Optimal. Leaf size=287 \[ 2 i b c^2 d^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-b^2 c^2 d^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac{1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-2 c^2 d^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{4} b^2 c^4 d^2 x^2+b^2 c^2 d^2 \log (x) \]
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Rubi [A] time = 0.474914, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4695, 4699, 4625, 3717, 2190, 2531, 2282, 6589, 4647, 4641, 30, 14} \[ 2 i b c^2 d^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-b^2 c^2 d^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac{1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-2 c^2 d^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{4} b^2 c^4 d^2 x^2+b^2 c^2 d^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 4695
Rule 4699
Rule 4625
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 4647
Rule 4641
Rule 30
Rule 14
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (2 c^2 d\right ) \int \frac{\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\left (b c d^2\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (2 c^2 d^2\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\left (b^2 c^2 d^2\right ) \int \frac{1-c^2 x^2}{x} \, dx+\left (2 b c^3 d^2\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\left (3 b c^3 d^2\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (2 c^2 d^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\left (b^2 c^2 d^2\right ) \int \left (\frac{1}{x}-c^2 x\right ) \, dx+\left (b c^3 d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{2} \left (3 b c^3 d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx-\left (b^2 c^4 d^2\right ) \int x \, dx+\frac{1}{2} \left (3 b^2 c^4 d^2\right ) \int x \, dx\\ &=-\frac{1}{4} b^2 c^4 d^2 x^2-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+b^2 c^2 d^2 \log (x)+\left (4 i c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{1}{4} b^2 c^4 d^2 x^2-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x)+\left (4 b c^2 d^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{1}{4} b^2 c^4 d^2 x^2-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x)+2 i b c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\left (2 i b^2 c^2 d^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{1}{4} b^2 c^4 d^2 x^2-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x)+2 i b c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\left (b^2 c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac{1}{4} b^2 c^4 d^2 x^2-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{4} c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac{2 i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^2 \log (x)+2 i b c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-b^2 c^2 d^2 \text{Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.881177, size = 343, normalized size = 1.2 \[ \frac{1}{2} d^2 \left (4 i a b c^2 \left (\sin ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )\right )+\frac{1}{6} i b^2 c^2 \left (-24 \sin ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(c x)}\right )+12 i \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(c x)}\right )-8 \sin ^{-1}(c x)^3+24 i \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )+\pi ^3\right )+a^2 c^4 x^2-4 a^2 c^2 \log (x)-\frac{a^2}{x^2}+2 a b c^4 x^2 \sin ^{-1}(c x)+a b c^2 \left (c x \sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right )-\frac{2 a b \left (c x \sqrt{1-c^2 x^2}+\sin ^{-1}(c x)\right )}{x^2}-8 a b c^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac{b^2 \left (-2 c^2 x^2 \log (c x)+2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+\sin ^{-1}(c x)^2\right )}{x^2}+\frac{1}{2} b^2 c^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )-\frac{1}{4} b^2 c^2 \left (2 \sin ^{-1}(c x)^2-1\right ) \cos \left (2 \sin ^{-1}(c x)\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.525, size = 767, normalized size = 2.7 \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}{2} \, a^{2} c^{4} d^{2} x^{2} - 2 \, a^{2} c^{2} d^{2} \log \left (x\right ) - a b d^{2}{\left (\frac{\sqrt{-c^{2} x^{2} + 1} c}{x} + \frac{\arcsin \left (c x\right )}{x^{2}}\right )} - \frac{a^{2} d^{2}}{2 \, x^{2}} + \int \frac{{\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{x^{3}}\,{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{a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} +{\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )}{x^{3}}, 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*} d^{2} \left (\int \frac{a^{2}}{x^{3}}\, dx + \int - \frac{2 a^{2} c^{2}}{x}\, dx + \int a^{2} c^{4} x\, dx + \int \frac{b^{2} \operatorname{asin}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{2 a b \operatorname{asin}{\left (c x \right )}}{x^{3}}\, dx + \int - \frac{2 b^{2} c^{2} \operatorname{asin}^{2}{\left (c x \right )}}{x}\, dx + \int b^{2} c^{4} x \operatorname{asin}^{2}{\left (c x \right )}\, dx + \int - \frac{4 a b c^{2} \operatorname{asin}{\left (c x \right )}}{x}\, dx + \int 2 a b c^{4} x \operatorname{asin}{\left (c x \right )}\, dx\right ) \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 (c^{2} d x^{2} - d\right )}^{2}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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