 x^2\right )^2} \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.39
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {c x}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {11}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}}{c}\) | \(248\) |
default | \(\frac {\frac {a \left (\frac {c x}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {11}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}}{c}\) | \(248\) |
parts | \(\frac {a \left (\frac {x}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 x}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \arctan \left (c x \right )}{8 c}\right )}{d^{3}}+\frac {b \left (\frac {c x \,\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {11}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {3 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {3 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3} c}\) | \(248\) |
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
[In]
[Out]