Optimal. Leaf size=111 \[ -\frac {1}{a c^3 x^2 \text {ArcTan}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)}+\frac {2 a \text {Si}(2 \text {ArcTan}(a x))}{c^3}+\frac {a \text {Si}(4 \text {ArcTan}(a x))}{2 c^3}-\frac {2 \text {Int}\left (\frac {1}{x^3 \text {ArcTan}(a x)},x\right )}{a c^3} \]
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Rubi [A]
time = 0.31, 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^2 \left (c+a^2 c x^2\right )^3 \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^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx\right )+\frac {\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\left (4 a^3\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx+\frac {\int \frac {1}{x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c^2}-\frac {a^2 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac {(4 a) \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac {\left (2 a^3\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac {(2 a) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac {(4 a) \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac {a \text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}+\frac {a \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac {(2 a) \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {a \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac {a \text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac {a \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac {1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac {a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {2 a \text {Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac {a \text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac {2 \int \frac {1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}\\ \end {align*}
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Mathematica [A]
time = 1.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \text {ArcTan}(a x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{2}}\, dx\]
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^{6} x^{8} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + x^{2} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \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^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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