Optimal. Leaf size=68 \[ -\frac {i \text {ArcCos}\left (a x^n\right )^2}{2 n}+\frac {\text {ArcCos}\left (a x^n\right ) \log \left (1+e^{2 i \text {ArcCos}\left (a x^n\right )}\right )}{n}-\frac {i \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}\left (a x^n\right )}\right )}{2 n} \]
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Rubi [A]
time = 0.04, antiderivative size = 68, 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 = {4915, 3800,
2221, 2317, 2438} \begin {gather*} -\frac {i \text {Li}_2\left (-e^{2 i \text {ArcCos}\left (a x^n\right )}\right )}{2 n}-\frac {i \text {ArcCos}\left (a x^n\right )^2}{2 n}+\frac {\text {ArcCos}\left (a x^n\right ) \log \left (1+e^{2 i \text {ArcCos}\left (a x^n\right )}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4915
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}\left (a x^n\right )}{x} \, dx &=-\frac {\text {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {(2 i) \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n}-\frac {\text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n}+\frac {i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{2 n}\\ &=-\frac {i \cos ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cos ^{-1}\left (a x^n\right ) \log \left (1+e^{2 i \cos ^{-1}\left (a x^n\right )}\right )}{n}-\frac {i \text {Li}_2\left (-e^{2 i \cos ^{-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. \(141\) vs. \(2(68)=136\).
time = 0.10, size = 141, normalized size = 2.07 \begin {gather*} \text {ArcCos}\left (a x^n\right ) \log (x)+\frac {a \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 84, normalized size = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {i \arccos \left (a \,x^{n}\right )^{2}}{2}+\arccos \left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )-\frac {i \polylog \left (2, -\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )}{2}}{n}\) | \(84\) |
default | \(\frac {-\frac {i \arccos \left (a \,x^{n}\right )^{2}}{2}+\arccos \left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )-\frac {i \polylog \left (2, -\left (a \,x^{n}+i \sqrt {1-a^{2} x^{2 n}}\right )^{2}\right )}{2}}{n}\) | \(84\) |
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 {acos}{\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 = -0.01 \begin {gather*} \int \frac {\mathrm {acos}\left (a\,x^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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