Optimal. Leaf size=88 \[ \frac {2^{-m-2} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{a f}+\frac {(c+d x)^{m+1}}{2 a d (m+1)} \]
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Rubi [A] time = 0.12, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3727, 2181} \[ \frac {2^{-m-2} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {2 f (c+d x)}{d}\right )}{a f}+\frac {(c+d x)^{m+1}}{2 a d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3727
Rubi steps
\begin {align*} \int \frac {(c+d x)^m}{a+a \coth (e+f x)} \, dx &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {\int e^{2 i \left (i e+\frac {\pi }{2}+i f x\right )} (c+d x)^m \, dx}{2 a}\\ &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {2^{-2-m} e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{a f}\\ \end {align*}
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Mathematica [B] time = 1.20, size = 186, normalized size = 2.11 \[ \frac {2^{-m-2} (c+d x)^m \text {csch}(e+f x) \left (-\frac {f (c+d x)}{d}\right )^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (\sinh \left (f \left (\frac {c}{d}+x\right )\right )+\cosh \left (f \left (\frac {c}{d}+x\right )\right )\right ) \left (f 2^{m+1} (c+d x) \left (f \left (\frac {c}{d}+x\right )\right )^m \left (\sinh \left (e-\frac {c f}{d}\right )+\cosh \left (e-\frac {c f}{d}\right )\right )+d (m+1) \left (\cosh \left (e-\frac {c f}{d}\right )-\sinh \left (e-\frac {c f}{d}\right )\right ) \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )\right )}{a d f (m+1) (\coth (e+f x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 148, normalized size = 1.68 \[ \frac {{\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (d m + d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) + 2 \, {\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) + 2 \, {\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{4 \, {\left (a d f m + a d f\right )}} \]
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 \[ \int \frac {{\left (d x + c\right )}^{m}}{a \coth \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{a +a \coth \left (f x +e \right )}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{m}}{a \coth \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^m}{a+a\,\mathrm {coth}\left (e+f\,x\right )} \,d x \]
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 \[ \frac {\int \frac {\left (c + d x\right )^{m}}{\coth {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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