Optimal. Leaf size=18 \[ \frac {2 \tan ^{-1}(a x)^{3/2}}{3 a c} \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4884} \[ \frac {2 \tan ^{-1}(a x)^{3/2}}{3 a c} \]
Antiderivative was successfully verified.
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Rule 4884
Rubi steps
\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=\frac {2 \tan ^{-1}(a x)^{3/2}}{3 a c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ \frac {2 \tan ^{-1}(a x)^{3/2}}{3 a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 14, normalized size = 0.78 \[ \frac {2 \, \arctan \left (a x\right )^{\frac {3}{2}}}{3 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 14, normalized size = 0.78 \[ \frac {2 \, \arctan \left (a x\right )^{\frac {3}{2}}}{3 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 15, normalized size = 0.83 \[ \frac {2 \arctan \left (a x \right )^{\frac {3}{2}}}{3 a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 14, normalized size = 0.78 \[ \frac {2\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{3\,a\,c} \]
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 \[ \frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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