Optimal. Leaf size=60 \[ -\frac {\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cosh ^{-1}\left (a x^n\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{n}+\frac {\text {PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n} \]
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Rubi [A]
time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6011, 3799,
2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (-e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac {\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cosh ^{-1}\left (a x^n\right ) \log \left (e^{2 \cosh ^{-1}\left (a x^n\right )}+1\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 6011
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}\left (a x^n\right )}{x} \, dx &=\frac {\text {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {2 \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cosh ^{-1}\left (a x^n\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{n}-\frac {\text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cosh ^{-1}\left (a x^n\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{n}-\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n}\\ &=-\frac {\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cosh ^{-1}\left (a x^n\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{n}+\frac {\text {Li}_2\left (-e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(179\) vs. \(2(60)=120\).
time = 0.38, size = 179, normalized size = 2.98 \begin {gather*} \cosh ^{-1}\left (a x^n\right ) \log (x)+\frac {a \sqrt {1-a^2 x^{2 n}} \left (\sinh ^{-1}\left (\sqrt {-a^2} x^n\right )^2+2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right )}\right )-2 n \log (x) \log \left (\sqrt {-a^2} x^n+\sqrt {1-a^2 x^{2 n}}\right )-\text {PolyLog}\left (2,e^{-2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right )}\right )\right )}{2 \sqrt {-a^2} n \sqrt {-1+a x^n} \sqrt {1+a x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 86, normalized size = 1.43
method | result | size |
derivativedivides | \(\frac {-\frac {\mathrm {arccosh}\left (a \,x^{n}\right )^{2}}{2}+\mathrm {arccosh}\left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )+\frac {\polylog \left (2, -\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )}{2}}{n}\) | \(86\) |
default | \(\frac {-\frac {\mathrm {arccosh}\left (a \,x^{n}\right )^{2}}{2}+\mathrm {arccosh}\left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )+\frac {\polylog \left (2, -\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )}{2}}{n}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}{\left (a x^{n} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {acosh}\left (a\,x^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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