Optimal. Leaf size=63 \[ -\frac{1}{2} i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+\log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right ) \]
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Rubi [A] time = 0.069963, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4625, 3717, 2190, 2279, 2391} \[ -\frac{1}{2} i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+\log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
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Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx &=\operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-b \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac{1}{2} (i b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+\left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{2} i b \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0321451, size = 52, normalized size = 0.83 \[ -\frac{1}{2} i b \left (\sin ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )\right )+a \log (x)+b \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.03, size = 122, normalized size = 1.9 \begin{align*} a\ln \left ( cx \right ) -{\frac{i}{2}}b \left ( \arcsin \left ( cx \right ) \right ) ^{2}+b\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) +b\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) -ib{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -ib{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{\arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{x}\,{d x} + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arcsin \left (c x\right ) + a}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asin}{\left (c x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arcsin \left (c x\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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