Optimal. Leaf size=157 \[ -\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}+\frac {\log (c+d x)}{2 a d}+\frac {\text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a d}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d} \]
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Rubi [A]
time = 0.20, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3807, 3384,
3379, 3382} \begin {gather*} \frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a d}-\frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{2 a d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{2 a d}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{2 a d}+\frac {\log (c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3807
Rubi steps
\begin {align*} \int \frac {1}{(c+d x) (a+a \coth (e+f x))} \, dx &=\frac {\log (c+d x)}{2 a d}+\frac {i \int \frac {\sin \left (2 \left (i e+\frac {\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{2 a}+\frac {\int \frac {\cos \left (2 \left (i e+\frac {\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{2 a}\\ &=\frac {\log (c+d x)}{2 a d}-\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}+\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}\\ &=-\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}+\frac {\log (c+d x)}{2 a d}+\frac {\text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a d}+\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}-\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 122, normalized size = 0.78 \begin {gather*} \frac {\text {csch}(e+f x) (\cosh (f x)+\sinh (f x)) \left (\log (f (c+d x)) (\cosh (e)+\sinh (e))+\text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (-\cosh \left (e-\frac {2 c f}{d}\right )+\sinh \left (e-\frac {2 c f}{d}\right )\right )+\left (\cosh \left (e-\frac {2 c f}{d}\right )-\sinh \left (e-\frac {2 c f}{d}\right )\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 a d (1+\coth (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 10.85, size = 61, normalized size = 0.39
method | result | size |
risch | \(\frac {\ln \left (d x +c \right )}{2 a d}+\frac {{\mathrm e}^{\frac {2 c f -2 d e}{d}} \expIntegral \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 a d}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.40, size = 49, normalized size = 0.31 \begin {gather*} \frac {e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} E_{1}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, a d} + \frac {\log \left (d x + c\right )}{2 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 88, normalized size = 0.56 \begin {gather*} -\frac {{\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - \log \left (d x + c\right )}{2 \, a d} \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 {1}{c \coth {\left (e + f x \right )} + c + d x \coth {\left (e + f x \right )} + d x}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 48, normalized size = 0.31 \begin {gather*} -\frac {{\left ({\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} - e^{\left (2 \, e\right )} \log \left (d x + c\right )\right )} e^{\left (-2 \, e\right )}}{2 \, a d} \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 {1}{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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