Optimal. Leaf size=155 \[ -\frac{i b \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}+\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac{\log \left (1-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^4 d^2}+\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2} \]
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Rubi [A] time = 0.18674, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4704, 4676, 3717, 2190, 2279, 2391, 288, 216} \[ -\frac{i b \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}+\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac{\log \left (1-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^4 d^2}+\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2} \]
Antiderivative was successfully verified.
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Rule 4704
Rule 4676
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rule 288
Rule 216
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b \int \frac{x^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}-\frac{\int \frac{x \left (a+b \cos ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{c^2 d}\\ &=\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}+\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{\operatorname{Subst}\left (\int (a+b x) \cot (x) \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d^2}-\frac{b \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{2 c^3 d^2}\\ &=\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}+\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2}-\frac{(2 i) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d^2}\\ &=\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}+\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2}+\frac{\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d^2}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d^2}\\ &=\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}+\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2}+\frac{\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d^2}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}\\ &=\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}+\frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2}+\frac{\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d^2}-\frac{i b \text{Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}\\ \end{align*}
Mathematica [A] time = 0.455747, size = 203, normalized size = 1.31 \[ \frac{-4 i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )-4 i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )-\frac{2 a}{c^2 x^2-1}+2 a \log \left (1-c^2 x^2\right )+\frac{b \sqrt{1-c^2 x^2}}{1-c x}-\frac{b \sqrt{1-c^2 x^2}}{c x+1}-2 i b \cos ^{-1}(c x)^2+\frac{b \cos ^{-1}(c x)}{1-c x}+\frac{b \cos ^{-1}(c x)}{c x+1}+4 b \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )+4 b \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{4 c^4 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.25, size = 312, normalized size = 2. \begin{align*} -{\frac{a}{4\,{d}^{2}{c}^{4} \left ( cx-1 \right ) }}+{\frac{a\ln \left ( cx-1 \right ) }{2\,{d}^{2}{c}^{4}}}+{\frac{a}{4\,{d}^{2}{c}^{4} \left ( cx+1 \right ) }}+{\frac{a\ln \left ( cx+1 \right ) }{2\,{d}^{2}{c}^{4}}}-{\frac{{\frac{i}{2}}b \left ( \arccos \left ( cx \right ) \right ) ^{2}}{{d}^{2}{c}^{4}}}-{\frac{{\frac{i}{2}}b{x}^{2}}{{c}^{2}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bx}{2\,{c}^{3}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arccos \left ( cx \right ) }{2\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{\frac{i}{2}}b}{{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b\arccos \left ( cx \right ) }{{d}^{2}{c}^{4}}\ln \left ( 1+cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{b\arccos \left ( cx \right ) }{{d}^{2}{c}^{4}}\ln \left ( 1-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{ib}{{d}^{2}{c}^{4}}{\it polylog} \left ( 2,-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{ib}{{d}^{2}{c}^{4}}{\it polylog} \left ( 2,cx+i\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{\left (\frac{1}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4} d^{2}}\right )} + \frac{{\left ({\left ({\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) +{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right ) - 1\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) -{\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )} \int \frac{{\left (c^{2} x^{2} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) +{\left (c^{2} x^{2} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right ) - e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}}{c^{9} d^{2} x^{6} - 2 \, c^{7} d^{2} x^{4} + c^{5} d^{2} x^{2} -{\left (c^{7} d^{2} x^{4} - 2 \, c^{5} d^{2} x^{2} + c^{3} d^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )}}\,{d x}\right )} b}{2 \,{\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}} \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 x^{3} \arccos \left (c x\right ) + a x^{3}}{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*} \frac{\int \frac{a x^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b x^{3} \operatorname{acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \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 \arccos \left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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