Optimal. Leaf size=77 \[ \frac {x \sqrt {\tan ^{-1}(a x)}}{2 c^2 \left (a^2 x^2+1\right )}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a c^2}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a c^2} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4892, 4970, 4406, 12, 3305, 3351} \[ \frac {x \sqrt {\tan ^{-1}(a x)}}{2 c^2 \left (a^2 x^2+1\right )}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a c^2}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4892
Rule 4970
Rubi steps
\begin {align*} \int \frac {\sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac {x \sqrt {\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac {1}{4} a \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=\frac {x \sqrt {\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2}\\ &=\frac {x \sqrt {\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2}\\ &=\frac {x \sqrt {\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a c^2}\\ &=\frac {x \sqrt {\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac {\operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a c^2}\\ &=\frac {x \sqrt {\tan ^{-1}(a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{3/2}}{3 a c^2}-\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a c^2}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 89, normalized size = 1.16 \[ \frac {16 \tan ^{-1}(a x)^2+3 \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 i \tan ^{-1}(a x)\right )+3 \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {3}{2},2 i \tan ^{-1}(a x)\right )}{48 a c^2 \sqrt {\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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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 = 0.57, size = 60, normalized size = 0.78 \[ \frac {8 \arctan \left (a x \right )^{2}+6 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-3 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{24 a \,c^{2} \sqrt {\arctan \left (a x \right )}} \]
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 \[ \int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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