Optimal. Leaf size=89 \[ \frac {(c+d x)^{1+m}}{2 a d (1+m)}-\frac {2^{-2-m} e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{a f} \]
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Rubi [A]
time = 0.08, antiderivative size = 89, 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 = {3808, 2212}
\begin {gather*} \frac {(c+d x)^{m+1}}{2 a d (m+1)}-\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3808
Rubi steps
\begin {align*} \int \frac {(c+d x)^m}{a+a \tanh (e+f x)} \, dx &=\frac {(c+d x)^{1+m}}{2 a d (1+m)}+\frac {\int e^{2 i (i e+i f x)} (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] Leaf count is larger than twice the leaf count of optimal. \(186\) vs. \(2(89)=178\).
time = 0.86, size = 186, normalized size = 2.09 \begin {gather*} \frac {2^{-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 {sech}(e+f x) \left (d (1+m) \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right ) \left (-\cosh \left (e-\frac {c f}{d}\right )+\sinh \left (e-\frac {c f}{d}\right )\right )+2^{1+m} f \left (f \left (\frac {c}{d}+x\right )\right )^m (c+d x) \left (\cosh \left (e-\frac {c f}{d}\right )+\sinh \left (e-\frac {c f}{d}\right )\right )\right ) \left (\cosh \left (f \left (\frac {c}{d}+x\right )\right )+\sinh \left (f \left (\frac {c}{d}+x\right )\right )\right )}{a d f (1+m) (1+\tanh (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{m}}{a +a \tanh \left (f x +e \right )}\, dx\]
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 \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 = 160, normalized size = 1.80 \begin {gather*} -\frac {{\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 {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 ) - 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 )}} \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}}{\tanh {\left (e + f x \right )} + 1}\, dx}{a} \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}{a+a\,\mathrm {tanh}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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