Integrand size = 10, antiderivative size = 108 \[ \int \frac {\text {arcsinh}(a x)^4}{x^3} \, dx=-2 a^2 \text {arcsinh}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}-\frac {\text {arcsinh}(a x)^4}{2 x^2}+6 a^2 \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+6 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5776, 5800, 5775, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arcsinh}(a x)^4}{x^3} \, dx=6 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )-\frac {2 a \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{x}-2 a^2 \text {arcsinh}(a x)^3+6 a^2 \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )-\frac {\text {arcsinh}(a x)^4}{2 x^2} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5775
Rule 5776
Rule 5800
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arcsinh}(a x)^4}{2 x^2}+(2 a) \int \frac {\text {arcsinh}(a x)^3}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}-\frac {\text {arcsinh}(a x)^4}{2 x^2}+\left (6 a^2\right ) \int \frac {\text {arcsinh}(a x)^2}{x} \, dx \\ & = -\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}-\frac {\text {arcsinh}(a x)^4}{2 x^2}+\left (6 a^2\right ) \text {Subst}\left (\int x^2 \coth (x) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 a^2 \text {arcsinh}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}-\frac {\text {arcsinh}(a x)^4}{2 x^2}-\left (12 a^2\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 a^2 \text {arcsinh}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}-\frac {\text {arcsinh}(a x)^4}{2 x^2}+6 a^2 \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )-\left (12 a^2\right ) \text {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 a^2 \text {arcsinh}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}-\frac {\text {arcsinh}(a x)^4}{2 x^2}+6 a^2 \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+6 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\left (6 a^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -2 a^2 \text {arcsinh}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}-\frac {\text {arcsinh}(a x)^4}{2 x^2}+6 a^2 \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+6 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-\left (3 a^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \text {arcsinh}(a x)}\right ) \\ & = -2 a^2 \text {arcsinh}(a x)^3-\frac {2 a \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{x}-\frac {\text {arcsinh}(a x)^4}{2 x^2}+6 a^2 \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+6 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05 \[ \int \frac {\text {arcsinh}(a x)^4}{x^3} \, dx=-\frac {\text {arcsinh}(a x)^4}{2 x^2}+\frac {1}{4} a^2 \left (i \pi ^3-8 \text {arcsinh}(a x)^3-\frac {8 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a x}+24 \text {arcsinh}(a x)^2 \log \left (1-e^{2 \text {arcsinh}(a x)}\right )+24 \text {arcsinh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(a x)}\right )-12 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(a x)}\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.84
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\operatorname {arcsinh}\left (a x \right )^{3} \left (-4 a^{2} x^{2}+4 a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2} x^{2}}-4 \operatorname {arcsinh}\left (a x \right )^{3}+6 \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+12 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-12 \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+12 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-12 \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(199\) |
default | \(a^{2} \left (-\frac {\operatorname {arcsinh}\left (a x \right )^{3} \left (-4 a^{2} x^{2}+4 a x \sqrt {a^{2} x^{2}+1}+\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{2} x^{2}}-4 \operatorname {arcsinh}\left (a x \right )^{3}+6 \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a^{2} x^{2}+1}\right )+12 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a^{2} x^{2}+1}\right )-12 \operatorname {polylog}\left (3, -a x -\sqrt {a^{2} x^{2}+1}\right )+6 \operatorname {arcsinh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+12 \,\operatorname {arcsinh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-12 \operatorname {polylog}\left (3, a x +\sqrt {a^{2} x^{2}+1}\right )\right )\) | \(199\) |
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\[ \int \frac {\text {arcsinh}(a x)^4}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{3}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^4}{x^3} \, dx=\int \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\text {arcsinh}(a x)^4}{x^3} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{4}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\text {arcsinh}(a x)^4}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^4}{x^3} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^4}{x^3} \,d x \]
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