Optimal. Leaf size=333 \[ \frac{2 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}-\frac{2 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}+\frac{7 i b^2 c^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac{7 i b^2 c^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac{2 b^2 c^3 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{d}+\frac{2 b^2 c^3 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac{2 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d}-\frac{14 b c^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{b^2 c^2}{3 d x} \]
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Rubi [A] time = 0.654227, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {4701, 4657, 4181, 2531, 2282, 6589, 4709, 4183, 2279, 2391, 30} \[ \frac{2 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}-\frac{2 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d}+\frac{7 i b^2 c^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac{7 i b^2 c^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac{2 b^2 c^3 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )}{d}+\frac{2 b^2 c^3 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac{2 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d}-\frac{14 b c^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 d}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{b^2 c^2}{3 d x} \]
Antiderivative was successfully verified.
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Rule 4701
Rule 4657
Rule 4181
Rule 2531
Rule 2282
Rule 6589
Rule 4709
Rule 4183
Rule 2279
Rule 2391
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )} \, dx &=-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+c^2 \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )} \, dx+\frac{(2 b c) \int \frac{a+b \sin ^{-1}(c x)}{x^3 \sqrt{1-c^2 x^2}} \, dx}{3 d}\\ &=-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}+c^4 \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{d-c^2 d x^2} \, dx+\frac{\left (b^2 c^2\right ) \int \frac{1}{x^2} \, dx}{3 d}+\frac{\left (b c^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx}{3 d}+\frac{\left (2 b c^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{1-c^2 x^2}} \, dx}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}+\frac{c^3 \operatorname{Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d}+\frac{\left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac{2 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{14 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac{\left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac{\left (2 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}-\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d}+\frac{\left (b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d}-\frac{\left (2 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac{\left (2 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac{2 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{14 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac{2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d}+\frac{\left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac{\left (i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac{\left (2 i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{\left (2 i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{\left (2 i b^2 c^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}+\frac{\left (2 i b^2 c^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac{2 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{14 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac{7 i b^2 c^3 \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac{2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{7 i b^2 c^3 \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac{\left (2 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}+\frac{\left (2 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d}\\ &=-\frac{b^2 c^2}{3 d x}-\frac{b c \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2}-\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac{c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x}-\frac{2 i c^3 \left (a+b \sin ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{14 b c^3 \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac{7 i b^2 c^3 \text{Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{3 d}+\frac{2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{2 i b c^3 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{d}-\frac{7 i b^2 c^3 \text{Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{3 d}-\frac{2 b^2 c^3 \text{Li}_3\left (-i e^{i \sin ^{-1}(c x)}\right )}{d}+\frac{2 b^2 c^3 \text{Li}_3\left (i e^{i \sin ^{-1}(c x)}\right )}{d}\\ \end{align*}
Mathematica [B] time = 7.8188, size = 849, normalized size = 2.55 \[ -\frac{a^2 \log (1-c x) c^3}{2 d}+\frac{a^2 \log (c x+1) c^3}{2 d}-\frac{b^2 \left (\frac{1}{2} c x \sin ^{-1}(c x)^2 \csc ^4\left (\frac{1}{2} \sin ^{-1}(c x)\right )+2 \sin ^{-1}(c x) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )+\frac{8 \sin ^{-1}(c x)^2 \sin ^4\left (\frac{1}{2} \sin ^{-1}(c x)\right )}{c^3 x^3}-2 \sin ^{-1}(c x) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)\right )+14 \sin ^{-1}(c x)^2 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )+4 \cot \left (\frac{1}{2} \sin ^{-1}(c x)\right )-56 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )-24 \sin ^{-1}(c x)^2 \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+24 \sin ^{-1}(c x)^2 \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+56 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )-56 i \text{PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )-48 i \sin ^{-1}(c x) \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+48 i \sin ^{-1}(c x) \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+56 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )+48 \text{PolyLog}\left (3,-i e^{i \sin ^{-1}(c x)}\right )-48 \text{PolyLog}\left (3,i e^{i \sin ^{-1}(c x)}\right )+14 \sin ^{-1}(c x)^2 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )+4 \tan \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) c^3}{24 d}-\frac{a^2 c^2}{d x}-\frac{2 a b \left (\frac{1}{2} \left (-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{2 \log \left (1+i e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)}{c}+\frac{3 i \pi \sin ^{-1}(c x)}{2 c}+\frac{2 \pi \log \left (1+e^{-i \sin ^{-1}(c x)}\right )}{c}-\frac{\pi \log \left (1+i e^{i \sin ^{-1}(c x)}\right )}{c}-\frac{2 \pi \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{c}+\frac{\pi \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{c}-\frac{2 i \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c}\right ) c^4-\frac{1}{2} \left (-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{2 \log \left (1-i e^{i \sin ^{-1}(c x)}\right ) \sin ^{-1}(c x)}{c}+\frac{i \pi \sin ^{-1}(c x)}{2 c}+\frac{2 \pi \log \left (1+e^{-i \sin ^{-1}(c x)}\right )}{c}+\frac{\pi \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c}-\frac{2 \pi \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{c}-\frac{\pi \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{c}-\frac{2 i \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c}\right ) c^4-\left (-\frac{\sin ^{-1}(c x)}{x}-c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )\right ) c^2+\frac{c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) x^3+c \sqrt{1-c^2 x^2} x+2 \sin ^{-1}(c x)}{6 x^3}\right )}{d}-\frac{a^2}{3 d x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.312, size = 725, normalized size = 2.2 \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{1}{6} \,{\left (\frac{3 \, c^{3} \log \left (c x + 1\right )}{d} - \frac{3 \, c^{3} \log \left (c x - 1\right )}{d} - \frac{2 \,{\left (3 \, c^{2} x^{2} + 1\right )}}{d x^{3}}\right )} a^{2} + \frac{3 \, b^{2} c^{3} x^{3} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (c x + 1\right ) - 3 \, b^{2} c^{3} x^{3} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} \log \left (-c x + 1\right ) - 2 \, d x^{3} \int \frac{6 \, a b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (3 \, b^{2} c^{4} x^{4} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) - 3 \, b^{2} c^{4} x^{4} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \,{\left (3 \, b^{2} c^{3} x^{3} + b^{2} c x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{2} d x^{6} - d x^{4}}\,{d x} - 2 \,{\left (3 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2}}{6 \, d x^{3}} \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^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}{c^{2} d x^{6} - d x^{4}}, 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^{2}}{c^{2} x^{6} - x^{4}}\, dx + \int \frac{b^{2} \operatorname{asin}^{2}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, dx + \int \frac{2 a b \operatorname{asin}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, 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 )}^{2}}{{\left (c^{2} d x^{2} - d\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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