Optimal. Leaf size=74 \[ -\frac {i a^2 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x (a x+i)},x\right )}{c}+\frac {\text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x^3},x\right )}{c}+\frac {2 i a^2 \tan ^{-1}(a x)^{3/2}}{3 c} \]
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Rubi [A] time = 0.19, 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 = {} \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\sqrt {\tan ^{-1}(a x)}}{x^3} \, dx}{c}\\ &=\frac {2 i a^2 \tan ^{-1}(a x)^{3/2}}{3 c}+\frac {\int \frac {\sqrt {\tan ^{-1}(a x)}}{x^3} \, dx}{c}-\frac {\left (i a^2\right ) \int \frac {\sqrt {\tan ^{-1}(a x)}}{x (i+a x)} \, dx}{c}\\ \end {align*}
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Mathematica [A] time = 2.31, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]
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 = 5.98, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arctan \left (a x \right )}}{x^{3} \left (a^{2} c \,x^{2}+c \right )}\, dx \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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 \[ \frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{2} x^{5} + x^{3}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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