Optimal. Leaf size=152 \[ \frac {(c+d x)^{1+m}}{4 a^2 d (1+m)}+\frac {2^{-2-m} e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{a^2 f}-\frac {4^{-2-m} e^{-4 e+\frac {4 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right )}{a^2 f} \]
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Rubi [A]
time = 0.12, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3810, 2212}
\begin {gather*} \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^2 f}-\frac {4^{-m-2} e^{\frac {4 c f}{d}-4 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {4 f (c+d x)}{d}\right )}{a^2 f}+\frac {(c+d x)^{m+1}}{4 a^2 d (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3810
Rubi steps
\begin {align*} \int \frac {(c+d x)^m}{(a+a \coth (e+f x))^2} \, dx &=\int \left (\frac {(c+d x)^m}{4 a^2}+\frac {e^{-4 e-4 f x} (c+d x)^m}{4 a^2}-\frac {e^{-2 e-2 f x} (c+d x)^m}{2 a^2}\right ) \, dx\\ &=\frac {(c+d x)^{1+m}}{4 a^2 d (1+m)}+\frac {\int e^{-4 e-4 f x} (c+d x)^m \, dx}{4 a^2}-\frac {\int e^{-2 e-2 f x} (c+d x)^m \, dx}{2 a^2}\\ &=\frac {(c+d x)^{1+m}}{4 a^2 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^2 f}-\frac {4^{-2-m} e^{-4 e+\frac {4 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right )}{a^2 f}\\ \end {align*}
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Mathematica [A]
time = 5.83, size = 194, normalized size = 1.28 \begin {gather*} \frac {4^{-2-m} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \text {csch}^2(e+f x) \left (2^{2+m} d e^{\frac {2 c f}{d}} (1+m) \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )+4^{1+m} d \left (\frac {f (c+d x)}{d}\right )^{1+m} (\cosh (e)+\sinh (e))^2-d e^{\frac {4 c f}{d}} (1+m) \Gamma \left (1+m,\frac {4 f (c+d x)}{d}\right ) (\cosh (2 e)-\sinh (2 e))\right ) (\cosh (2 f x)+\sinh (2 f x))}{a^2 d f (1+m) (1+\coth (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{m}}{\left (a +a \coth \left (f x +e \right )\right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.09, size = 272, normalized size = 1.79 \begin {gather*} -\frac {{\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) - 4 \, c f + 4 \, d \cosh \left (1\right ) + 4 \, d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, \frac {4 \, {\left (d f x + c f\right )}}{d}\right ) - 4 \, {\left (d m + d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) - 2 \, c f + 2 \, d \cosh \left (1\right ) + 2 \, d \sinh \left (1\right )}{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 {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {4 \, f}{d}\right ) - 4 \, c f + 4 \, d \cosh \left (1\right ) + 4 \, d \sinh \left (1\right )}{d}\right ) + 4 \, {\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 \, c f + 2 \, d \cosh \left (1\right ) + 2 \, d \sinh \left (1\right )}{d}\right ) - 4 \, {\left (d f x + c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 4 \, {\left (d f x + c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{16 \, {\left (a^{2} d f m + a^{2} d f\right )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\left (c + d x\right )^{m}}{\coth ^{2}{\left (e + f x \right )} + 2 \coth {\left (e + f x \right )} + 1}\, dx}{a^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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