Optimal. Leaf size=133 \[ -\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{2 a^4 c^2}-\frac {i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac {\tan ^{-1}(a x)}{4 a^4 c^2}-\frac {\log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^4 c^2}+\frac {\tan ^{-1}(a x)}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac {x}{4 a^3 c^2 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4964, 4920, 4854, 2402, 2315, 4930, 199, 205} \[ -\frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c^2}-\frac {x}{4 a^3 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac {i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac {\tan ^{-1}(a x)}{4 a^4 c^2}-\frac {\log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^4 c^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 2315
Rule 2402
Rule 4854
Rule 4920
Rule 4930
Rule 4964
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {\int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2}+\frac {\int \frac {x \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^2 c}\\ &=\frac {\tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^3}-\frac {\int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{a^3 c^2}\\ &=-\frac {x}{4 a^3 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}+\frac {\int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c^2}-\frac {\int \frac {1}{c+a^2 c x^2} \, dx}{4 a^3 c}\\ &=-\frac {x}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{4 a^4 c^2}+\frac {\tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {i \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^4 c^2}\\ &=-\frac {x}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{4 a^4 c^2}+\frac {\tan ^{-1}(a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {i \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 77, normalized size = 0.58 \[ \frac {4 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+4 i \tan ^{-1}(a x)^2-\sin \left (2 \tan ^{-1}(a x)\right )+2 \tan ^{-1}(a x) \left (\cos \left (2 \tan ^{-1}(a x)\right )-4 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{8 a^4 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \arctan \left (a x\right )}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 257, normalized size = 1.93 \[ \frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{4} c^{2}}+\frac {\arctan \left (a x \right )}{2 a^{4} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {x}{4 a^{3} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4 a^{4} c^{2}}+\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{4 a^{4} c^{2}}-\frac {i \ln \left (a x -i\right )^{2}}{8 a^{4} c^{2}}-\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{4 a^{4} c^{2}}-\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{4 a^{4} c^{2}}-\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{4 a^{4} c^{2}}+\frac {i \ln \left (a x +i\right )^{2}}{8 a^{4} c^{2}}+\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{4 a^{4} c^{2}}+\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{4 a^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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