Optimal. Leaf size=35 \[ \frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{4 a^2 c^3}+\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{8 a^2 c^3} \]
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Rubi [A] time = 0.09, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4970, 4406, 3299} \[ \frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{4 a^2 c^3}+\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{8 a^2 c^3} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4406
Rule 4970
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}\\ &=\frac {\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^2 c^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^3}\\ &=\frac {\text {Si}\left (2 \tan ^{-1}(a x)\right )}{4 a^2 c^3}+\frac {\text {Si}\left (4 \tan ^{-1}(a x)\right )}{8 a^2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 27, normalized size = 0.77 \[ \frac {2 \text {Si}\left (2 \tan ^{-1}(a x)\right )+\text {Si}\left (4 \tan ^{-1}(a x)\right )}{8 a^2 c^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.44, size = 171, normalized size = 4.89 \[ \frac {i \, \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - i \, \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + 2 i \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 i \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{16 \, a^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 32, normalized size = 0.91 \[ \frac {\Si \left (2 \arctan \left (a x \right )\right )}{4 a^{2} c^{3}}+\frac {\Si \left (4 \arctan \left (a x \right )\right )}{8 a^{2} c^{3}} \]
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 \[ \int \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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 \[ \frac {\int \frac {x}{a^{6} x^{6} \operatorname {atan}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}{\left (a x \right )} + \operatorname {atan}{\left (a x \right )}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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