Optimal. Leaf size=69 \[ a \log (x)-\frac {1}{4} i b \text {Li}_2\left (e^{2 i \sin ^{-1}\left (c x^2\right )}\right )-\frac {1}{4} i b \sin ^{-1}\left (c x^2\right )^2+\frac {1}{2} b \sin ^{-1}\left (c x^2\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^2\right )}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 69, 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 = {6742, 4830, 3717, 2190, 2279, 2391} \[ -\frac {1}{4} i b \text {PolyLog}\left (2,e^{2 i \sin ^{-1}\left (c x^2\right )}\right )+a \log (x)-\frac {1}{4} i b \sin ^{-1}\left (c x^2\right )^2+\frac {1}{2} b \sin ^{-1}\left (c x^2\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^2\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4830
Rule 6742
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c x^2\right )}{x} \, dx &=\int \left (\frac {a}{x}+\frac {b \sin ^{-1}\left (c x^2\right )}{x}\right ) \, dx\\ &=a \log (x)+b \int \frac {\sin ^{-1}\left (c x^2\right )}{x} \, dx\\ &=a \log (x)+\frac {1}{2} b \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (c x^2\right )\right )\\ &=-\frac {1}{4} i b \sin ^{-1}\left (c x^2\right )^2+a \log (x)-(i b) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (c x^2\right )\right )\\ &=-\frac {1}{4} i b \sin ^{-1}\left (c x^2\right )^2+\frac {1}{2} b \sin ^{-1}\left (c x^2\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^2\right )}\right )+a \log (x)-\frac {1}{2} b \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (c x^2\right )\right )\\ &=-\frac {1}{4} i b \sin ^{-1}\left (c x^2\right )^2+\frac {1}{2} b \sin ^{-1}\left (c x^2\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^2\right )}\right )+a \log (x)+\frac {1}{4} (i b) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (c x^2\right )}\right )\\ &=-\frac {1}{4} i b \sin ^{-1}\left (c x^2\right )^2+\frac {1}{2} b \sin ^{-1}\left (c x^2\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^2\right )}\right )+a \log (x)-\frac {1}{4} i b \text {Li}_2\left (e^{2 i \sin ^{-1}\left (c x^2\right )}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 64, normalized size = 0.93 \[ a \log (x)+\frac {1}{2} b \left (\sin ^{-1}\left (c x^2\right ) \log \left (1-e^{2 i \sin ^{-1}\left (c x^2\right )}\right )-\frac {1}{2} i \left (\sin ^{-1}\left (c x^2\right )^2+\text {Li}_2\left (e^{2 i \sin ^{-1}\left (c x^2\right )}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (c x^{2}\right ) + a}{x}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x^{2}\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arcsin \left (c \,x^{2}\right )}{x}\, dx \]
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 \[ b \int \frac {\arctan \left (c x^{2}, \sqrt {c x^{2} + 1} \sqrt {-c x^{2} + 1}\right )}{x}\,{d x} + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 57, normalized size = 0.83 \[ a\,\ln \relax (x)-\frac {b\,{\mathrm {asin}\left (c\,x^2\right )}^2\,1{}\mathrm {i}}{4}-\frac {b\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (c\,x^2\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{4}+\frac {b\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (c\,x^2\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (c\,x^2\right )}{2} \]
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 \[ \int \frac {a + b \operatorname {asin}{\left (c x^{2} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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