Optimal. Leaf size=113 \[ -\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}-\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{a^2 c^3}-\frac{x}{2 a c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}+\frac{3}{2 a^2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{2}{a^2 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.449375, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {4968, 4964, 4902, 4970, 4406, 12, 3299} \[ -\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}-\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{a^2 c^3}-\frac{x}{2 a c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}+\frac{3}{2 a^2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{2}{a^2 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4968
Rule 4964
Rule 4902
Rule 4970
Rule 4406
Rule 12
Rule 3299
Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx &=-\frac{x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{2 a}-\frac{1}{2} (3 a) \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx\\ &=-\frac{x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{1}{2 a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-2 \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx+\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{2 a}-\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{2 a c}\\ &=-\frac{x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-6 \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac{2 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}+\frac{3 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac{x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{2 \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{3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}-\frac{6 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}\\ &=-\frac{x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3}{2 a^2 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 )}{4 a^2 c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}-\frac{6 \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{x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}-\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{4 a^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^3}\\ &=-\frac{x}{2 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{3}{2 a^2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}-\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{a^2 c^3}\\ \end{align*}
Mathematica [A] time = 0.162507, size = 98, normalized size = 0.87 \[ -\frac{\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2 \text{Si}\left (2 \tan ^{-1}(a x)\right )+2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2 \text{Si}\left (4 \tan ^{-1}(a x)\right )-3 a^2 x^2 \tan ^{-1}(a x)+a x+\tan ^{-1}(a x)}{2 a^2 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 88, normalized size = 0.8 \begin{align*} -{\frac{8\,{\it Si} \left ( 2\,\arctan \left ( ax \right ) \right ) \left ( \arctan \left ( ax \right ) \right ) ^{2}+16\,{\it Si} \left ( 4\,\arctan \left ( ax \right ) \right ) \left ( \arctan \left ( ax \right ) \right ) ^{2}+4\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +4\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +2\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) +\sin \left ( 4\,\arctan \left ( ax \right ) \right ) }{16\,{a}^{2}{c}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-a x +{\left (3 \, a^{2} x^{2} - 1\right )} \arctan \left (a x\right ) + \frac{2 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}{\left (3 \, a^{2} \int \frac{x^{3}}{a^{6} x^{6} \arctan \left (a x\right ) + 3 \, a^{4} x^{4} \arctan \left (a x\right ) + 3 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x} - 5 \, \int \frac{x}{a^{6} x^{6} \arctan \left (a x\right ) + 3 \, a^{4} x^{4} \arctan \left (a x\right ) + 3 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2}}{c^{3}}}{2 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )} \arctan \left (a x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.87409, size = 784, normalized size = 6.94 \begin{align*} \frac{{\left (-2 i \, a^{4} x^{4} - 4 i \, a^{2} x^{2} - 2 i\right )} \arctan \left (a x\right )^{2} \logintegral \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 (2 i \, a^{4} x^{4} + 4 i \, a^{2} x^{2} + 2 i\right )} \arctan \left (a x\right )^{2} \logintegral \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 )^{2} \logintegral \left (-\frac{a^{2} x^{2} + 2 i \, a x - 1}{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 )^{2} \logintegral \left (-\frac{a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \, a x + 2 \,{\left (3 \, a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{4 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )} \arctan \left (a x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{a^{6} x^{6} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atan}^{3}{\left (a x \right )} + \operatorname{atan}^{3}{\left (a x \right )}}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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