Optimal. Leaf size=61 \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}+\frac{\text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{2 a^2 c^3}-\frac{x}{a c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24642, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4968, 4970, 4406, 3302, 4904, 3312} \[ \frac{\text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}+\frac{\text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )}{2 a^2 c^3}-\frac{x}{a c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4968
Rule 4970
Rule 4406
Rule 3302
Rule 4904
Rule 3312
Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2} \, dx &=-\frac{x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx}{a}-(3 a) \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx\\ &=-\frac{x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cos ^4(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}\\ &=-\frac{x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\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^2 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^2 c^3}\\ &=-\frac{x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}+\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^3}\\ &=-\frac{x}{a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}+\frac{\text{Ci}\left (2 \tan ^{-1}(a x)\right )}{2 a^2 c^3}+\frac{\text{Ci}\left (4 \tan ^{-1}(a x)\right )}{2 a^2 c^3}\\ \end{align*}
Mathematica [A] time = 0.066865, size = 75, normalized size = 1.23 \[ \frac{\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \text{CosIntegral}\left (2 \tan ^{-1}(a x)\right )+\left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \text{CosIntegral}\left (4 \tan ^{-1}(a x)\right )-2 a x}{2 c^3 \left (a^3 x^2+a\right )^2 \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.061, size = 60, normalized size = 1. \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\,{a}^{2}{c}^{3}\arctan \left ( ax \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{x + \frac{{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}{\left (3 \, a^{2} \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} - \int \frac{1}{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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 2.0782, size = 693, normalized size = 11.36 \begin{align*} \frac{{\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 \, a x}{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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{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.
[In]
[Out]
________________________________________________________________________________________
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 )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]