Optimal. Leaf size=157 \[ \frac{\text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}-\frac{\text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}-\frac{\sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}+\frac{\cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}+\frac{\log (c+d x)}{2 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.251087, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3726, 3303, 3298, 3301} \[ \frac{\text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}-\frac{\text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}-\frac{\sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}+\frac{\cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}+\frac{\log (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3726
Rule 3303
Rule 3298
Rule 3301
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*}
Mathematica [A] time = 0.250206, size = 122, normalized size = 0.78 \[ \frac{\text{csch}(e+f x) (\sinh (f x)+\cosh (f x)) \left (\text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\sinh \left (e-\frac{2 c f}{d}\right )-\cosh \left (e-\frac{2 c f}{d}\right )\right )+\text{Shi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac{2 c f}{d}\right )-\sinh \left (e-\frac{2 c f}{d}\right )\right )+(\sinh (e)+\cosh (e)) \log (f (c+d x))\right )}{2 a d (\coth (e+f x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.182, size = 61, normalized size = 0.4 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{2\,da}}+{\frac{1}{2\,da}{{\rm e}^{2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.75167, size = 65, normalized size = 0.41 \begin{align*} \frac{e^{\left (-2 \, e + \frac{2 \, c f}{d}\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.18103, size = 166, normalized size = 1.06 \begin{align*} -\frac{{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) +{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) - \log \left (d x + c\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14969, size = 68, normalized size = 0.43 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]