Optimal. Leaf size=94 \[ -\frac {2}{a c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a c^3}-\frac {2 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a c^3} \]
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Rubi [A] time = 0.14, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4902, 4970, 4406, 3305, 3351} \[ -\frac {2}{a c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a c^3}-\frac {2 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a c^3} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4406
Rule 4902
Rule 4970
Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-(8 a) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=-\frac {2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {8 \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=-\frac {2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}-\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=-\frac {2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2 \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a c^3}-\frac {4 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a c^3}\\ &=-\frac {2}{a c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a c^3}-\frac {2 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a c^3}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 144, normalized size = 1.53 \[ \frac {-\frac {8}{\left (a^2 x^2+1\right )^2}+2 \sqrt {2} \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )+2 \sqrt {2} \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )+\sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )+\sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )}{4 a c^3 \sqrt {\tan ^{-1}(a x)}} \]
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 = 0.64, size = 85, normalized size = 0.90 \[ -\frac {2 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arctan \left (a x \right )}\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+8 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+4 \cos \left (2 \arctan \left (a x \right )\right )+\cos \left (4 \arctan \left (a x \right )\right )+3}{4 a \,c^{3} \sqrt {\arctan \left (a x \right )}} \]
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 \[ \int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {1}{a^{6} x^{6} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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