Optimal. Leaf size=62 \[ \frac {x}{4 a c^2 \left (a^2 x^2+1\right )}-\frac {\tan ^{-1}(a x)}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)}{4 a^2 c^2} \]
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Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4930, 199, 205} \[ \frac {x}{4 a c^2 \left (a^2 x^2+1\right )}-\frac {\tan ^{-1}(a x)}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)}{4 a^2 c^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {\tan ^{-1}(a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a}\\ &=\frac {x}{4 a c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\int \frac {1}{c+a^2 c x^2} \, dx}{4 a c}\\ &=\frac {x}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)}{2 a^2 c^2 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 0.63 \[ \frac {\left (a^2 x^2-1\right ) \tan ^{-1}(a x)+a x}{4 a^2 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 40, normalized size = 0.65 \[ \frac {a x + {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{4 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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 = 0.03, size = 57, normalized size = 0.92 \[ \frac {x}{4 a \,c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{4 a^{2} c^{2}}-\frac {\arctan \left (a x \right )}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 59, normalized size = 0.95 \[ \frac {\frac {x}{a^{2} c x^{2} + c} + \frac {\arctan \left (a x\right )}{a c}}{4 \, a c} - \frac {\arctan \left (a x\right )}{2 \, {\left (a^{2} c x^{2} + c\right )} a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 40, normalized size = 0.65 \[ \frac {a\,x-\mathrm {atan}\left (a\,x\right )+a^2\,x^2\,\mathrm {atan}\left (a\,x\right )}{4\,a^2\,c^2\,\left (a^2\,x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.42, size = 107, normalized size = 1.73 \[ \begin {cases} \frac {a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} + \frac {a x}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} - \frac {\operatorname {atan}{\left (a x \right )}}{4 a^{4} c^{2} x^{2} + 4 a^{2} c^{2}} & \text {for}\: c \neq 0 \\\tilde {\infty } \left (\frac {x^{2} \operatorname {atan}{\left (a x \right )}}{2} - \frac {x}{2 a} + \frac {\operatorname {atan}{\left (a x \right )}}{2 a^{2}}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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