Optimal. Leaf size=52 \[ -\frac {a \log \left (a^2 x^2+1\right )}{2 c}+\frac {a \log (x)}{c}-\frac {a \tan ^{-1}(a x)^2}{2 c}-\frac {\tan ^{-1}(a x)}{c x} \]
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Rubi [A] time = 0.09, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4918, 4852, 266, 36, 29, 31, 4884} \[ -\frac {a \log \left (a^2 x^2+1\right )}{2 c}+\frac {a \log (x)}{c}-\frac {a \tan ^{-1}(a x)^2}{2 c}-\frac {\tan ^{-1}(a x)}{c x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4852
Rule 4884
Rule 4918
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)}{c x}-\frac {a \tan ^{-1}(a x)^2}{2 c}+\frac {a \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)}{c x}-\frac {a \tan ^{-1}(a x)^2}{2 c}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {\tan ^{-1}(a x)}{c x}-\frac {a \tan ^{-1}(a x)^2}{2 c}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {\tan ^{-1}(a x)}{c x}-\frac {a \tan ^{-1}(a x)^2}{2 c}+\frac {a \log (x)}{c}-\frac {a \log \left (1+a^2 x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 1.00 \[ -\frac {a \log \left (a^2 x^2+1\right )}{2 c}+\frac {a \log (x)}{c}-\frac {a \tan ^{-1}(a x)^2}{2 c}-\frac {\tan ^{-1}(a x)}{c x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 43, normalized size = 0.83 \[ -\frac {a x \arctan \left (a x\right )^{2} + a x \log \left (a^{2} x^{2} + 1\right ) - 2 \, a x \log \relax (x) + 2 \, \arctan \left (a x\right )}{2 \, c x} \]
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 = 51, normalized size = 0.98 \[ -\frac {\arctan \left (a x \right )}{c x}-\frac {a \arctan \left (a x \right )^{2}}{2 c}+\frac {a \ln \left (a x \right )}{c}-\frac {a \ln \left (a^{2} x^{2}+1\right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 53, normalized size = 1.02 \[ -{\left (\frac {a \arctan \left (a x\right )}{c} + \frac {1}{c x}\right )} \arctan \left (a x\right ) + \frac {{\left (\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right ) + 2 \, \log \relax (x)\right )} a}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 48, normalized size = 0.92 \[ \frac {a\,\ln \relax (x)}{c}-\frac {a\,\ln \left (a^2\,x^2+1\right )}{2\,c}-\frac {a\,{\mathrm {atan}\left (a\,x\right )}^2}{2\,c}-\frac {\mathrm {atan}\left (a\,x\right )}{c\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.47, size = 68, normalized size = 1.31 \[ \begin {cases} \frac {a \log {\relax (x )}}{c} - \frac {a \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 c} - \frac {a \operatorname {atan}^{2}{\left (a x \right )}}{2 c} - \frac {\operatorname {atan}{\left (a x \right )}}{c x} & \text {for}\: c \neq 0 \\\tilde {\infty } \left (a \log {\relax (x )} - \frac {a \log {\left (a^{2} x^{2} + 1 \right )}}{2} - \frac {\operatorname {atan}{\left (a x \right )}}{x}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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