Optimal. Leaf size=207 \[ -\frac {3 (3 n-4 a x) e^{n \cot ^{-1}(a x)}}{a c \left (9 n^2+16\right ) \left (a^2 c x^2+c\right )^{2/3}}-\frac {12 x \left (\frac {1}{a^2 x^2}+1\right )^{2/3} \left (\frac {a-\frac {i}{x}}{a+\frac {i}{x}}\right )^{\frac {1}{6} (4-3 i n)} \left (1-\frac {i}{a x}\right )^{\frac {1}{6} (-4+3 i n)} \left (1+\frac {i}{a x}\right )^{\frac {1}{6} (2-3 i n)} \, _2F_1\left (\frac {1}{3},\frac {1}{6} (4-3 i n);\frac {4}{3};\frac {2 i}{\left (a+\frac {i}{x}\right ) x}\right )}{c \left (9 n^2+16\right ) \left (a^2 c x^2+c\right )^{2/3}} \]
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Rubi [A] time = 0.25, antiderivative size = 207, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 (3 n-4 a x) e^{n \cot ^{-1}(a x)}}{a c \left (9 n^2+16\right ) \left (a^2 c x^2+c\right )^{2/3}}-\frac {12 x \left (\frac {1}{a^2 x^2}+1\right )^{2/3} \left (\frac {a-\frac {i}{x}}{a+\frac {i}{x}}\right )^{\frac {1}{6} (4-3 i n)} \left (1-\frac {i}{a x}\right )^{\frac {1}{6} (-4+3 i n)} \left (1+\frac {i}{a x}\right )^{\frac {1}{6} (2-3 i n)} \, _2F_1\left (\frac {1}{3},\frac {1}{6} (4-3 i n);\frac {4}{3};\frac {2 i}{\left (a+\frac {i}{x}\right ) x}\right )}{c \left (9 n^2+16\right ) \left (a^2 c x^2+c\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 132
Rule 5115
Rule 5122
Rule 5126
Rubi steps
\begin {align*} \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/3}} \, dx &=-\frac {3 e^{n \cot ^{-1}(a x)} (3 n-4 a x)}{a c \left (16+9 n^2\right ) \left (c+a^2 c x^2\right )^{2/3}}+\frac {4 \int \frac {e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx}{c \left (16+9 n^2\right )}\\ &=-\frac {3 e^{n \cot ^{-1}(a x)} (3 n-4 a x)}{a c \left (16+9 n^2\right ) \left (c+a^2 c x^2\right )^{2/3}}+\frac {\left (4 \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}{c \left (16+9 n^2\right ) \left (c+a^2 c x^2\right )^{2/3}}\\ &=-\frac {3 e^{n \cot ^{-1}(a x)} (3 n-4 a x)}{a c \left (16+9 n^2\right ) \left (c+a^2 c x^2\right )^{2/3}}-\frac {\left (4 \left (1+\frac {1}{a^2 x^2}\right )^{2/3}\right ) \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 )}{c \left (16+9 n^2\right ) \left (\frac {1}{x}\right )^{4/3} \left (c+a^2 c x^2\right )^{2/3}}\\ &=-\frac {3 e^{n \cot ^{-1}(a x)} (3 n-4 a x)}{a c \left (16+9 n^2\right ) \left (c+a^2 c x^2\right )^{2/3}}-\frac {12 \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 )}{c \left (16+9 n^2\right ) \left (c+a^2 c x^2\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 89, normalized size = 0.43 \[ -\frac {3 \left (-1+e^{2 i \cot ^{-1}(a x)}\right ) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left (1,\frac {i n}{2}+\frac {5}{3};\frac {i n}{2}+\frac {1}{3};e^{-2 i \cot ^{-1}(a x)}\right )}{a c (3 n+4 i) \left (a^2 c x^2+c\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {1}{3}} e^{\left (n \operatorname {arccot}\left (a x\right )\right )}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {e^{\left (n \operatorname {arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccot}\left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {e^{\left (n \operatorname {arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {acot}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{5/3}} \,d x \]
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 \[ \int \frac {e^{n \operatorname {acot}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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