Optimal. Leaf size=14 \[ -\frac{1}{a c \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.0246541, antiderivative size = 14, normalized size of antiderivative = 1., 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 = {4884} \[ -\frac{1}{a c \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4884
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx &=-\frac{1}{a c \tan ^{-1}(a x)}\\ \end{align*}
Mathematica [A] time = 0.0034184, size = 14, normalized size = 1. \[ -\frac{1}{a c \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 15, normalized size = 1.1 \begin{align*} -{\frac{1}{ac\arctan \left ( ax \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00949, size = 19, normalized size = 1.36 \begin{align*} -\frac{1}{a c \arctan \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60263, size = 30, normalized size = 2.14 \begin{align*} -\frac{1}{a c \arctan \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15975, size = 19, normalized size = 1.36 \begin{align*} -\frac{1}{a c \arctan \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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