Optimal. Leaf size=410 \[ -\frac{i b^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 0.572884, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {4777, 4763, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 4657, 4181} \[ -\frac{i b^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}+\frac{g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 4777
Rule 4763
Rule 4651
Rule 4675
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rule 4677
Rule 4657
Rule 4181
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{f \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (f \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}+\frac{\left (g \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b c f \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b g \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 i b f \sqrt{1-c^2 x^2}\right ) \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 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^2 g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{\left (i b^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{f x \left (a+b \sin ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 1.36067, size = 237, normalized size = 0.58 \[ \frac{\sqrt{1-c^2 x^2} \left ((c f-g) \left (-\cot \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i \left (4 b^2 \text{PolyLog}\left (2,-i e^{-i \sin ^{-1}(c x)}\right )+\left (a+b \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)-4 i b \log \left (1+i e^{-i \sin ^{-1}(c x)}\right )\right )\right )\right )-(c f+g) \left (-\tan \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i \left (4 b^2 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+\left (a+b \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)+4 i b \log \left (1+i e^{i \sin ^{-1}(c x)}\right )\right )\right )\right )\right )}{2 c^2 d \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.312, size = 1047, normalized size = 2.6 \begin{align*} \text{result too large to display} \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{a b c f \sqrt{\frac{1}{c^{4} d}} \log \left (x^{2} - \frac{1}{c^{2}}\right )}{d} + \frac{2 \, a b f x \arcsin \left (c x\right )}{\sqrt{-c^{2} d x^{2} + d} d} + \frac{a^{2} f x}{\sqrt{-c^{2} d x^{2} + d} d} - \sqrt{d} \int \frac{2 \, a b g x \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) +{\left (b^{2} g x + b^{2} f\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2}}{{\left (c^{2} d^{2} x^{2} - d^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}\,{d x} + \frac{a^{2} g}{\sqrt{-c^{2} d x^{2} + d} 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{\sqrt{-c^{2} d x^{2} + d}{\left (a^{2} g x + a^{2} f +{\left (b^{2} g x + b^{2} f\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b g x + a b f\right )} \arcsin \left (c x\right )\right )}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, 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{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}\, 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{{\left (g x + f\right )}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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