Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x \left (a^2 c x^2+c\right )^3},x\right ) \]
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Rubi [A] time = 0.06, 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 = {} \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^3} \, dx &=\int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^3} \, dx\\ \end {align*}
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Mathematica [A] time = 2.46, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.49, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arctan \left (a x \right )}}{x \left (a^{2} c \,x^{2}+c \right )^{3}}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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