Optimal. Leaf size=125 \[ \frac {16 x}{3 c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2}{3 a c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac {4 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 a c^3}-\frac {8 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a c^3} \]
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Rubi [A] time = 0.30, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {4902, 4968, 4970, 4406, 3304, 3352, 4904, 3312} \[ \frac {16 x}{3 c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2}{3 a c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac {4 \sqrt {2 \pi } \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 a c^3}-\frac {8 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a c^3} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4406
Rule 4902
Rule 4904
Rule 4968
Rule 4970
Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac {1}{3} (8 a) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {16 x}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {16}{3} \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\left (16 a^2\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {16 x}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a c^3}+\frac {16 \operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}\\ &=-\frac {2}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {16 x}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {16 \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 )}{3 a c^3}+\frac {16 \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 c^3}\\ &=-\frac {2}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {16 x}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2 \operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a c^3}-\frac {2 \operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^3}-\frac {8 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a c^3}\\ &=-\frac {2}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {16 x}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {4 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 a c^3}-\frac {4 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a c^3}-\frac {16 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 a c^3}\\ &=-\frac {2}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {16 x}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {4 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 a c^3}-\frac {8 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a c^3}\\ \end {align*}
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Mathematica [C] time = 0.72, size = 186, normalized size = 1.49 \[ \frac {2 \left (-\frac {1}{a \left (a^2 x^2+1\right )^2}+\frac {8 x \tan ^{-1}(a x)}{\left (a^2 x^2+1\right )^2}+\frac {\sqrt {2} \tan ^{-1}(a x)^2 \Gamma \left (\frac {1}{2},2 i \tan ^{-1}(a x)\right )}{a \sqrt {i \tan ^{-1}(a x)}}+\frac {\tan ^{-1}(a x)^2 \Gamma \left (\frac {1}{2},4 i \tan ^{-1}(a x)\right )}{a \sqrt {i \tan ^{-1}(a x)}}-\frac {\sqrt {2} \left (-i \tan ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-2 i \tan ^{-1}(a x)\right )}{a}-\frac {\left (-i \tan ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-4 i \tan ^{-1}(a x)\right )}{a}\right )}{3 c^3 \tan ^{-1}(a x)^{3/2}} \]
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 = 0.70, size = 113, normalized size = 0.90 \[ \frac {-16 \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}-32 \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \arctan \left (a x \right )^{\frac {3}{2}}+16 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+8 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-4 \cos \left (2 \arctan \left (a x \right )\right )-\cos \left (4 \arctan \left (a x \right )\right )-3}{12 a \,c^{3} \arctan \left (a x \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {1}{a^{6} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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