Optimal. Leaf size=140 \[ -\frac {2 \text {Int}\left (\frac {1}{x^2 \left (a^2 c x^2+c\right )^3 \sqrt {\tan ^{-1}(a x)}},x\right )}{a}-\frac {2}{a c^3 x \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 c^3}-\frac {5 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{c^3}-\frac {15 \sqrt {\tan ^{-1}(a x)}}{2 c^3} \]
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Rubi [A] time = 0.23, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-(10 a) \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-\frac {10 \operatorname {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-\frac {10 \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 )}{c^3}\\ &=-\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {15 \sqrt {\tan ^{-1}(a x)}}{2 c^3}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-\frac {5 \operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}-\frac {5 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {15 \sqrt {\tan ^{-1}(a x)}}{2 c^3}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-\frac {5 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2 c^3}-\frac {10 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{c^3}\\ &=-\frac {2}{a c^3 x \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {15 \sqrt {\tan ^{-1}(a x)}}{2 c^3}-\frac {5 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 c^3}-\frac {5 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{c^3}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}\\ \end {align*}
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Mathematica [A] time = 5.35, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx \]
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 = 4.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{6} x^{7} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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