Optimal. Leaf size=67 \[ \frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 a^3 c^3}-\frac {1}{a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {1}{a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.28, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4964, 4902, 4970, 4406, 12, 3299} \[ \frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 a^3 c^3}-\frac {1}{a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac {1}{a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 4406
Rule 4902
Rule 4964
Rule 4970
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{a^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{a^2 c}\\ &=\frac {1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {4 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a}-\frac {2 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a c}\\ &=\frac {1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}+\frac {4 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}+\frac {4 \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^3}\\ &=\frac {1}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac {1}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{2 a^3 c^3}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 59, normalized size = 0.88 \[ \frac {\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \text {Si}\left (4 \tan ^{-1}(a x)\right )-2 a^2 x^2}{2 a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.51, size = 196, normalized size = 2.93 \[ -\frac {4 \, a^{2} x^{2} - {\left (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - {\left (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right )}{4 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )} \arctan \left (a x\right )} \]
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 = 0.21, size = 37, normalized size = 0.55 \[ \frac {4 \Si \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (4 \arctan \left (a x \right )\right )-1}{8 a^{3} c^{3} \arctan \left (a x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \arctan \left (a x\right ) \int \frac {a^{2} x^{3} - x}{{\left (a^{7} c^{3} x^{6} + 3 \, a^{5} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \arctan \left (a x\right )}\,{d x} + x^{2}}{{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \arctan \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{2}}{a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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