Integrand size = 19, antiderivative size = 174 \[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \operatorname {PolyLog}\left (4,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \operatorname {PolyLog}\left (4,i e^{\text {arcsinh}(a x)}\right )}{a c} \]
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Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5789, 4265, 2611, 6744, 2320, 6724} \[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \operatorname {PolyLog}\left (4,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \operatorname {PolyLog}\left (4,i e^{\text {arcsinh}(a x)}\right )}{a c} \]
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Rule 2320
Rule 2611
Rule 4265
Rule 5789
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\text {arcsinh}(a x)\right )}{a c} \\ & = \frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {(3 i) \text {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c}+\frac {(3 i) \text {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c} \\ & = \frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {(6 i) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c}-\frac {(6 i) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c} \\ & = \frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {(6 i) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c}+\frac {(6 i) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a c} \\ & = \frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {(6 i) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {(6 i) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )}{a c} \\ & = \frac {2 \text {arcsinh}(a x)^3 \arctan \left (e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {3 i \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(a x)}\right )}{a c}-\frac {6 i \operatorname {PolyLog}\left (4,-i e^{\text {arcsinh}(a x)}\right )}{a c}+\frac {6 i \operatorname {PolyLog}\left (4,i e^{\text {arcsinh}(a x)}\right )}{a c} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.28 \[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\frac {-\text {arcsinh}(a x)^3 \log \left (1+\frac {a e^{\text {arcsinh}(a x)}}{\sqrt {-a^2}}\right )+\text {arcsinh}(a x)^3 \log \left (1+\frac {\sqrt {-a^2} e^{\text {arcsinh}(a x)}}{a}\right )+3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {a e^{\text {arcsinh}(a x)}}{\sqrt {-a^2}}\right )-3 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-a^2} e^{\text {arcsinh}(a x)}}{a}\right )-6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,\frac {a e^{\text {arcsinh}(a x)}}{\sqrt {-a^2}}\right )+6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,\frac {\sqrt {-a^2} e^{\text {arcsinh}(a x)}}{a}\right )+6 \operatorname {PolyLog}\left (4,\frac {a e^{\text {arcsinh}(a x)}}{\sqrt {-a^2}}\right )-6 \operatorname {PolyLog}\left (4,\frac {\sqrt {-a^2} e^{\text {arcsinh}(a x)}}{a}\right )}{\sqrt {-a^2} c} \]
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\[\int \frac {\operatorname {arcsinh}\left (a x \right )^{3}}{a^{2} c \,x^{2}+c}d x\]
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\[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{c+a^2 c x^2} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{c\,a^2\,x^2+c} \,d x \]
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