Optimal. Leaf size=24 \[ \text {Int}\left ((c+d x)^m (a+i a \cot (e+f x)),x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (c+d x)^m (a+i a \cot (e+f x)) \, dx &=\int (c+d x)^m (a+i a \cot (e+f x)) \, dx\\ \end {align*}
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Mathematica [A] time = 5.74, size = 0, normalized size = 0.00 \[ \int (c+d x)^m (a+i a \cot (e+f x)) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {2 \, {\left (d x + c\right )}^{m} a}{e^{\left (2 i \, f x + 2 i \, e\right )} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (i \, a \cot \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.01, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +i a \cot \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ 2 i \, a \int \frac {{\left (d x + c\right )}^{m} \sin \left (2 \, f x + 2 \, e\right )}{\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} - 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1}\,{d x} + \frac {{\left (d x + c\right )}^{m + 1} a}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ i a \left (\int \left (- i \left (c + d x\right )^{m}\right )\, dx + \int \left (c + d x\right )^{m} \cot {\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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