3.21 \(\int \frac{1}{(c+d x)^3 (a+a \coth (e+f x))} \, dx\)

Optimal. Leaf size=211 \[ \frac{f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{a d^3}-\frac{f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{a d^3}-\frac{f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a d^3}+\frac{f^2 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a d^3}+\frac{f}{d^2 (c+d x) (a \coth (e+f x)+a)}-\frac{f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a \coth (e+f x)+a)} \]

[Out]

-f/(2*a*d^2*(c + d*x)) - (f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - 1/(2*d*(c + d*x
)^2*(a + a*Coth[e + f*x])) + f/(d^2*(c + d*x)*(a + a*Coth[e + f*x])) + (f^2*CoshIntegral[(2*c*f)/d + 2*f*x]*Si
nh[2*e - (2*c*f)/d])/(a*d^3) + (f^2*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - (f^2*Sinh
[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.300802, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3725, 3724, 3303, 3298, 3301} \[ \frac{f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{a d^3}-\frac{f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{a d^3}-\frac{f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a d^3}+\frac{f^2 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a d^3}+\frac{f}{d^2 (c+d x) (a \coth (e+f x)+a)}-\frac{f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a \coth (e+f x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^3*(a + a*Coth[e + f*x])),x]

[Out]

-f/(2*a*d^2*(c + d*x)) - (f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - 1/(2*d*(c + d*x
)^2*(a + a*Coth[e + f*x])) + f/(d^2*(c + d*x)*(a + a*Coth[e + f*x])) + (f^2*CoshIntegral[(2*c*f)/d + 2*f*x]*Si
nh[2*e - (2*c*f)/d])/(a*d^3) + (f^2*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - (f^2*Sinh
[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3)

Rule 3725

Int[((c_.) + (d_.)*(x_))^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(f*(c + d*x)^(m + 2))/
(b*d^2*(m + 1)*(m + 2)), x] + (Dist[(2*b*f)/(a*d*(m + 1)), Int[(c + d*x)^(m + 1)/(a + b*Tan[e + f*x]), x], x]
+ Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^
2, 0] && LtQ[m, -1] && NeQ[m, -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)^3 (a+a \coth (e+f x))} \, dx &=-\frac{f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+a \coth (e+f x))}-\frac{f \int \frac{1}{(c+d x)^2 (a+a \coth (e+f x))} \, dx}{d}\\ &=-\frac{f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac{f}{d^2 (c+d x) (a+a \coth (e+f x))}+\frac{\left (i f^2\right ) \int \frac{\sin \left (2 \left (i e+\frac{\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d^2}+\frac{f^2 \int \frac{\cos \left (2 \left (i e+\frac{\pi }{2}\right )+2 i f x\right )}{c+d x} \, dx}{a d^2}\\ &=-\frac{f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac{f}{d^2 (c+d x) (a+a \coth (e+f x))}-\frac{\left (f^2 \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^2}+\frac{\left (f^2 \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^2}+\frac{\left (f^2 \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^2}-\frac{\left (f^2 \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^2}\\ &=-\frac{f}{2 a d^2 (c+d x)}-\frac{f^2 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{a d^3}-\frac{1}{2 d (c+d x)^2 (a+a \coth (e+f x))}+\frac{f}{d^2 (c+d x) (a+a \coth (e+f x))}+\frac{f^2 \text{Chi}\left (\frac{2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{a d^3}+\frac{f^2 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{a d^3}-\frac{f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{a d^3}\\ \end{align*}

