Optimal. Leaf size=122 \[ -\frac{i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{i b \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac{2 \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}+\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}} \]
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Rubi [A] time = 0.170644, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {4706, 4680, 4419, 4183, 2279, 2391, 191} \[ -\frac{i b \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{i b \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac{2 \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}+\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 4706
Rule 4680
Rule 4419
Rule 4183
Rule 2279
Rule 2391
Rule 191
Rubi steps
\begin{align*} \int \frac{a+b \cos ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx &=\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac{(b c) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac{\int \frac{a+b \cos ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d}\\ &=\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac{\operatorname{Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\cos ^{-1}(c x)\right )}{d^2}\\ &=\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac{2 \operatorname{Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\cos ^{-1}(c x)\right )}{d^2}\\ &=\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac{2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{d^2}+\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{d^2}-\frac{b \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{d^2}\\ &=\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac{2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{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 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 d^2}\\ &=\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac{2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{d^2}-\frac{i b \text{Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}+\frac{i b \text{Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 d^2}\\ \end{align*}
Mathematica [A] time = 0.485909, size = 152, normalized size = 1.25 \[ \frac{b \left (-i \text{PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )+i \text{PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )+\frac{c x}{\sqrt{1-c^2 x^2}}+\frac{\cos ^{-1}(c x)}{1-c^2 x^2}-2 \cos ^{-1}(c x) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )+2 \cos ^{-1}(c x) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )\right )+\frac{a}{1-c^2 x^2}-a \log \left (1-c^2 x^2\right )+2 a \log (x)}{2 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.188, size = 340, normalized size = 2.8 \begin{align*} -{\frac{a}{4\,{d}^{2} \left ( cx-1 \right ) }}-{\frac{a\ln \left ( cx-1 \right ) }{2\,{d}^{2}}}+{\frac{a}{4\,{d}^{2} \left ( cx+1 \right ) }}-{\frac{a\ln \left ( cx+1 \right ) }{2\,{d}^{2}}}+{\frac{a\ln \left ( cx \right ) }{{d}^{2}}}-{\frac{{\frac{i}{2}}b{c}^{2}{x}^{2}}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{xbc}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arccos \left ( cx \right ) }{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{\frac{i}{2}}b}{{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b\arccos \left ( cx \right ) }{{d}^{2}}\ln \left ( 1+cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{ib}{{d}^{2}}{\it polylog} \left ( 2,-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{b\arccos \left ( cx \right ) }{{d}^{2}}\ln \left ( 1-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{ib}{{d}^{2}}{\it polylog} \left ( 2,cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{b\arccos \left ( cx \right ) }{{d}^{2}}\ln \left ( 1+ \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \right ) }-{\frac{{\frac{i}{2}}b}{{d}^{2}}{\it polylog} \left ( 2,- \left ( cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) ^{2} \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^{2} d^{2} x^{2} - d^{2}} + \frac{\log \left (c x + 1\right )}{d^{2}} + \frac{\log \left (c x - 1\right )}{d^{2}} - \frac{2 \, \log \left (x\right )}{d^{2}}\right )} + b \int \frac{\arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} 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 \arccos \left (c x\right ) + a}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} 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{b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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