Optimal. Leaf size=296 \[ \frac{i b \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac{i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac{b^2 \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac{b^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac{4 b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt{1-c^2 x^2}}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^3}+\frac{b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac{2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3} \]
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Rubi [A] time = 0.49134, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {4705, 4679, 4419, 4183, 2531, 2282, 6589, 4651, 260, 4655, 261} \[ \frac{i b \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac{i b \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^3}-\frac{b^2 \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac{b^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}-\frac{4 b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt{1-c^2 x^2}}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d^3}+\frac{b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac{2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3} \]
Antiderivative was successfully verified.
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Rule 4705
Rule 4679
Rule 4419
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rule 4651
Rule 260
Rule 4655
Rule 261
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^3} \, dx &=\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{(b c) \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}+\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^2} \, dx}{d}\\ &=-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac{(b c) \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac{(b c) \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^3}+\frac{\left (b^2 c^2\right ) \int \frac{x}{\left (1-c^2 x^2\right )^2} \, dx}{6 d^3}+\frac{\int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )} \, dx}{d^2}\\ &=\frac{b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}+\frac{\left (b^2 c^2\right ) \int \frac{x}{1-c^2 x^2} \, dx}{3 d^3}+\frac{\left (b^2 c^2\right ) \int \frac{x}{1-c^2 x^2} \, dx}{d^3}\\ &=\frac{b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac{2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac{2 \operatorname{Subst}\left (\int (a+b x)^2 \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=\frac{b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac{2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}+\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=\frac{b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac{2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=\frac{b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac{2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ &=\frac{b^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac{b c x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{4 b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac{2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{2 b^2 \log \left (1-c^2 x^2\right )}{3 d^3}+\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{i b \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac{b^2 \text{Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac{b^2 \text{Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ \end{align*}
Mathematica [A] time = 3.68108, size = 459, normalized size = 1.55 \[ -\frac{4 a b \left (-6 i \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )+6 i \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\frac{8 c x}{\sqrt{1-c^2 x^2}}+\frac{c x}{\left (1-c^2 x^2\right )^{3/2}}+\frac{6 \sin ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \sin ^{-1}(c x)}{\left (c^2 x^2-1\right )^2}-12 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+12 \sin ^{-1}(c x) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )\right )+b^2 \left (-24 i \sin ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(c x)}\right )-24 i \sin ^{-1}(c x) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )-12 \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(c x)}\right )+12 \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(c x)}\right )+\frac{2}{c^2 x^2-1}+16 \log \left (1-c^2 x^2\right )+\frac{12 \sin ^{-1}(c x)^2}{c^2 x^2-1}-\frac{6 \sin ^{-1}(c x)^2}{\left (c^2 x^2-1\right )^2}+\frac{32 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{4 c x \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}}-16 i \sin ^{-1}(c x)^3-24 \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )+24 \sin ^{-1}(c x)^2 \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )+i \pi ^3\right )+\frac{12 a^2}{c^2 x^2-1}-\frac{6 a^2}{\left (c^2 x^2-1\right )^2}+12 a^2 \log \left (1-c^2 x^2\right )-24 a^2 \log (c x)}{24 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.31, size = 1224, normalized size = 4.1 \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}{4} \, a^{2}{\left (\frac{2 \, c^{2} x^{2} - 3}{c^{4} d^{3} x^{4} - 2 \, c^{2} d^{3} x^{2} + d^{3}} + \frac{2 \, \log \left (c x + 1\right )}{d^{3}} + \frac{2 \, \log \left (c x - 1\right )}{d^{3}} - \frac{4 \, \log \left (x\right )}{d^{3}}\right )} - \int \frac{b^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \, a b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{c^{6} d^{3} x^{7} - 3 \, c^{4} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{3} - d^{3} x}\,{d x} \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^{6} d^{3} x^{7} - 3 \, c^{4} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{3} - d^{3} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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