3.19 \(\int \frac{1}{(c+d x) (a+a \coth (e+f x))} \, dx\)

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]

-(Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*a*d) + Log[c + d*x]/(2*a*d) + (CoshIntegral[(2*c*f
)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(2*a*d) + (Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a*d)
- (Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a*d)

________________________________________________________________________________________

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]

Int[1/((c + d*x)*(a + a*Coth[e + f*x])),x]

[Out]

-(Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*a*d) + Log[c + d*x]/(2*a*d) + (CoshIntegral[(2*c*f
)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(2*a*d) + (Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a*d)
- (Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a*d)

Rule 3726

Int[1/(((c_.) + (d_.)*(x_))*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Log[c + d*x]/(2*a*d), x
] + (Dist[1/(2*a), Int[Cos[2*e + 2*f*x]/(c + d*x), x], x] + Dist[1/(2*b), Int[Sin[2*e + 2*f*x]/(c + d*x), x],
x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

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]

Integrate[1/((c + d*x)*(a + a*Coth[e + f*x])),x]

[Out]

(Csch[e + f*x]*(Cosh[f*x] + Sinh[f*x])*(Log[f*(c + d*x)]*(Cosh[e] + Sinh[e]) + CoshIntegral[(2*f*(c + d*x))/d]
*(-Cosh[e - (2*c*f)/d] + Sinh[e - (2*c*f)/d]) + (Cosh[e - (2*c*f)/d] - Sinh[e - (2*c*f)/d])*SinhIntegral[(2*f*
(c + d*x))/d]))/(2*a*d*(1 + Coth[e + f*x]))

________________________________________________________________________________________

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]

int(1/(d*x+c)/(a+a*coth(f*x+e)),x)

[Out]

1/2*ln(d*x+c)/d/a+1/2/a/d*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/d)

________________________________________________________________________________________

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]

integrate(1/(d*x+c)/(a+a*coth(f*x+e)),x, algorithm="maxima")

[Out]

1/2*e^(-2*e + 2*c*f/d)*exp_integral_e(1, 2*(d*x + c)*f/d)/(a*d) + 1/2*log(d*x + c)/(a*d)

________________________________________________________________________________________

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]

integrate(1/(d*x+c)/(a+a*coth(f*x+e)),x, algorithm="fricas")

[Out]

-1/2*(Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) + Ei(-2*(d*f*x + c*f)/d)*sinh(-2*(d*e - c*f)/d) - log(d*x
+ c))/(a*d)

________________________________________________________________________________________

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]

integrate(1/(d*x+c)/(a+a*coth(f*x+e)),x)

[Out]

Integral(1/(c*coth(e + f*x) + c + d*x*coth(e + f*x) + d*x), x)/a

________________________________________________________________________________________

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]

integrate(1/(d*x+c)/(a+a*coth(f*x+e)),x, algorithm="giac")

[Out]

-1/2*(Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d) - e^(2*e)*log(d*x + c))*e^(-2*e)/(a*d)