Optimal. Leaf size=101 \[ -\frac {8 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^2 c^2}-\frac {2 x}{3 a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4932, 4970, 4406, 12, 3305, 3351} \[ -\frac {8 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^2 c^2}-\frac {2 x}{3 a c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4932
Rule 4970
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {16}{3} \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c^2}\\ &=-\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c^2}\\ &=-\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^2 c^2}\\ &=-\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 a^2 c^2}\\ &=-\frac {2 x}{3 a c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \left (1-a^2 x^2\right )}{3 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {8 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 88, normalized size = 0.87 \[ -\frac {2 \left (4 \sqrt {\pi } \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2} S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )+\left (2-2 a^2 x^2\right ) \tan ^{-1}(a x)+a x\right )}{3 a^2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}} \]
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(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 59, normalized size = 0.58 \[ -\frac {8 \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}+4 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sin \left (2 \arctan \left (a x \right )\right )}{3 a^{2} c^{2} \arctan \left (a x \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {x}{a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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