Optimal. Leaf size=110 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^3 \tan ^{-1}(a x)},x\right )}{a c^3}+\frac{a}{c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}+\frac{a}{c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}+\frac{2 a \text{Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac{a \text{Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{1}{a c^3 x^2 \tan ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.425717, 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 = {} \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx\right )+\frac{\int \frac{1}{x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=\frac{a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\left (4 a^3\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx+\frac{\int \frac{1}{x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx}{c^2}-\frac{a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac{1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{2 \int \frac{1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{\left (2 a^3\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{c}\\ &=-\frac{1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{2 \int \frac{1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{(4 a) \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{1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}-\frac{2 \int \frac{1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac{a \operatorname{Subst}\left (\int \frac{\sin (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^3}+\frac{a \operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{a \text{Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac{a \text{Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{2 \int \frac{1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}+\frac{a \operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^3}\\ &=-\frac{1}{a c^3 x^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{a}{c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 a \text{Si}\left (2 \tan ^{-1}(a x)\right )}{c^3}+\frac{a \text{Si}\left (4 \tan ^{-1}(a x)\right )}{2 c^3}-\frac{2 \int \frac{1}{x^3 \tan ^{-1}(a x)} \, dx}{a c^3}\\ \end{align*}
Mathematica [A] time = 2.12322, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.343, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left (a^{5} c^{3} x^{6} + 2 \, a^{3} c^{3} x^{4} + a c^{3} x^{2}\right )} \arctan \left (a x\right ) \int \frac{3 \, a^{2} x^{2} + 1}{{\left (a^{7} c^{3} x^{9} + 3 \, a^{5} c^{3} x^{7} + 3 \, a^{3} c^{3} x^{5} + a c^{3} x^{3}\right )} \arctan \left (a x\right )}\,{d x} + 1}{{\left (a^{5} c^{3} x^{6} + 2 \, a^{3} c^{3} x^{4} + a c^{3} x^{2}\right )} \arctan \left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{6} x^{8} \operatorname{atan}^{2}{\left (a x \right )} + 3 a^{4} x^{6} \operatorname{atan}^{2}{\left (a x \right )} + 3 a^{2} x^{4} \operatorname{atan}^{2}{\left (a x \right )} + x^{2} \operatorname{atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]