Optimal. Leaf size=12 \[ \frac {\log \left (\tan ^{-1}(a x)\right )}{a c} \]
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Rubi [A] time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {4882} \[ \frac {\log \left (\tan ^{-1}(a x)\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 4882
Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)} \, dx &=\frac {\log \left (\tan ^{-1}(a x)\right )}{a c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 1.00 \[ \frac {\log \left (\tan ^{-1}(a x)\right )}{a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 12, normalized size = 1.00 \[ \frac {\log \left (\arctan \left (a x\right )\right )}{a c} \]
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 = 0.08, size = 13, normalized size = 1.08 \[ \frac {\ln \left (\arctan \left (a x \right )\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 15, normalized size = 1.25 \[ \frac {\log \left (2 \, {\left | \arctan \left (a x\right ) \right |}\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 12, normalized size = 1.00 \[ \frac {\ln \left (\mathrm {atan}\left (a\,x\right )\right )}{a\,c} \]
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 \[ \begin {cases} \frac {\log {\left (\operatorname {atan}{\left (a x \right )} \right )}}{a c} & \text {for}\: c \neq 0 \\\tilde {\infty } \int \frac {1}{\operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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