Optimal. Leaf size=207 \[ -\frac{3 (3 n-2 a x) e^{n \cot ^{-1}(a x)}}{a c \left (9 n^2+4\right ) \sqrt [3]{a^2 c x^2+c}}-\frac{6 x \sqrt [3]{\frac{1}{a^2 x^2}+1} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-2+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (4-3 i n)} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (2-3 i n)} \, _2F_1\left (-\frac{1}{3},\frac{1}{6} (2-3 i n);\frac{2}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{c \left (9 n^2+4\right ) \sqrt [3]{a^2 c x^2+c}} \]
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Rubi [A] time = 0.235156, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {5115, 5122, 5126, 132} \[ -\frac{3 (3 n-2 a x) e^{n \cot ^{-1}(a x)}}{a c \left (9 n^2+4\right ) \sqrt [3]{a^2 c x^2+c}}-\frac{6 x \sqrt [3]{\frac{1}{a^2 x^2}+1} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-2+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (4-3 i n)} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (2-3 i n)} \, _2F_1\left (-\frac{1}{3},\frac{1}{6} (2-3 i n);\frac{2}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{c \left (9 n^2+4\right ) \sqrt [3]{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5115
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 )^{4/3}} \, dx &=-\frac{3 e^{n \cot ^{-1}(a x)} (3 n-2 a x)}{a c \left (4+9 n^2\right ) \sqrt [3]{c+a^2 c x^2}}-\frac{2 \int \frac{e^{n \cot ^{-1}(a x)}}{\sqrt [3]{c+a^2 c x^2}} \, dx}{c \left (4+9 n^2\right )}\\ &=-\frac{3 e^{n \cot ^{-1}(a x)} (3 n-2 a x)}{a c \left (4+9 n^2\right ) \sqrt [3]{c+a^2 c x^2}}-\frac{\left (2 \sqrt [3]{1+\frac{1}{a^2 x^2}} x^{2/3}\right ) \int \frac{e^{n \cot ^{-1}(a x)}}{\sqrt [3]{1+\frac{1}{a^2 x^2}} x^{2/3}} \, dx}{c \left (4+9 n^2\right ) \sqrt [3]{c+a^2 c x^2}}\\ &=-\frac{3 e^{n \cot ^{-1}(a x)} (3 n-2 a x)}{a c \left (4+9 n^2\right ) \sqrt [3]{c+a^2 c x^2}}+\frac{\left (2 \sqrt [3]{1+\frac{1}{a^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{i x}{a}\right )^{-\frac{1}{3}+\frac{i n}{2}} \left (1+\frac{i x}{a}\right )^{-\frac{1}{3}-\frac{i n}{2}}}{x^{4/3}} \, dx,x,\frac{1}{x}\right )}{c \left (4+9 n^2\right ) \left (\frac{1}{x}\right )^{2/3} \sqrt [3]{c+a^2 c x^2}}\\ &=-\frac{3 e^{n \cot ^{-1}(a x)} (3 n-2 a x)}{a c \left (4+9 n^2\right ) \sqrt [3]{c+a^2 c x^2}}-\frac{6 \sqrt [3]{1+\frac{1}{a^2 x^2}} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (2-3 i n)} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-2+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (4-3 i n)} x \, _2F_1\left (-\frac{1}{3},\frac{1}{6} (2-3 i n);\frac{2}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{c \left (4+9 n^2\right ) \sqrt [3]{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.163702, size = 89, normalized size = 0.43 \[ -\frac{3 \left (-1+e^{2 i \cot ^{-1}(a x)}\right ) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left (1,\frac{i n}{2}+\frac{4}{3};\frac{i n}{2}+\frac{2}{3};e^{-2 i \cot ^{-1}(a x)}\right )}{a c (3 n+2 i) \sqrt [3]{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.297, 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{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}} e^{\left (n \operatorname{arccot}\left (a x\right )\right )}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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