Optimal. Leaf size=47 \[ \frac {\sqrt {\tan ^{-1}(a x)}}{a^3 c^2}-\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2 a^3 c^2} \]
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Rubi [A] time = 0.11, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4970, 3312, 3304, 3352} \[ \frac {\sqrt {\tan ^{-1}(a x)}}{a^3 c^2}-\frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2 a^3 c^2} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2}\\ &=\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 )}{a^3 c^2}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{a^3 c^2}-\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^2}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{a^3 c^2}-\frac {\operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a^3 c^2}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{a^3 c^2}-\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2 a^3 c^2}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 122, normalized size = 2.60 \[ \frac {-4 \sqrt {\pi } \sqrt {\tan ^{-1}(a x)} C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )+16 \tan ^{-1}(a x)+i \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )-i \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )}{16 a^3 c^2 \sqrt {\tan ^{-1}(a x)}} \]
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(-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 = 0.43, size = 38, normalized size = 0.81 \[ -\frac {\FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }}{2 a^{3} c^{2}}+\frac {\sqrt {\arctan \left (a x \right )}}{a^{3} c^{2}} \]
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 = -0.02 \[ \int \frac {x^2}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {x^{2}}{a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}} + 2 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}} + \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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