Optimal. Leaf size=74 \[ -\frac {1}{a c^2 x \text {ArcTan}(a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)}-\frac {\text {CosIntegral}(2 \text {ArcTan}(a x))}{c^2}-\frac {\text {Int}\left (\frac {1}{x^2 \text {ArcTan}(a x)},x\right )}{a c^2} \]
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Rubi [A]
time = 0.25, 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)^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)^2} \, dx &=-\left (a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx\right )+\frac {\int \frac {1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac {1}{a c^2 x \tan ^{-1}(a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-a \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx+a^3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)} \, dx}{a c^2}\\ &=-\frac {1}{a c^2 x \tan ^{-1}(a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}+\frac {\text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)} \, dx}{a c^2}\\ &=-\frac {1}{a c^2 x \tan ^{-1}(a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2}-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)} \, dx}{a c^2}\\ &=-\frac {1}{a c^2 x \tan ^{-1}(a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-2 \frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^2}-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)} \, dx}{a c^2}\\ &=-\frac {1}{a c^2 x \tan ^{-1}(a x)}+\frac {a x}{c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {\text {Ci}\left (2 \tan ^{-1}(a x)\right )}{c^2}-\frac {\int \frac {1}{x^2 \tan ^{-1}(a x)} \, dx}{a c^2}\\ \end {align*}
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Mathematica [A]
time = 0.98, 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)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{4} x^{5} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{3} \operatorname {atan}^{2}{\left (a x \right )} + x \operatorname {atan}^{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 )}^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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