Integrand size = 14, antiderivative size = 348 \[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=-\frac {\text {arcsinh}(c x)^4}{4 e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \]
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Time = 0.29 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5827, 5680, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e}-\frac {\text {arcsinh}(c x)^4}{4 e} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 5680
Rule 5827
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^3 \cosh (x)}{c d+e \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {\text {arcsinh}(c x)^4}{4 e}+\text {Subst}\left (\int \frac {e^x x^3}{c d-\sqrt {c^2 d^2+e^2}+e e^x} \, dx,x,\text {arcsinh}(c x)\right )+\text {Subst}\left (\int \frac {e^x x^3}{c d+\sqrt {c^2 d^2+e^2}+e e^x} \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {\text {arcsinh}(c x)^4}{4 e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {3 \text {Subst}\left (\int x^2 \log \left (1+\frac {e e^x}{c d-\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e}-\frac {3 \text {Subst}\left (\int x^2 \log \left (1+\frac {e e^x}{c d+\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e} \\ & = -\frac {\text {arcsinh}(c x)^4}{4 e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-\frac {e e^x}{c d-\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e}-\frac {6 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-\frac {e e^x}{c d+\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e} \\ & = -\frac {\text {arcsinh}(c x)^4}{4 e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-\frac {e e^x}{c d-\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e}+\frac {6 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-\frac {e e^x}{c d+\sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e} \\ & = -\frac {\text {arcsinh}(c x)^4}{4 e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {e x}{-c d+\sqrt {c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{e}+\frac {6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {e x}{c d+\sqrt {c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{e} \\ & = -\frac {\text {arcsinh}(c x)^4}{4 e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {3 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.93 \[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=\frac {-\text {arcsinh}(c x)^4+4 \text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )+4 \text {arcsinh}(c x)^3 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )+12 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )+12 \text {arcsinh}(c x)^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )-24 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-24 \text {arcsinh}(c x) \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )+24 \operatorname {PolyLog}\left (4,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )+24 \operatorname {PolyLog}\left (4,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{4 e} \]
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\[\int \frac {\operatorname {arcsinh}\left (c x \right )^{3}}{e x +d}d x\]
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\[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=\int { \frac {\operatorname {arsinh}\left (c x\right )^{3}}{e x + d} \,d x } \]
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\[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (c x \right )}}{d + e x}\, dx \]
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\[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=\int { \frac {\operatorname {arsinh}\left (c x\right )^{3}}{e x + d} \,d x } \]
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\[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=\int { \frac {\operatorname {arsinh}\left (c x\right )^{3}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(c x)^3}{d+e x} \, dx=\int \frac {{\mathrm {asinh}\left (c\,x\right )}^3}{d+e\,x} \,d x \]
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