Optimal. Leaf size=157 \[ \frac{2 a \text{Unintegrable}\left (\frac{1}{x^2 \tan ^{-1}(a x)},x\right )}{c^3}-\frac{3 \text{Unintegrable}\left (\frac{1}{x^4 \tan ^{-1}(a x)},x\right )}{a c^3}+\frac{5 a^2 \text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}+\frac{a^2 \text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{2 a^3 x}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{a^3 x}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}-\frac{1}{a c^3 x^3 \tan ^{-1}(a x)}+\frac{2 a}{c^3 x \tan ^{-1}(a x)} \]
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Rubi [A] time = 1.2204, 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 = {} \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx\right )+\frac{\int \frac{1}{x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=a^4 \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx+\frac{\int \frac{1}{x^3 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \frac{a^2 \int \frac{1}{x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac{1}{a c^3 x^3 \tan ^{-1}(a x)}-\frac{a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+a^3 \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\left (3 a^5\right ) \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^3}-2 \left (\frac{a^2 \int \frac{1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c^2}-\frac{a^4 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\right )\\ &=-\frac{1}{a c^3 x^3 \tan ^{-1}(a x)}-\frac{a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-2 \left (-\frac{a}{c^3 x \tan ^{-1}(a x)}+\frac{a^3 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^3}-\frac{a^3 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}+\frac{a^5 \int \frac{x^2}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\right )\\ &=-\frac{1}{a c^3 x^3 \tan ^{-1}(a x)}-\frac{a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac{a^2 \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 )}{c^3}-2 \left (-\frac{a}{c^3 x \tan ^{-1}(a x)}+\frac{a^3 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^3}-\frac{a^2 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\right )-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{8 x}-\frac{\cos (4 x)}{8 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{1}{a c^3 x^3 \tan ^{-1}(a x)}-\frac{a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 c^3}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 c^3}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}-2 \left (-\frac{a}{c^3 x \tan ^{-1}(a x)}+\frac{a^3 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^3}+\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}-\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\right )\\ &=-\frac{1}{a c^3 x^3 \tan ^{-1}(a x)}-\frac{a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{a^2 \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}+\frac{a^2 \text{Ci}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^3}-2 \left (-\frac{a}{c^3 x \tan ^{-1}(a x)}+\frac{a^3 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^3}-2 \frac{a^2 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}\right )\\ &=-\frac{1}{a c^3 x^3 \tan ^{-1}(a x)}-\frac{a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{a^2 \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{2 c^3}+\frac{a^2 \text{Ci}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{3 \int \frac{1}{x^4 \tan ^{-1}(a x)} \, dx}{a c^3}-2 \left (-\frac{a}{c^3 x \tan ^{-1}(a x)}+\frac{a^3 x}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{a^2 \text{Ci}\left (2 \tan ^{-1}(a x)\right )}{c^3}-\frac{a \int \frac{1}{x^2 \tan ^{-1}(a x)} \, dx}{c^3}\right )\\ \end{align*}
Mathematica [A] time = 2.91253, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.204, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ({a}^{2}c{x}^{2}+c \right ) ^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}}\, 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{\frac{{\left (a^{5} c^{3} x^{7} + 2 \, a^{3} c^{3} x^{5} + a c^{3} x^{3}\right )}{\left (7 \, a^{2} \int \frac{x^{2}}{a^{6} x^{10} \arctan \left (a x\right ) + 3 \, a^{4} x^{8} \arctan \left (a x\right ) + 3 \, a^{2} x^{6} \arctan \left (a x\right ) + x^{4} \arctan \left (a x\right )}\,{d x} + 3 \, \int \frac{1}{a^{6} x^{10} \arctan \left (a x\right ) + 3 \, a^{4} x^{8} \arctan \left (a x\right ) + 3 \, a^{2} x^{6} \arctan \left (a x\right ) + x^{4} \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )}{a c^{3}} + 1}{{\left (a^{5} c^{3} x^{7} + 2 \, a^{3} c^{3} x^{5} + a c^{3} x^{3}\right )} \arctan \left (a x\right )} \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^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )} \arctan \left (a x\right )^{2}}, 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^{9} \operatorname{atan}^{2}{\left (a x \right )} + 3 a^{4} x^{7} \operatorname{atan}^{2}{\left (a x \right )} + 3 a^{2} x^{5} \operatorname{atan}^{2}{\left (a x \right )} + x^{3} \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{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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