Optimal. Leaf size=116 \[ -\frac{1}{6} a^5 c^3 x^2-\frac{8}{3} a^3 c^3 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac{a c^3}{6 x^2}-\frac{c^3 \tan ^{-1}(a x)}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.157906, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4948, 4846, 260, 4852, 266, 44, 36, 29, 31, 43} \[ -\frac{1}{6} a^5 c^3 x^2-\frac{8}{3} a^3 c^3 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac{a c^3}{6 x^2}-\frac{c^3 \tan ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4948
Rule 4846
Rule 260
Rule 4852
Rule 266
Rule 44
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^4} \, dx &=\int \left (3 a^4 c^3 \tan ^{-1}(a x)+\frac{c^3 \tan ^{-1}(a x)}{x^4}+\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x^2}+a^6 c^3 x^2 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)}{x^4} \, dx+\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x^2} \, dx+\left (3 a^4 c^3\right ) \int \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{1}{3} \left (a c^3\right ) \int \frac{1}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (3 a^3 c^3\right ) \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx-\left (3 a^5 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^7 c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)-\frac{3}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+\frac{1}{6} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )+\frac{1}{2} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{6} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)-\frac{3}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+\frac{1}{6} \left (a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{a^2}{x}+\frac{a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )+\frac{1}{2} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (3 a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{6} \left (a^7 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{a c^3}{6 x^2}-\frac{1}{6} a^5 c^3 x^2-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)-\frac{8}{3} a^3 c^3 \log \left (1+a^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0706868, size = 83, normalized size = 0.72 \[ \frac{c^3 \left (2 \left (a^6 x^6+9 a^4 x^4-9 a^2 x^2-1\right ) \tan ^{-1}(a x)-a x \left (a^4 x^4-16 a^2 x^2 \log (x)+16 a^2 x^2 \log \left (a^2 x^2+1\right )+1\right )\right )}{6 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 107, normalized size = 0.9 \begin{align*}{\frac{{a}^{6}{c}^{3}{x}^{3}\arctan \left ( ax \right ) }{3}}+3\,{a}^{4}{c}^{3}x\arctan \left ( ax \right ) -3\,{\frac{{a}^{2}{c}^{3}\arctan \left ( ax \right ) }{x}}-{\frac{{c}^{3}\arctan \left ( ax \right ) }{3\,{x}^{3}}}-{\frac{{a}^{5}{c}^{3}{x}^{2}}{6}}-{\frac{8\,{a}^{3}{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{a{c}^{3}}{6\,{x}^{2}}}+{\frac{8\,{a}^{3}{c}^{3}\ln \left ( ax \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12865, size = 130, normalized size = 1.12 \begin{align*} -\frac{1}{6} \,{\left (a^{4} c^{3} x^{2} + 16 \, a^{2} c^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{2} c^{3} \log \left (x\right ) + \frac{c^{3}}{x^{2}}\right )} a + \frac{1}{3} \,{\left (a^{6} c^{3} x^{3} + 9 \, a^{4} c^{3} x - \frac{9 \, a^{2} c^{3} x^{2} + c^{3}}{x^{3}}\right )} \arctan \left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.74709, size = 216, normalized size = 1.86 \begin{align*} -\frac{a^{5} c^{3} x^{5} + 16 \, a^{3} c^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{3} c^{3} x^{3} \log \left (x\right ) + a c^{3} x - 2 \,{\left (a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} - 9 \, a^{2} c^{3} x^{2} - c^{3}\right )} \arctan \left (a x\right )}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.94068, size = 117, normalized size = 1.01 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{3} \operatorname{atan}{\left (a x \right )}}{3} - \frac{a^{5} c^{3} x^{2}}{6} + 3 a^{4} c^{3} x \operatorname{atan}{\left (a x \right )} + \frac{8 a^{3} c^{3} \log{\left (x \right )}}{3} - \frac{8 a^{3} c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{3} - \frac{3 a^{2} c^{3} \operatorname{atan}{\left (a x \right )}}{x} - \frac{a c^{3}}{6 x^{2}} - \frac{c^{3} \operatorname{atan}{\left (a x \right )}}{3 x^{3}} & \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 = 1.18822, size = 149, normalized size = 1.28 \begin{align*} -\frac{1}{6} \, a^{5} c^{3} x^{2} - \frac{8}{3} \, a^{3} c^{3} \log \left (a^{2} x^{2} + 1\right ) + \frac{4}{3} \, a^{3} c^{3} \log \left (x^{2}\right ) + \frac{1}{3} \,{\left (a^{6} c^{3} x^{3} + 9 \, a^{4} c^{3} x - \frac{9 \, a^{2} c^{3} x^{2} + c^{3}}{x^{3}}\right )} \arctan \left (a x\right ) - \frac{8 \, a^{3} c^{3} x^{2} + a c^{3}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]