Optimal. Leaf size=36 \[ -\frac {2 a \text {ArcTan}(a x)^{3/2}}{3 c}+\frac {\text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{x^2},x\right )}{c} \]
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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 {\sqrt {\text {ArcTan}(a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\sqrt {\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx\right )+\frac {\int \frac {\sqrt {\tan ^{-1}(a x)}}{x^2} \, dx}{c}\\ &=-\frac {2 a \tan ^{-1}(a x)^{3/2}}{3 c}+\frac {\int \frac {\sqrt {\tan ^{-1}(a x)}}{x^2} \, dx}{c}\\ \end {align*}
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Mathematica [A]
time = 1.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\text {ArcTan}(a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.38, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arctan \left (a x \right )}}{x^{2} \left (a^{2} c \,x^{2}+c \right )}\, dx\]
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*} \frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{2} x^{4} + x^{2}}\, dx}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^2\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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