Optimal. Leaf size=168 \[ \frac {3 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a^4 c^3}-\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{64 a^4 c^3}-\frac {3 \tan ^{-1}(a x)^{3/2}}{32 a^4 c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^4 c^3}-\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2} \]
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Rubi [A] time = 0.25, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4944, 4970, 3312, 3296, 3305, 3351} \[ \frac {3 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a^4 c^3}-\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{64 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 \tan ^{-1}(a x)^{3/2}}{32 a^4 c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^4 c^3}-\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^4 c^3} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3305
Rule 3312
Rule 3351
Rule 4944
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{8} (3 a) \int \frac {x^4 \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \operatorname {Subst}\left (\int \sqrt {x} \sin ^4(x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^4 c^3}\\ &=\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \operatorname {Subst}\left (\int \left (\frac {3 \sqrt {x}}{8}-\frac {1}{2} \sqrt {x} \cos (2 x)+\frac {1}{8} \sqrt {x} \cos (4 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^4 c^3}\\ &=-\frac {3 \tan ^{-1}(a x)^{3/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \operatorname {Subst}\left (\int \sqrt {x} \cos (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^4 c^3}+\frac {3 \operatorname {Subst}\left (\int \sqrt {x} \cos (2 x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^4 c^3}\\ &=-\frac {3 \tan ^{-1}(a x)^{3/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^4 c^3}-\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^4 c^3}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a^4 c^3}\\ &=-\frac {3 \tan ^{-1}(a x)^{3/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^4 c^3}-\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^4 c^3}+\frac {3 \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{256 a^4 c^3}-\frac {3 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{32 a^4 c^3}\\ &=-\frac {3 \tan ^{-1}(a x)^{3/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a^4 c^3}-\frac {3 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{64 a^4 c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{32 a^4 c^3}-\frac {3 \sqrt {\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{256 a^4 c^3}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 350, normalized size = 2.08 \[ -\frac {9 \left (-2 \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )-2 \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )-\sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )-\sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )\right )}{4096 a^4 c^3 \sqrt {\tan ^{-1}(a x)}}-\frac {15 \left (-2 \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )-2 \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )+\sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )+\sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )\right )}{4096 a^4 c^3 \sqrt {\tan ^{-1}(a x)}}+\sqrt {\tan ^{-1}(a x)} \left (\frac {\left (5 a^4 x^4-6 a^2 x^2-3\right ) \tan ^{-1}(a x)}{32 a^4 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x \left (5 a^2 x^2+3\right )}{64 a^3 c^3 \left (a^2 x^2+1\right )^2}\right ) \]
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 = 0.56, size = 124, normalized size = 0.74 \[ -\frac {-3 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arctan \left (a x \right )}\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+128 \arctan \left (a x \right )^{2} \cos \left (2 \arctan \left (a x \right )\right )-32 \arctan \left (a x \right )^{2} \cos \left (4 \arctan \left (a x \right )\right )+48 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-96 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+12 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{1024 a^{4} c^{3} \sqrt {\arctan \left (a x \right )}} \]
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 \[ \frac {\int \frac {x^{3} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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