Integrand size = 10, antiderivative size = 84 \[ \int \frac {\text {arcsinh}(a x)^3}{x^2} \, dx=-\frac {\text {arcsinh}(a x)^3}{x}-6 a \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 a \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-6 a \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5776, 5816, 4267, 2611, 2320, 6724} \[ \int \frac {\text {arcsinh}(a x)^3}{x^2} \, dx=-6 a \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 a \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-6 a \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-\frac {\text {arcsinh}(a x)^3}{x} \]
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Rule 2320
Rule 2611
Rule 4267
Rule 5776
Rule 5816
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}(a x)^3}{x}+(3 a) \int \frac {\text {arcsinh}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\text {arcsinh}(a x)^3}{x}+(3 a) \text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {\text {arcsinh}(a x)^3}{x}-6 a \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-(6 a) \text {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+(6 a) \text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {\text {arcsinh}(a x)^3}{x}-6 a \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+(6 a) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )-(6 a) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {\text {arcsinh}(a x)^3}{x}-6 a \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+(6 a) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )-(6 a) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right ) \\ & = -\frac {\text {arcsinh}(a x)^3}{x}-6 a \text {arcsinh}(a x)^2 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+6 a \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+6 a \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )-6 a \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.39 \[ \int \frac {\text {arcsinh}(a x)^3}{x^2} \, dx=a \left (-\frac {\text {arcsinh}(a x)^3}{a x}+3 \text {arcsinh}(a x)^2 \log \left (1-e^{-\text {arcsinh}(a x)}\right )-3 \text {arcsinh}(a x)^2 \log \left (1+e^{-\text {arcsinh}(a x)}\right )+6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-6 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(a x)}\right )-6 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(a x)}\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.92
method | result | size |
derivativedivides | \(a \left (-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{a x}-3 \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+6 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(161\) |
default | \(a \left (-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{a x}-3 \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+3 \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+6 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(161\) |
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^2} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{x^2} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^2} \,d x \]
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