Optimal. Leaf size=96 \[ -\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^4 c^3}+\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a^4 c^3}-\frac {2 x^3}{a c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}} \]
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Rubi [A] time = 0.33, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4968, 4970, 3312, 3304, 3352, 4406} \[ -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^4 c^3}+\frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a^4 c^3}-\frac {2 x^3}{a c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4406
Rule 4968
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {6 \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-(2 a) \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2 \operatorname {Subst}\left (\int \frac {\sin ^4(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac {6 \operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=-\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2 \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 )}{a^4 c^3}+\frac {6 \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^4 c^3}\\ &=-\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=-\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {\operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2 a^4 c^3}-\frac {3 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2 a^4 c^3}+\frac {2 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a^4 c^3}\\ &=-\frac {2 x^3}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a^4 c^3}+\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a^4 c^3}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 148, normalized size = 1.54 \[ \frac {\frac {-\frac {32 a^3 x^3}{\left (a^2 x^2+1\right )^2}+3 i \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )-3 i \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )}{\sqrt {\tan ^{-1}(a x)}}-2 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )+16 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{16 a^4 c^3} \]
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(-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 = 86, normalized size = 0.90 \[ -\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-4 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+2 \sin \left (2 \arctan \left (a x \right )\right )-\sin \left (4 \arctan \left (a x \right )\right )}{4 a^{4} c^{3} \sqrt {\arctan \left (a x \right )}} \]
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 \[ \int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {x^{3}}{a^{6} x^{6} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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