Optimal. Leaf size=58 \[ \frac {\sqrt {\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{8 a^3 c^3} \]
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Rubi [A] time = 0.12, antiderivative size = 58, 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, 4406, 3304, 3352} \[ \frac {\sqrt {\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{8 a^3 c^3} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4406
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{8 \sqrt {x}}-\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^3}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac {\operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a^3 c^3}\\ &=\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{8 a^3 c^3}\\ \end {align*}
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Mathematica [C] time = 0.48, size = 229, normalized size = 3.95 \[ \frac {-2 \sqrt {2 \pi } \sqrt {\tan ^{-1}(a x)^2} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )+16 \sqrt {\pi } \sqrt {\tan ^{-1}(a x)^2} C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )+\sqrt {\tan ^{-1}(a x)} \left (64 \sqrt {\tan ^{-1}(a x)^2}+4 \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )+4 \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )+7 \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )+7 \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )\right )}{256 a^3 c^3 \sqrt {\tan ^{-1}(a x)^2}} \]
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(-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 = 0.50, size = 45, normalized size = 0.78 \[ -\frac {\FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }}{16 a^{3} c^{3}}+\frac {\sqrt {\arctan \left (a x \right )}}{4 a^{3} c^{3}} \]
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 \[ \int \frac {x^2}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {x^{2}}{a^{6} x^{6} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}} + \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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