Optimal. Leaf size=33 \[ \frac {\log \left (\tan ^{-1}(a x)\right )}{2 a^3 c^2}-\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^3 c^2} \]
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Rubi [A] time = 0.11, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4970, 3312, 3302} \[ \frac {\log \left (\tan ^{-1}(a x)\right )}{2 a^3 c^2}-\frac {\text {CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a^3 c^2} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 3312
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=\frac {\log \left (\tan ^{-1}(a x)\right )}{2 a^3 c^2}-\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^2}\\ &=-\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^3 c^2}+\frac {\log \left (\tan ^{-1}(a x)\right )}{2 a^3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 25, normalized size = 0.76 \[ \frac {\log \left (\tan ^{-1}(a x)\right )-\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^3 c^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.44, size = 74, normalized size = 2.24 \[ \frac {2 \, \log \left (\arctan \left (a x\right )\right ) - \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{4 \, a^{3} c^{2}} \]
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 [A] time = 0.24, size = 30, normalized size = 0.91 \[ -\frac {\Ci \left (2 \arctan \left (a x \right )\right )}{2 a^{3} c^{2}}+\frac {\ln \left (\arctan \left (a x \right )\right )}{2 a^{3} c^{2}} \]
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 \[ \int \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {x^{2}}{a^{4} x^{4} \operatorname {atan}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}{\left (a x \right )} + \operatorname {atan}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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