Optimal. Leaf size=147 \[ -\frac{3 x \left (\frac{1}{a^2 x^2}+1\right )^{2/3} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (4-3 i n)} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-4+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (2-3 i n)} \, _2F_1\left (\frac{1}{3},\frac{1}{6} (4-3 i n);\frac{4}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{\left (a^2 c x^2+c\right )^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.196872, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5122, 5126, 132} \[ -\frac{3 x \left (\frac{1}{a^2 x^2}+1\right )^{2/3} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (4-3 i n)} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-4+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (2-3 i n)} \, _2F_1\left (\frac{1}{3},\frac{1}{6} (4-3 i n);\frac{4}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{\left (a^2 c x^2+c\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5122
Rule 5126
Rule 132
Rubi steps
\begin{align*} \int \frac{e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx &=\frac{\left (\left (1+\frac{1}{a^2 x^2}\right )^{2/3} x^{4/3}\right ) \int \frac{e^{n \cot ^{-1}(a x)}}{\left (1+\frac{1}{a^2 x^2}\right )^{2/3} x^{4/3}} \, dx}{\left (c+a^2 c x^2\right )^{2/3}}\\ &=-\frac{\left (1+\frac{1}{a^2 x^2}\right )^{2/3} \operatorname{Subst}\left (\int \frac{\left (1-\frac{i x}{a}\right )^{-\frac{2}{3}+\frac{i n}{2}} \left (1+\frac{i x}{a}\right )^{-\frac{2}{3}-\frac{i n}{2}}}{x^{2/3}} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{4/3} \left (c+a^2 c x^2\right )^{2/3}}\\ &=-\frac{3 \left (1+\frac{1}{a^2 x^2}\right )^{2/3} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (4-3 i n)} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-4+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (2-3 i n)} x \, _2F_1\left (\frac{1}{3},\frac{1}{6} (4-3 i n);\frac{4}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{\left (c+a^2 c x^2\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.153477, size = 89, normalized size = 0.61 \[ -\frac{3 \sqrt [3]{a^2 c x^2+c} \left (-1+e^{2 i \cot ^{-1}(a x)}\right ) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left (1,\frac{i n}{2}+\frac{2}{3};\frac{i n}{2}+\frac{4}{3};e^{-2 i \cot ^{-1}(a x)}\right )}{a c (3 n-2 i)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.291, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccot} \left (ax\right )}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{2}{3}}}}\, dx \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*} \int \frac{e^{\left (n \operatorname{arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (n \operatorname{arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}, 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*} \int \frac{e^{n \operatorname{acot}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{2}{3}}}\, dx \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{e^{\left (n \operatorname{arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]