3.20 \(\int \frac{1}{(c+d x)^2 (a+a \coth (e+f x))} \, dx\)

Optimal. Leaf size=159 \[ -\frac{f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{a d^2}+\frac{f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{a d^2}+\frac{f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a d^2}-\frac{f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a d^2}-\frac{1}{d (c+d x) (a \coth (e+f x)+a)} \]

[Out]

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

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Rubi [A]  time = 0.213013, antiderivative size = 159, 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 = {3724, 3303, 3298, 3301} \[ -\frac{f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{a d^2}+\frac{f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{a d^2}+\frac{f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a d^2}-\frac{f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a d^2}-\frac{1}{d (c+d x) (a \coth (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^2*(a + a*Coth[e + f*x])),x]

[Out]

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

Rule 3724

Int[1/(((c_.) + (d_.)*(x_))^2*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> -Simp[(d*(c + d*x)*(a + b*
Tan[e + f*x]))^(-1), x] + (-Dist[f/(a*d), Int[Sin[2*e + 2*f*x]/(c + d*x), x], x] + Dist[f/(b*d), Int[Cos[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)^2 (a+a \coth (e+f x))} \, dx &=-\frac{1}{d (c+d x) (a+a \coth (e+f x))}-\frac{(i f) \int \frac{\sin \left (2 \left (i e+\frac{\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d}-\frac{f \int \frac{\cos \left (2 \left (i e+\frac{\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d}\\ &=-\frac{1}{d (c+d x) (a+a \coth (e+f x))}+\frac{\left (f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d}-\frac{\left (f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d}-\frac{\left (f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d}+\frac{\left (f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{a d}\\ &=\frac{f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{a d^2}-\frac{1}{d (c+d x) (a+a \coth (e+f x))}-\frac{f \text{Chi}\left (\frac{2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{a d^2}-\frac{f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{a d^2}+\frac{f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{a d^2}\\ \end{align*}

Mathematica [A]  time = 0.764114, size = 206, normalized size = 1.3 \[ -\frac{\text{csch}(e+f x) \left (\sinh \left (\frac{c f}{d}\right )+\cosh \left (\frac{c f}{d}\right )\right ) \left (2 f (c+d x) \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\sinh \left (e-\frac{f (c+d x)}{d}\right )-\cosh \left (e-\frac{f (c+d x)}{d}\right )\right )+2 f (c+d x) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac{f (c+d x)}{d}\right )-\sinh \left (e-\frac{f (c+d x)}{d}\right )\right )+d \left (\sinh \left (f \left (x-\frac{c}{d}\right )+e\right )+\sinh \left (f \left (\frac{c}{d}+x\right )+e\right )+\cosh \left (f \left (x-\frac{c}{d}\right )+e\right )-\cosh \left (f \left (\frac{c}{d}+x\right )+e\right )\right )\right )}{2 a d^2 (c+d x) (\coth (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^2*(a + a*Coth[e + f*x])),x]

[Out]

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

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Maple [A]  time = 0.203, size = 91, normalized size = 0.6 \begin{align*} -{\frac{1}{2\, \left ( dx+c \right ) ad}}+{\frac{f{{\rm e}^{-2\,fx-2\,e}}}{2\,da \left ( dfx+cf \right ) }}-{\frac{f}{a{d}^{2}}{{\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)^2/(a+a*coth(f*x+e)),x)

[Out]

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

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Maxima [A]  time = 2.25829, size = 76, normalized size = 0.48 \begin{align*} -\frac{1}{2 \,{\left (a d^{2} x + a c d\right )}} + \frac{e^{\left (-2 \, e + \frac{2 \, c f}{d}\right )} E_{2}\left (\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{2 \,{\left (d x + c\right )} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*coth(f*x+e)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.13375, size = 498, normalized size = 3.13 \begin{align*} \frac{{\left (d f x + c f\right )}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \cosh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) +{\left (d f x + c f\right )}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \sinh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) +{\left ({\left (d f x + c f\right )}{\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 ) +{\left (d f x + c f\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 ) - d\right )} \sinh \left (f x + e\right )}{{\left (a d^{3} x + a c d^{2}\right )} \cosh \left (f x + e\right ) +{\left (a d^{3} x + a c d^{2}\right )} \sinh \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*coth(f*x+e)),x, algorithm="fricas")

[Out]

((d*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(f*x + e)*cosh(-2*(d*e - c*f)/d) + (d*f*x + c*f)*Ei(-2*(d*f*x + c*f)
/d)*cosh(f*x + e)*sinh(-2*(d*e - c*f)/d) + ((d*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) + (d*f
*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*sinh(-2*(d*e - c*f)/d) - d)*sinh(f*x + e))/((a*d^3*x + a*c*d^2)*cosh(f*x + e)
+ (a*d^3*x + a*c*d^2)*sinh(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c^{2} \coth{\left (e + f x \right )} + c^{2} + 2 c d x \coth{\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth{\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**2/(a+a*coth(f*x+e)),x)

[Out]

Integral(1/(c**2*coth(e + f*x) + c**2 + 2*c*d*x*coth(e + f*x) + 2*c*d*x + d**2*x**2*coth(e + f*x) + d**2*x**2)
, x)/a

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Giac [A]  time = 1.33347, size = 149, normalized size = 0.94 \begin{align*} \frac{2 \, d f x{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d} - 2 \, e\right )} + 2 \, c f{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d} - 2 \, e\right )} + d e^{\left (-2 \, f x - 2 \, e\right )}}{2 \,{\left (d^{3} x + c d^{2}\right )} a} - \frac{1}{2 \,{\left (d x + c\right )} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*coth(f*x+e)),x, algorithm="giac")

[Out]

1/2*(2*d*f*x*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 2*c*f*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + d*e^(
-2*f*x - 2*e))/((d^3*x + c*d^2)*a) - 1/2/((d*x + c)*a*d)