Optimal. Leaf size=198 \[ \frac {56}{3} \text {Int}\left (\frac {1}{x^2 \left (a^2 c x^2+c\right )^2 \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {8 \text {Int}\left (\frac {1}{x^4 \left (a^2 c x^2+c\right )^2 \sqrt {\tan ^{-1}(a x)}},x\right )}{a^2}+\frac {16}{3 c^2 x \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {2}{3 a c^2 x^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {8 \sqrt {\pi } a C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{c^2}+\frac {16 a \sqrt {\tan ^{-1}(a x)}}{c^2} \]
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Rubi [A] time = 0.48, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}-\frac {4 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (8 a) \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2}+\left (16 a^2\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2}+\frac {(16 a) \operatorname {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2}+\frac {(16 a) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 a \sqrt {\tan ^{-1}(a x)}}{c^2}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2}+\frac {(8 a) \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 a \sqrt {\tan ^{-1}(a x)}}{c^2}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2}+\frac {(16 a) \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{c^2}\\ &=-\frac {2}{3 a c^2 x^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}+\frac {8}{3 a^2 c^2 x^3 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16}{3 c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}+\frac {16 a \sqrt {\tan ^{-1}(a x)}}{c^2}+\frac {8 a \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{c^2}+\frac {16}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {40}{3} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^4 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a^2}\\ \end {align*}
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Mathematica [A] time = 6.88, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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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 = 2.40, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx \]
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 [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{4} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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