Optimal. Leaf size=16 \[ \frac {2 \sqrt {\tan ^{-1}(a x)}}{a c} \]
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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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4884} \[ \frac {2 \sqrt {\tan ^{-1}(a x)}}{a c} \]
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 ) \sqrt {\tan ^{-1}(a x)}} \, dx &=\frac {2 \sqrt {\tan ^{-1}(a x)}}{a c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \frac {2 \sqrt {\tan ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 14, normalized size = 0.88 \[ \frac {2 \, \sqrt {\arctan \left (a x\right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 14, normalized size = 0.88 \[ \frac {2 \, \sqrt {\arctan \left (a x\right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 15, normalized size = 0.94 \[ \frac {2 \sqrt {\arctan \left (a x \right )}}{a c} \]
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 = 0.34, size = 14, normalized size = 0.88 \[ \frac {2\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{a\,c} \]
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 \[ \begin {cases} \frac {2 \sqrt {\operatorname {atan}{\left (a x \right )}}}{a c} & \text {for}\: c \neq 0 \\\tilde {\infty } \int \frac {1}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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