Optimal. Leaf size=188 \[ \frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}-\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}+\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}-\frac {i e^{c (a+b x)}}{b c} \]
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Rubi [A] time = 0.19, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4443, 2194, 2251} \[ \frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}-\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}+\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}-\frac {i e^{c (a+b x)}}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 4443
Rubi steps
\begin {align*} \int e^{c (a+b x)} \cot ^3(d+e x) \, dx &=i \int \left (-e^{c (a+b x)}-\frac {8 e^{c (a+b x)}}{\left (-1+e^{2 i (d+e x)}\right )^3}-\frac {12 e^{c (a+b x)}}{\left (-1+e^{2 i (d+e x)}\right )^2}-\frac {6 e^{c (a+b x)}}{-1+e^{2 i (d+e x)}}\right ) \, dx\\ &=-\left (i \int e^{c (a+b x)} \, dx\right )-6 i \int \frac {e^{c (a+b x)}}{-1+e^{2 i (d+e x)}} \, dx-8 i \int \frac {e^{c (a+b x)}}{\left (-1+e^{2 i (d+e x)}\right )^3} \, dx-12 i \int \frac {e^{c (a+b x)}}{\left (-1+e^{2 i (d+e x)}\right )^2} \, dx\\ &=-\frac {i e^{c (a+b x)}}{b c}+\frac {6 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}-\frac {12 i e^{c (a+b x)} \, _2F_1\left (2,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}+\frac {8 i e^{c (a+b x)} \, _2F_1\left (3,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}\\ \end {align*}
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Mathematica [A] time = 2.18, size = 210, normalized size = 1.12 \[ \frac {1}{2} e^{c (a+b x)} \left (\frac {2 e^{2 i d} \left (b^2 c^2-2 e^2\right ) \left (i b c e^{2 i e x} \, _2F_1\left (1,1-\frac {i b c}{2 e};2-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )+(2 e-i b c) \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )\right )}{b c \left (-1+e^{2 i d}\right ) e^2 (b c+2 i e)}+\frac {b c \csc (d) \sin (e x) \csc (d+e x)}{e^2}-\frac {2 \cot (d)}{b c}-\frac {\csc ^2(d+e x)}{e}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cot \left (e x + d\right )^{3} e^{\left (b c x + a c\right )}, x\right ) \]
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 \[ \int \cot \left (e x + d\right )^{3} e^{\left ({\left (b x + a\right )} c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{c \left (b x +a \right )} \left (\cot ^{3}\left (e x +d \right )\right )\, dx \]
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 \[ \text {result too large to display} \]
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 \[ \int {\mathrm {cot}\left (d+e\,x\right )}^3\,{\mathrm {e}}^{c\,\left (a+b\,x\right )} \,d x \]
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 \[ e^{a c} \int e^{b c x} \cot ^{3}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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