Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^m \sqrt {c+a^2 c x^2}}{\text {ArcTan}(a x)},x\right ) \]
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Rubi [A]
time = 0.07, 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 \frac {x^m \sqrt {c+a^2 c x^2}}{\text {ArcTan}(a x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^m \sqrt {c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx &=\int \frac {x^m \sqrt {c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \sqrt {c+a^2 c x^2}}{\text {ArcTan}(a x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \sqrt {a^{2} c \,x^{2}+c}}{\arctan \left (a x \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {x^{m} \sqrt {c \left (a^{2} x^{2} + 1\right )}}{\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 \frac {x^m\,\sqrt {c\,a^2\,x^2+c}}{\mathrm {atan}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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