Optimal. Leaf size=118 \[ \frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{64 a^2 c^3}+\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{16 a^2 c^3}-\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^3} \]
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Rubi [A] time = 0.15, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{64 a^2 c^3}+\frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{16 a^2 c^3}-\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^3} \]
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 )^3} \, dx &=-\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{8 a}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}+\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^3}-\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a^2 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a^2 c^3}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^3}-\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {\operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{32 a^2 c^3}+\frac {\operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{8 a^2 c^3}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a^2 c^3}-\frac {\sqrt {\tan ^{-1}(a x)}}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{64 a^2 c^3}+\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{16 a^2 c^3}\\ \end {align*}
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Mathematica [C] time = 0.69, size = 230, normalized size = 1.95 \[ \frac {\frac {\frac {64 \left (3 a^4 x^4+6 a^2 x^2-5\right ) \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2}-20 i \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )+20 i \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )-11 i \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )+11 i \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )}{\sqrt {\tan ^{-1}(a x)}}-6 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )+48 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2048 a^2 c^3} \]
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 = 0.52, size = 94, normalized size = 0.80 \[ -\frac {-\sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-8 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+16 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{128 a^{2} c^{3} \sqrt {\arctan \left (a x \right )}} \]
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 \[ \int \frac {x\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {x \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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