Optimal. Leaf size=89 \[ -\frac{i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 b d e}+\frac{\log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e} \]
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Rubi [A] time = 0.104629, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4805, 12, 4625, 3717, 2190, 2279, 2391} \[ -\frac{i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 b d e}+\frac{\log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 12
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c+d x)}{c e+d e x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}(x)}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 b d e}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 b d e}+\frac{\left (a+b \sin ^{-1}(c+d x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d e}\\ &=-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 b d e}+\frac{\left (a+b \sin ^{-1}(c+d x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}\\ &=-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 b d e}+\frac{\left (a+b \sin ^{-1}(c+d x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e}-\frac{i b \text{Li}_2\left (e^{2 i \sin ^{-1}(c+d x)}\right )}{2 d e}\\ \end{align*}
Mathematica [A] time = 0.0589282, size = 71, normalized size = 0.8 \[ \frac{-\frac{1}{2} i b \left (\sin ^{-1}(c+d x)^2+\text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c+d x)}\right )\right )+a \log (c+d x)+b \sin ^{-1}(c+d x) \log \left (1-e^{2 i \sin ^{-1}(c+d x)}\right )}{d e} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.038, size = 182, normalized size = 2. \begin{align*}{\frac{a\ln \left ( dx+c \right ) }{de}}-{\frac{{\frac{i}{2}}b \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}}{de}}+{\frac{b\arcsin \left ( dx+c \right ) }{de}\ln \left ( 1+i \left ( dx+c \right ) +\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) }+{\frac{b\arcsin \left ( dx+c \right ) }{de}\ln \left ( 1-i \left ( dx+c \right ) -\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) }-{\frac{ib}{de}{\it polylog} \left ( 2,i \left ( dx+c \right ) +\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) }-{\frac{ib}{de}{\it polylog} \left ( 2,-i \left ( dx+c \right ) -\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (d x + c\right ) + a}{d e x + c e}, 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*} \frac{\int \frac{a}{c + d x}\, dx + \int \frac{b \operatorname{asin}{\left (c + d x \right )}}{c + d x}\, dx}{e} \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 (d x + c\right ) + a}{d e x + c e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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