Optimal. Leaf size=108 \[ -\frac{1}{20} a^5 c^3 x^4-\frac{2}{5} a^3 c^3 x^2-\frac{8}{5} a c^3 \log \left (a^2 x^2+1\right )+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)+3 a^2 c^3 x \tan ^{-1}(a x)+a c^3 \log (x)-\frac{c^3 \tan ^{-1}(a x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155943, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {4948, 4846, 260, 4852, 266, 36, 29, 31, 43} \[ -\frac{1}{20} a^5 c^3 x^4-\frac{2}{5} a^3 c^3 x^2-\frac{8}{5} a c^3 \log \left (a^2 x^2+1\right )+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)+3 a^2 c^3 x \tan ^{-1}(a x)+a c^3 \log (x)-\frac{c^3 \tan ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4948
Rule 4846
Rule 260
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)}{x^2} \, dx &=\int \left (3 a^2 c^3 \tan ^{-1}(a x)+\frac{c^3 \tan ^{-1}(a x)}{x^2}+3 a^4 c^3 x^2 \tan ^{-1}(a x)+a^6 c^3 x^4 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)}{x^2} \, dx+\left (3 a^2 c^3\right ) \int \tan ^{-1}(a x) \, dx+\left (3 a^4 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)}{x}+3 a^2 c^3 x \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)+\left (a c^3\right ) \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx-\left (3 a^3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\left (a^5 c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{5} \left (a^7 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)}{x}+3 a^2 c^3 x \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)-\frac{3}{2} a c^3 \log \left (1+a^2 x^2\right )+\frac{1}{2} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{2} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{10} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{c^3 \tan ^{-1}(a x)}{x}+3 a^2 c^3 x \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)-\frac{3}{2} a c^3 \log \left (1+a^2 x^2\right )+\frac{1}{2} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{2} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{10} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{2}{5} a^3 c^3 x^2-\frac{1}{20} a^5 c^3 x^4-\frac{c^3 \tan ^{-1}(a x)}{x}+3 a^2 c^3 x \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)+a c^3 \log (x)-\frac{8}{5} a c^3 \log \left (1+a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0702488, size = 78, normalized size = 0.72 \[ \frac{c^3 \left (4 \left (a^6 x^6+5 a^4 x^4+15 a^2 x^2-5\right ) \tan ^{-1}(a x)-a x \left (a^4 x^4+8 a^2 x^2+32 \log \left (a^2 x^2+1\right )-20 \log (x)\right )\right )}{20 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.031, size = 103, normalized size = 1. \begin{align*}{\frac{{a}^{6}{c}^{3}{x}^{5}\arctan \left ( ax \right ) }{5}}+{a}^{4}{c}^{3}{x}^{3}\arctan \left ( ax \right ) +3\,{a}^{2}{c}^{3}x\arctan \left ( ax \right ) -{\frac{{c}^{3}\arctan \left ( ax \right ) }{x}}-{\frac{{a}^{5}{c}^{3}{x}^{4}}{20}}-{\frac{2\,{a}^{3}{c}^{3}{x}^{2}}{5}}-{\frac{8\,a{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{5}}+a{c}^{3}\ln \left ( ax \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.971987, size = 126, normalized size = 1.17 \begin{align*} -\frac{1}{20} \,{\left (a^{4} c^{3} x^{4} + 8 \, a^{2} c^{3} x^{2} + 32 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 20 \, c^{3} \log \left (x\right )\right )} a + \frac{1}{5} \,{\left (a^{6} c^{3} x^{5} + 5 \, a^{4} c^{3} x^{3} + 15 \, a^{2} c^{3} x - \frac{5 \, c^{3}}{x}\right )} \arctan \left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60866, size = 216, normalized size = 2. \begin{align*} -\frac{a^{5} c^{3} x^{5} + 8 \, a^{3} c^{3} x^{3} + 32 \, a c^{3} x \log \left (a^{2} x^{2} + 1\right ) - 20 \, a c^{3} x \log \left (x\right ) - 4 \,{\left (a^{6} c^{3} x^{6} + 5 \, a^{4} c^{3} x^{4} + 15 \, a^{2} c^{3} x^{2} - 5 \, c^{3}\right )} \arctan \left (a x\right )}{20 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.10596, size = 110, normalized size = 1.02 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{5} \operatorname{atan}{\left (a x \right )}}{5} - \frac{a^{5} c^{3} x^{4}}{20} + a^{4} c^{3} x^{3} \operatorname{atan}{\left (a x \right )} - \frac{2 a^{3} c^{3} x^{2}}{5} + 3 a^{2} c^{3} x \operatorname{atan}{\left (a x \right )} + a c^{3} \log{\left (x \right )} - \frac{8 a c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{5} - \frac{c^{3} \operatorname{atan}{\left (a x \right )}}{x} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16316, size = 134, normalized size = 1.24 \begin{align*} -\frac{8}{5} \, a c^{3} \log \left (a^{2} x^{2} + 1\right ) + \frac{1}{2} \, a c^{3} \log \left (x^{2}\right ) + \frac{1}{5} \,{\left (a^{6} c^{3} x^{5} + 5 \, a^{4} c^{3} x^{3} + 15 \, a^{2} c^{3} x - \frac{5 \, c^{3}}{x}\right )} \arctan \left (a x\right ) - \frac{a^{9} c^{3} x^{4} + 8 \, a^{7} c^{3} x^{2}}{20 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]