Integrand size = 10, antiderivative size = 87 \[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=-\frac {1}{4} \text {arccosh}(a x)^4+\text {arccosh}(a x)^3 \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {3}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{2 \text {arccosh}(a x)}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5882, 3799, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=\frac {3}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{2 \text {arccosh}(a x)}\right )-\frac {1}{4} \text {arccosh}(a x)^4+\text {arccosh}(a x)^3 \log \left (e^{2 \text {arccosh}(a x)}+1\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 5882
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^3 \tanh (x) \, dx,x,\text {arccosh}(a x)\right ) \\ & = -\frac {1}{4} \text {arccosh}(a x)^4+2 \text {Subst}\left (\int \frac {e^{2 x} x^3}{1+e^{2 x}} \, dx,x,\text {arccosh}(a x)\right ) \\ & = -\frac {1}{4} \text {arccosh}(a x)^4+\text {arccosh}(a x)^3 \log \left (1+e^{2 \text {arccosh}(a x)}\right )-3 \text {Subst}\left (\int x^2 \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = -\frac {1}{4} \text {arccosh}(a x)^4+\text {arccosh}(a x)^3 \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {3}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-3 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = -\frac {1}{4} \text {arccosh}(a x)^4+\text {arccosh}(a x)^3 \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {3}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+\frac {3}{2} \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = -\frac {1}{4} \text {arccosh}(a x)^4+\text {arccosh}(a x)^3 \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {3}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 \text {arccosh}(a x)}\right ) \\ & = -\frac {1}{4} \text {arccosh}(a x)^4+\text {arccosh}(a x)^3 \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {3}{2} \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,-e^{2 \text {arccosh}(a x)}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=\frac {1}{4} \left (\text {arccosh}(a x)^4+4 \text {arccosh}(a x)^3 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )-6 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )-6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(a x)}\right )-3 \operatorname {PolyLog}\left (4,-e^{-2 \text {arccosh}(a x)}\right )\right ) \]
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Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52
method | result | size |
derivativedivides | \(-\frac {\operatorname {arccosh}\left (a x \right )^{4}}{4}+\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{4}\) | \(132\) |
default | \(-\frac {\operatorname {arccosh}\left (a x \right )^{4}}{4}+\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{4}\) | \(132\) |
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\[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{x} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x} \,d x \]
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