Optimal. Leaf size=199 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right )}{2 a c^3}+\frac{a x}{2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{3}{2 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{2}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}+\frac{3 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}+\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{1}{2 a c^3 x \tan ^{-1}(a x)^2} \]
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Rubi [A] time = 0.765388, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx\right )+\frac{\int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac{a x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}-\frac{1}{2} a \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx+\frac{1}{2} \left (3 a^3\right ) \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx+\frac{\int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{c^2}-\frac{a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{1-a^2 x^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{1}{2} (3 a) \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx+\left (2 a^2\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}+\frac{(3 a) \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{2 c}+\frac{\left (2 a^2\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{1-a^2 x^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\left (6 a^2\right ) \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 (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}-\frac{\left (3 a^2\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac{1}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{1-a^2 x^2}{2 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 )}{c^3}+\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 )}{c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{6 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}\\ &=-\frac{1}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{1-a^2 x^2}{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 c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}+\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{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 )}{c^3}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}\\ &=-\frac{1}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{1-a^2 x^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{3 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}+\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{4 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^3}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}\\ &=-\frac{1}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{2}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{3}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{1-a^2 x^2}{2 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{3 \text{Si}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}+\frac{\text{Si}\left (4 \tan ^{-1}(a x)\right )}{c^3}-\frac{\int \frac{1}{x^2 \tan ^{-1}(a x)^2} \, dx}{2 a c^3}\\ \end{align*}
Mathematica [A] time = 2.57424, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.436, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) ^{3} \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-a x +{\left (5 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) + \frac{2 \,{\left (a^{6} c^{3} x^{6} + 2 \, a^{4} c^{3} x^{4} + a^{2} c^{3} x^{2}\right )}{\left (10 \, a^{4} \int \frac{x^{4}}{a^{6} x^{9} \arctan \left (a x\right ) + 3 \, a^{4} x^{7} \arctan \left (a x\right ) + 3 \, a^{2} x^{5} \arctan \left (a x\right ) + x^{3} \arctan \left (a x\right )}\,{d x} + 3 \, a^{2} \int \frac{x^{2}}{a^{6} x^{9} \arctan \left (a x\right ) + 3 \, a^{4} x^{7} \arctan \left (a x\right ) + 3 \, a^{2} x^{5} \arctan \left (a x\right ) + x^{3} \arctan \left (a x\right )}\,{d x} + \int \frac{1}{{\left (a^{2} x^{2} + 1\right )}^{3} x^{3} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2}}{a^{2} c^{3}}}{2 \,{\left (a^{6} c^{3} x^{6} + 2 \, a^{4} c^{3} x^{4} + a^{2} c^{3} x^{2}\right )} \arctan \left (a x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{6} x^{7} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname{atan}^{3}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname{atan}^{3}{\left (a x \right )} + x \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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