Optimal. Leaf size=82 \[ \frac{i b \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^2 d}+\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac{\log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d} \]
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Rubi [A] time = 0.105294, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4675, 3719, 2190, 2279, 2391} \[ \frac{i b \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^2 d}+\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac{\log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d} \]
Antiderivative was successfully verified.
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Rule 4675
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d}\\ &=\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\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 x)\right )}{c^2 d}\\ &=\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^2 d}+\frac{b \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d}\\ &=\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^2 d}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 c^2 d}\\ &=\frac{i \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c^2 d}-\frac{\left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^2 d}+\frac{i b \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 c^2 d}\\ \end{align*}
Mathematica [B] time = 0.0762788, size = 244, normalized size = 2.98 \[ -\frac{-2 i b \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )-2 i b \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+a \log \left (1-c^2 x^2\right )-i b \sin ^{-1}(c x)^2+2 i \pi b \sin ^{-1}(c x)+2 b \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+2 b \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+4 \pi b \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+\pi b \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-\pi b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-\pi b \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-4 \pi b \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+\pi b \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{2 c^2 d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.042, size = 118, normalized size = 1.4 \begin{align*} -{\frac{a\ln \left ( cx-1 \right ) }{2\,{c}^{2}d}}-{\frac{a\ln \left ( cx+1 \right ) }{2\,{c}^{2}d}}+{\frac{{\frac{i}{2}}b \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{{c}^{2}d}}-{\frac{b\arcsin \left ( cx \right ) }{{c}^{2}d}\ln \left ( 1+ \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }+{\frac{{\frac{i}{2}}b}{{c}^{2}d}{\it polylog} \left ( 2,- \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \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*} -\frac{{\left (c^{2} d \int \frac{e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) + e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right )}{c^{5} d x^{4} - c^{3} d x^{2} -{\left (c^{3} d x^{2} - c d\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x} + \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) + \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right )\right )} b}{2 \, c^{2} d} - \frac{a \log \left (c^{2} d x^{2} - d\right )}{2 \, c^{2} d} \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 x \arcsin \left (c x\right ) + a x}{c^{2} d x^{2} - d}, 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 x}{c^{2} x^{2} - 1}\, dx + \int \frac{b x \operatorname{asin}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \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{{\left (b \arcsin \left (c x\right ) + a\right )} x}{c^{2} d x^{2} - d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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