Optimal. Leaf size=109 \[ -\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2 c^2}-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {3 x \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2} \]
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Rubi [A] time = 0.15, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4930, 4892, 4970, 4406, 12, 3305, 3351} \[ -\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2 c^2}-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {3 x \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (a^2 x^2+1\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4892
Rule 4930
Rule 4970
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {3 \int \frac {\sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a}\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3}{16} \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a^2 c^2}\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a^2 c^2}\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{32 a^2 c^2}\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{16 a^2 c^2}\\ &=\frac {3 x \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}-\frac {\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left (1+a^2 x^2\right )}-\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 75, normalized size = 0.69 \[ \frac {\frac {4 \sqrt {\tan ^{-1}(a x)} \left (2 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)+3 a x\right )}{a^2 x^2+1}-3 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a^2 c^2} \]
Antiderivative was successfully verified.
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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 [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 [A] time = 0.31, size = 67, normalized size = 0.61 \[ -\frac {8 \arctan \left (a x \right )^{2} \cos \left (2 \arctan \left (a x \right )\right )+3 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-6 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{32 a^{2} c^{2} \sqrt {\arctan \left (a x \right )}} \]
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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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