Optimal. Leaf size=49 \[ \frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\tan ^{-1}(a x)}{a^2 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4930, 191} \[ \frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\tan ^{-1}(a x)}{a^2 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\tan ^{-1}(a x)}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a}\\ &=\frac {x}{a c \sqrt {c+a^2 c x^2}}-\frac {\tan ^{-1}(a x)}{a^2 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 42, normalized size = 0.86 \[ \frac {\sqrt {a^2 c x^2+c} \left (a x-\tan ^{-1}(a x)\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 43, normalized size = 0.88 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x - \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] 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 [C] time = 0.86, size = 100, normalized size = 2.04 \[ -\frac {\left (i+\arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2} a^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 28, normalized size = 0.57 \[ \frac {a x - \arctan \left (a x\right )}{\sqrt {a^{2} x^{2} + 1} a^{2} c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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