Optimal. Leaf size=216 \[ 8 \text {Int}\left (\frac {1}{x \left (a^2 c x^2+c\right )^3 \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {8 \text {Int}\left (\frac {1}{x^3 \left (a^2 c x^2+c\right )^3 \sqrt {\tan ^{-1}(a x)}},x\right )}{3 a^2}+\frac {4}{3 a^2 c^3 x^2 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {20}{3 c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}-\frac {2}{3 a c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {5 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 c^3}+\frac {20 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 c^3} \]
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Rubi [A] time = 0.41, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx \]
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)^{5/2}} \, dx &=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (10 a) \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {1}{3} \left (80 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {80 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 c^3}\\ &=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {80 \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 )}{3 c^3}\\ &=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {10 \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 c^3}+\frac {20 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 c^3}\\ &=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}+\frac {20 \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 c^3}+\frac {40 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{3 c^3}\\ &=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}+\frac {5 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{3 c^3}+\frac {20 \sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 c^3}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^2}\\ \end {align*}
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Mathematica [A] time = 6.23, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx \]
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 = 3.81, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx \]
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 = -0.00 \[ \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{6} x^{7} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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