Optimal. Leaf size=123 \[ \frac {3\ 2^{-\frac {1}{3}-\frac {i n}{2}} \sqrt [3]{a^2 x^2+1} (1-i a x)^{\frac {1}{6} (-2+3 i n)} \, _2F_1\left (\frac {1}{6} (3 i n-2),\frac {1}{6} (3 i n+8);\frac {1}{6} (3 i n+4);\frac {1}{2} (1-i a x)\right )}{a c (3 n+2 i) \sqrt [3]{a^2 c x^2+c}} \]
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Rubi [A] time = 0.12, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5076, 5073, 69} \[ \frac {3\ 2^{-\frac {1}{3}-\frac {i n}{2}} \sqrt [3]{a^2 x^2+1} (1-i a x)^{\frac {1}{6} (-2+3 i n)} \, _2F_1\left (\frac {1}{6} (3 i n-2),\frac {1}{6} (3 i n+8);\frac {1}{6} (3 i n+4);\frac {1}{2} (1-i a x)\right )}{a c (3 n+2 i) \sqrt [3]{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{4/3}} \, dx &=\frac {\sqrt [3]{1+a^2 x^2} \int \frac {e^{n \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{4/3}} \, dx}{c \sqrt [3]{c+a^2 c x^2}}\\ &=\frac {\sqrt [3]{1+a^2 x^2} \int (1-i a x)^{-\frac {4}{3}+\frac {i n}{2}} (1+i a x)^{-\frac {4}{3}-\frac {i n}{2}} \, dx}{c \sqrt [3]{c+a^2 c x^2}}\\ &=\frac {3\ 2^{-\frac {1}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{6} (-2+3 i n)} \sqrt [3]{1+a^2 x^2} \, _2F_1\left (\frac {1}{6} (-2+3 i n),\frac {1}{6} (8+3 i n);\frac {1}{6} (4+3 i n);\frac {1}{2} (1-i a x)\right )}{a c (2 i+3 n) \sqrt [3]{c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 123, normalized size = 1.00 \[ \frac {3\ 2^{-\frac {1}{3}-\frac {i n}{2}} \sqrt [3]{a^2 x^2+1} (1-i a x)^{-\frac {1}{3}+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}-\frac {1}{3},\frac {i n}{2}+\frac {4}{3};\frac {i n}{2}+\frac {2}{3};\frac {1}{2}-\frac {i a x}{2}\right )}{a c (3 n+2 i) \sqrt [3]{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {2}{3}} e^{\left (n \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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