Optimal. Leaf size=120 \[ -\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{6} (2+3 i n)} \left (1+a^2 x^2\right )^{2/3} \, _2F_1\left (\frac {1}{6} (2+3 i n),\frac {1}{6} (4+3 i n);\frac {1}{6} (8+3 i n);\frac {1}{2} (1-i a x)\right )}{a (2 i-3 n) \left (c+a^2 c x^2\right )^{2/3}} \]
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Rubi [A]
time = 0.08, antiderivative size = 120, 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 = {5184, 5181, 71}
\begin {gather*} -\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} \left (a^2 x^2+1\right )^{2/3} (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+4);\frac {1}{6} (3 i n+8);\frac {1}{2} (1-i a x)\right )}{a (-3 n+2 i) \left (a^2 c x^2+c\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 5181
Rule 5184
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx &=\frac {\left (1+a^2 x^2\right )^{2/3} \int \frac {e^{n \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{2/3}} \, dx}{\left (c+a^2 c x^2\right )^{2/3}}\\ &=\frac {\left (1+a^2 x^2\right )^{2/3} \int (1-i a x)^{-\frac {2}{3}+\frac {i n}{2}} (1+i a x)^{-\frac {2}{3}-\frac {i n}{2}} \, dx}{\left (c+a^2 c x^2\right )^{2/3}}\\ &=-\frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{6} (2+3 i n)} \left (1+a^2 x^2\right )^{2/3} \, _2F_1\left (\frac {1}{6} (2+3 i n),\frac {1}{6} (4+3 i n);\frac {1}{6} (8+3 i n);\frac {1}{2} (1-i a x)\right )}{a (2 i-3 n) \left (c+a^2 c x^2\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 120, normalized size = 1.00 \begin {gather*} \frac {3\ 2^{\frac {1}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{3}+\frac {i n}{2}} \left (1+a^2 x^2\right )^{2/3} \, _2F_1\left (\frac {1}{3}+\frac {i n}{2},\frac {2}{3}+\frac {i n}{2};\frac {4}{3}+\frac {i n}{2};\frac {1}{2}-\frac {i a x}{2}\right )}{a (-2 i+3 n) \left (c+a^2 c x^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {2}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {2}{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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