Optimal. Leaf size=103 \[ -\frac {1}{2} a^2 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {a^2 c x^2+c}},x\right )+\frac {1}{4} a \text {Int}\left (\frac {1}{x^2 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )-\frac {\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{2 c x^2} \]
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Rubi [A] time = 0.33, 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 = {} \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^3 \sqrt {c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^3 \sqrt {c+a^2 c x^2}} \, dx &=-\frac {\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{2 c x^2}+\frac {1}{4} a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx-\frac {1}{2} a^2 \int \frac {\sqrt {\tan ^{-1}(a x)}}{x \sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 3.74, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\tan ^{-1}(a x)}}{x^3 \sqrt {c+a^2 c x^2}} \, dx \]
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 = 5.08, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\arctan \left (a x \right )}}{x^{3} \sqrt {a^{2} c \,x^{2}+c}}\, dx \]
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 \[ \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{x^3\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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