Optimal. Leaf size=201 \[ i b c^2 d^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{4} b c^3 d^2 x \sqrt{1-c^2 x^2}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \sin ^{-1}(c x) \]
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Rubi [A] time = 0.207758, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4685, 277, 195, 216, 4683, 4625, 3717, 2190, 2279, 2391} \[ i b c^2 d^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{4} b c^3 d^2 x \sqrt{1-c^2 x^2}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 4685
Rule 277
Rule 195
Rule 216
Rule 4683
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{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 )}{x} \, dx+\frac{1}{2} \left (b c d^2\right ) \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (2 c^2 d^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x} \, dx+\left (b c^3 d^2\right ) \int \sqrt{1-c^2 x^2} \, dx-\frac{1}{2} \left (3 b c^3 d^2\right ) \int \sqrt{1-c^2 x^2} \, dx\\ &=-\frac{1}{4} b c^3 d^2 x \sqrt{1-c^2 x^2}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\left (2 c^2 d^2\right ) \operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac{1}{2} \left (b c^3 d^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{4} \left (3 b c^3 d^2\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{1}{4} b c^3 d^2 x \sqrt{1-c^2 x^2}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \sin ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}+\left (4 i c^2 d^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 )\\ &=-\frac{1}{4} b c^3 d^2 x \sqrt{1-c^2 x^2}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \sin ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\left (2 b c^2 d^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac{1}{4} b c^3 d^2 x \sqrt{1-c^2 x^2}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \sin ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\left (i b c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac{1}{4} b c^3 d^2 x \sqrt{1-c^2 x^2}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \sin ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac{d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac{i c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+i b c^2 d^2 \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.166968, size = 162, normalized size = 0.81 \[ \frac{d^2 \left (4 i b c^2 x^2 \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+2 a c^4 x^4-8 a c^2 x^2 \log (x)-2 a+b c^3 x^3 \sqrt{1-c^2 x^2}-2 b c x \sqrt{1-c^2 x^2}+4 i b c^2 x^2 \sin ^{-1}(c x)^2+b \sin ^{-1}(c x) \left (2 c^4 x^4-c^2 x^2-8 c^2 x^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-2\right )\right )}{4 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.375, size = 278, normalized size = 1.4 \begin{align*}{\frac{{c}^{4}{d}^{2}a{x}^{2}}{2}}-{\frac{{d}^{2}a}{2\,{x}^{2}}}-2\,{c}^{2}{d}^{2}a\ln \left ( cx \right ) +i{c}^{2}{d}^{2}b \left ( \arcsin \left ( cx \right ) \right ) ^{2}+{\frac{b{c}^{3}{d}^{2}x}{4}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{{c}^{4}{d}^{2}b\arcsin \left ( cx \right ){x}^{2}}{2}}-{\frac{b{c}^{2}{d}^{2}\arcsin \left ( cx \right ) }{4}}+{\frac{i}{2}}{c}^{2}{d}^{2}b-{\frac{{d}^{2}bc}{2\,x}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b{d}^{2}\arcsin \left ( cx \right ) }{2\,{x}^{2}}}-2\,{c}^{2}{d}^{2}b\arcsin \left ( cx \right ) \ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) -2\,{c}^{2}{d}^{2}b\arcsin \left ( cx \right ) \ln \left ( 1-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,i{c}^{2}{d}^{2}b{\it polylog} \left ( 2,-icx-\sqrt{-{c}^{2}{x}^{2}+1} \right ) +2\,i{c}^{2}{d}^{2}b{\it polylog} \left ( 2,icx+\sqrt{-{c}^{2}{x}^{2}+1} \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{1}{2} \, a c^{4} d^{2} x^{2} - 2 \, a c^{2} d^{2} \log \left (x\right ) - \frac{1}{2} \, b d^{2}{\left (\frac{\sqrt{-c^{2} x^{2} + 1} c}{x} + \frac{\arcsin \left (c x\right )}{x^{2}}\right )} - \frac{a d^{2}}{2 \, x^{2}} + \int \frac{{\left (b c^{4} d^{2} x^{2} - 2 \, b c^{2} d^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{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{a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + 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}{x^{3}}\, dx + \int - \frac{2 a c^{2}}{x}\, dx + \int a c^{4} x\, dx + \int \frac{b \operatorname{asin}{\left (c x \right )}}{x^{3}}\, dx + \int - \frac{2 b c^{2} \operatorname{asin}{\left (c x \right )}}{x}\, dx + \int 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 )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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