Optimal. Leaf size=86 \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac{\text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac{x}{a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{x}{a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.517154, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4964, 4968, 4970, 3312, 3302, 4904, 4406} \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac{\text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac{x}{a^3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{x}{a^3 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4964
Rule 4968
Rule 4970
Rule 3312
Rule 3302
Rule 4904
Rule 4406
Rubi steps
\begin{align*} \int \frac{x^3}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\frac{\int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx}{a^2}+\frac{\int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{a^2 c}\\ &=\frac{x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a^3}+\frac{3 \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a^3 c}-\frac{\int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a c}\\ &=\frac{x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac{\operatorname{Subst}\left (\int \frac{\cos ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=\frac{x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cos (2 x)}{2 x}+\frac{\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{1}{8 x}-\frac{\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=\frac{x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^4 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^4 c^3}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 c^3}\\ &=\frac{x}{a^3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{x}{a^3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\text{Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^4 c^3}-\frac{\text{Ci}\left (4 \tan ^{-1}(a x)\right )}{2 a^4 c^3}\\ \end{align*}
Mathematica [A] time = 0.133276, size = 83, normalized size = 0.97 \[ \frac{\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )-\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )-2 a^3 x^3}{2 a^4 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 58, normalized size = 0.7 \begin{align*}{\frac{4\,{\it Ci} \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -4\,{\it Ci} \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 ) }{8\,{c}^{3}{a}^{4}\arctan \left ( ax \right ) }} \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{x^{3} + \frac{{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}{\left (a^{2} \int \frac{x^{4}}{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} - 3 \, \int \frac{x^{2}}{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 )}{a c^{3}}}{{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )} \arctan \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.01264, size = 699, normalized size = 8.13 \begin{align*} -\frac{4 \, a^{3} x^{3} +{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \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 (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \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 (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \logintegral \left (-\frac{a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) -{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \logintegral \left (-\frac{a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{4 \,{\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )} \arctan \left (a x\right )} \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^{3}}{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}} \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^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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