Optimal. Leaf size=26 \[ \text {Int}\left (\sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)},x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)} \, dx &=\int \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.50, size = 0, normalized size = 0.00 \[\int \sqrt {a^{2} c \,x^{2}+c}\, \sqrt {\arctan \left (a x \right )}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c \left (a^{2} x^{2} + 1\right )} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \sqrt {\mathrm {atan}\left (a\,x\right )}\,\sqrt {c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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