Optimal. Leaf size=16 \[ \frac {\tan ^{-1}(a x)^3}{3 a c} \]
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Rubi [A] time = 0.02, antiderivative size = 16, 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 = {4884} \[ \frac {\tan ^{-1}(a x)^3}{3 a c} \]
Antiderivative was successfully verified.
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Rule 4884
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx &=\frac {\tan ^{-1}(a x)^3}{3 a c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \frac {\tan ^{-1}(a x)^3}{3 a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 14, normalized size = 0.88 \[ \frac {\arctan \left (a x\right )^{3}}{3 \, a c} \]
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 = 0.03, size = 15, normalized size = 0.94 \[ \frac {\arctan \left (a x \right )^{3}}{3 a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 14, normalized size = 0.88 \[ \frac {\arctan \left (a x\right )^{3}}{3 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 14, normalized size = 0.88 \[ \frac {{\mathrm {atan}\left (a\,x\right )}^3}{3\,a\,c} \]
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 \[ \frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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