Optimal. Leaf size=26 \[ \text {Int}\left ((c+d x)^m (a+i a \cot (e+f x))^2,x\right ) \]
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Rubi [A] time = 0.05, 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 = {} \[ \int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx \]
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))^2 \, dx &=\int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx\\ \end {align*}
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Mathematica [A] time = 10.38, size = 0, normalized size = 0.00 \[ \int (c+d x)^m (a+i a \cot (e+f x))^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.71, size = 0, normalized size = 0.00 \[ \frac {2 i \, {\left (d x + c\right )}^{m} a^{2} + {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )} {\rm integral}\left (\frac {{\left (4 \, a^{2} d f x + 4 \, a^{2} c f + 2 i \, a^{2} d m\right )} {\left (d x + c\right )}^{m}}{d f x + c f - {\left (d f x + c f\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}, x\right )}{f e^{\left (2 i \, f x + 2 i \, e\right )} - f} \]
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 \[ \int {\left (i \, a \cot \left (f x + e\right ) + a\right )}^{2} {\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +i a \cot \left (f x +e \right )\right )^{2}\, dx \]
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 \[ \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} - \int -\frac {{\left (d x + c\right )}^{m} a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (d x + c\right )}^{m} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (d x + c\right )}^{m} a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} - 4 \, {\left (d x + c\right )}^{m} a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, {\left (d x + c\right )}^{m} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, {\left (d x + c\right )}^{m} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (d x + c\right )}^{m} a^{2} - 2 \, {\left (2 \, {\left (d x + c\right )}^{m} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + {\left (d x + c\right )}^{m} a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right )}{2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \cos \left (4 \, f x + 4 \, e\right ) - \cos \left (4 \, f x + 4 \, e\right )^{2} - 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} - \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) - 1}\,{d x} - i \, \int -\frac {4 \, {\left ({\left (d x + c\right )}^{m} a^{2} \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (d x + c\right )}^{m} a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )}}{2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \cos \left (4 \, f x + 4 \, e\right ) - \cos \left (4 \, f x + 4 \, e\right )^{2} - 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} - \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) - 1}\,{d x} \]
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 \[ \int {\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]
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 \[ - a^{2} \left (\int \left (c + d x\right )^{m} \cot ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \left (c + d x\right )^{m} \cot {\left (e + f x \right )}\right )\, dx + \int \left (- \left (c + d x\right )^{m}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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