Optimal. Leaf size=102 \[ -\frac {2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt {\text {ArcTan}(a x)}}-\frac {6 \sqrt {\text {ArcTan}(a x)}}{c^2}-\frac {3 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{c^2}-\frac {2 \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\text {ArcTan}(a x)}},x\right )}{a} \]
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Rubi [A]
time = 0.15, 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 = {}
\begin {gather*} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}
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 )^2 \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac {2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-(6 a) \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-\frac {6 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-\frac {6 \text {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}{a c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {6 \sqrt {\tan ^{-1}(a x)}}{c^2}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-\frac {3 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}\\ &=-\frac {2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {6 \sqrt {\tan ^{-1}(a x)}}{c^2}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}-\frac {6 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{c^2}\\ &=-\frac {2}{a c^2 x \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}-\frac {6 \sqrt {\tan ^{-1}(a x)}}{c^2}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{c^2}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{a}\\ \end {align*}
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Mathematica [A]
time = 3.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 1.04, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \frac {\int \frac {1}{a^{4} x^{5} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 2 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^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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