Optimal. Leaf size=79 \[ \frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^2 c^2}-\frac {\sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4930, 4904, 3312, 3304, 3352} \[ \frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^2 c^2}-\frac {\sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (a^2 x^2+1\right )}+\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^2} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4904
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {\sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{4 a}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^2}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^2}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^2}-\frac {\sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^2}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^2}-\frac {\sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c^2}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^2}-\frac {\sqrt {\tan ^{-1}(a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a^2 c^2}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 136, normalized size = 1.72 \[ \frac {4 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )+\frac {\frac {16 \left (a^2 x^2-1\right ) \tan ^{-1}(a x)}{a^2 x^2+1}-i \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )+i \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )}{\sqrt {\tan ^{-1}(a x)}}}{64 a^2 c^2} \]
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 = 0.39, size = 46, normalized size = 0.58 \[ -\frac {\sqrt {\arctan \left (a x \right )}\, \cos \left (2 \arctan \left (a x \right )\right )}{4 a^{2} c^{2}}+\frac {\FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }}{8 a^{2} c^{2}} \]
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 \[ \int \frac {x\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {x \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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