Optimal. Leaf size=75 \[ -\frac {i b \text {ArcSin}\left (c x^n\right )^2}{2 n}+\frac {b \text {ArcSin}\left (c x^n\right ) \log \left (1-e^{2 i \text {ArcSin}\left (c x^n\right )}\right )}{n}+a \log (x)-\frac {i b \text {PolyLog}\left (2,e^{2 i \text {ArcSin}\left (c x^n\right )}\right )}{2 n} \]
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Rubi [A]
time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6874, 4914,
3798, 2221, 2317, 2438} \begin {gather*} a \log (x)-\frac {i b \text {Li}_2\left (e^{2 i \text {ArcSin}\left (c x^n\right )}\right )}{2 n}-\frac {i b \text {ArcSin}\left (c x^n\right )^2}{2 n}+\frac {b \text {ArcSin}\left (c x^n\right ) \log \left (1-e^{2 i \text {ArcSin}\left (c x^n\right )}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4914
Rule 6874
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c x^n\right )}{x} \, dx &=\int \left (\frac {a}{x}+\frac {b \sin ^{-1}\left (c x^n\right )}{x}\right ) \, dx\\ &=a \log (x)+b \int \frac {\sin ^{-1}\left (c x^n\right )}{x} \, dx\\ &=a \log (x)+\frac {b \text {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (c x^n\right )\right )}{n}\\ &=-\frac {i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+a \log (x)-\frac {(2 i b) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (c x^n\right )\right )}{n}\\ &=-\frac {i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac {b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n}+a \log (x)-\frac {b \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (c x^n\right )\right )}{n}\\ &=-\frac {i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac {b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n}+a \log (x)+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{2 n}\\ &=-\frac {i b \sin ^{-1}\left (c x^n\right )^2}{2 n}+\frac {b \sin ^{-1}\left (c x^n\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{n}+a \log (x)-\frac {i b \text {Li}_2\left (e^{2 i \sin ^{-1}\left (c x^n\right )}\right )}{2 n}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(157\) vs. \(2(75)=150\).
time = 0.12, size = 157, normalized size = 2.09 \begin {gather*} a \log (x)+b \text {ArcSin}\left (c x^n\right ) \log (x)-\frac {b c \left (\log (x) \log \left (\sqrt {-c^2} x^n+\sqrt {1-c^2 x^{2 n}}\right )+\frac {i \left (i \sinh ^{-1}\left (\sqrt {-c^2} x^n\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt {-c^2} x^n\right )}\right )-\frac {1}{2} i \left (-\sinh ^{-1}\left (\sqrt {-c^2} x^n\right )^2+\text {PolyLog}\left (2,e^{-2 \sinh ^{-1}\left (\sqrt {-c^2} x^n\right )}\right )\right )\right )}{n}\right )}{\sqrt {-c^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 150, normalized size = 2.00
method | result | size |
derivativedivides | \(\frac {a \ln \left (c \,x^{n}\right )-\frac {i b \arcsin \left (c \,x^{n}\right )^{2}}{2}+b \arcsin \left (c \,x^{n}\right ) \ln \left (1+i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )+b \arcsin \left (c \,x^{n}\right ) \ln \left (1-i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )-i b \polylog \left (2, i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )-i b \polylog \left (2, -i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )}{n}\) | \(150\) |
default | \(\frac {a \ln \left (c \,x^{n}\right )-\frac {i b \arcsin \left (c \,x^{n}\right )^{2}}{2}+b \arcsin \left (c \,x^{n}\right ) \ln \left (1+i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )+b \arcsin \left (c \,x^{n}\right ) \ln \left (1-i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )-i b \polylog \left (2, i c \,x^{n}+\sqrt {1-c^{2} x^{2 n}}\right )-i b \polylog \left (2, -i c \,x^{n}-\sqrt {1-c^{2} x^{2 n}}\right )}{n}\) | \(150\) |
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 {a + b \operatorname {asin}{\left (c 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 {a+b\,\mathrm {asin}\left (c\,x^n\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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