Optimal. Leaf size=61 \[ \frac {1}{4 a c^2 \left (a^2 x^2+1\right )}+\frac {x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^2}{4 a c^2} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4892, 261} \[ \frac {1}{4 a c^2 \left (a^2 x^2+1\right )}+\frac {x \tan ^{-1}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^2}{4 a c^2} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4892
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac {x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^2}{4 a c^2}-\frac {1}{2} a \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=\frac {1}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^2}{4 a c^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.72 \[ \frac {\left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+1}{4 c^2 \left (a^3 x^2+a\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 46, normalized size = 0.75 \[ \frac {2 \, a x \arctan \left (a x\right ) + {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 1}{4 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \]
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 [A] time = 0.04, size = 56, normalized size = 0.92 \[ \frac {1}{4 a \,c^{2} \left (a^{2} x^{2}+1\right )}+\frac {x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{2}}{4 a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 78, normalized size = 1.28 \[ \frac {1}{2} \, {\left (\frac {x}{a^{2} c^{2} x^{2} + c^{2}} + \frac {\arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right ) - \frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a}{4 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 48, normalized size = 0.79 \[ \frac {a^2\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+2\,a\,x\,\mathrm {atan}\left (a\,x\right )+{\mathrm {atan}\left (a\,x\right )}^2+1}{4\,a\,c^2\,\left (a^2\,x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
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