Mathematica [A]  time = 1.07348, size = 265, normalized size = 1.26 \[ -\frac{\text{csch}(e+f x) \left (\sinh \left (\frac{c f}{d}\right )+\cosh \left (\frac{c f}{d}\right )\right ) \left (4 f^2 (c+d x)^2 \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac{f (c+d x)}{d}\right )-\sinh \left (e-\frac{f (c+d x)}{d}\right )\right )+4 f^2 (c+d x)^2 \text{Shi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\sinh \left (e-\frac{f (c+d x)}{d}\right )-\cosh \left (e-\frac{f (c+d x)}{d}\right )\right )+d \left (d \sinh \left (f \left (x-\frac{c}{d}\right )+e\right )+d \sinh \left (f \left (\frac{c}{d}+x\right )+e\right )-2 c f \sinh \left (f \left (\frac{c}{d}+x\right )+e\right )-2 d f x \sinh \left (f \left (\frac{c}{d}+x\right )+e\right )+d \cosh \left (f \left (x-\frac{c}{d}\right )+e\right )+(2 c f+2 d f x-d) \cosh \left (f \left (\frac{c}{d}+x\right )+e\right )\right )\right )}{4 a d^3 (c+d x)^2 (\coth (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^3*(a + a*Coth[e + f*x])),x]

[Out]

-(Csch[e + f*x]*(Cosh[(c*f)/d] + Sinh[(c*f)/d])*(d*(d*Cosh[e + f*(-(c/d) + x)] + (-d + 2*c*f + 2*d*f*x)*Cosh[e
+ f*(c/d + x)] + d*Sinh[e + f*(-(c/d) + x)] + d*Sinh[e + f*(c/d + x)] - 2*c*f*Sinh[e + f*(c/d + x)] - 2*d*f*x
*Sinh[e + f*(c/d + x)]) + 4*f^2*(c + d*x)^2*CoshIntegral[(2*f*(c + d*x))/d]*(Cosh[e - (f*(c + d*x))/d] - Sinh[
e - (f*(c + d*x))/d]) + 4*f^2*(c + d*x)^2*(-Cosh[e - (f*(c + d*x))/d] + Sinh[e - (f*(c + d*x))/d])*SinhIntegra
l[(2*f*(c + d*x))/d]))/(4*a*d^3*(c + d*x)^2*(1 + Coth[e + f*x]))

________________________________________________________________________________________

Maple [A]  time = 0.219, size = 210, normalized size = 1. \begin{align*} -{\frac{1}{4\,da \left ( dx+c \right ) ^{2}}}-{\frac{{f}^{3}{{\rm e}^{-2\,fx-2\,e}}x}{2\,da \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{f}^{3}{{\rm e}^{-2\,fx-2\,e}}c}{2\,a{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{f}^{2}{{\rm e}^{-2\,fx-2\,e}}}{4\,da \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{f}^{2}}{a{d}^{3}}{{\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)^3/(a+a*coth(f*x+e)),x)

[Out]

-1/4/d/a/(d*x+c)^2-1/2/a*f^3*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x-1/2/a*f^3*exp(-2*f*x-2*e)/d
^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c+1/4/a*f^2*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)+1/a*f^2/d
^3*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/d)

________________________________________________________________________________________

Maxima [A]  time = 3.05869, size = 92, normalized size = 0.44 \begin{align*} -\frac{1}{4 \,{\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} + \frac{e^{\left (-2 \, e + \frac{2 \, c f}{d}\right )} E_{3}\left (\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{2 \,{\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)^3/(a+a*coth(f*x+e)),x, algorithm="maxima")

[Out]

-1/4/(a*d^3*x^2 + 2*a*c*d^2*x + a*c^2*d) + 1/2*e^(-2*e + 2*c*f/d)*exp_integral_e(3, 2*(d*x + c)*f/d)/((d*x + c
)^2*a*d)

________________________________________________________________________________________

Fricas [A]  time = 2.16983, size = 748, normalized size = 3.55 \begin{align*} -\frac{2 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\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 (d^{2} f x + c d f + 2 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\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 )\right )} \cosh \left (f x + e\right ) -{\left (d^{2} f x + c d f - 2 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\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 ) - 2 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\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^{2}\right )} \sinh \left (f x + e\right )}{2 \,{\left ({\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \cosh \left (f x + e\right ) +{\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \sinh \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+a*coth(f*x+e)),x, algorithm="fricas")

[Out]

-1/2*(2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*cosh(f*x + e)*sinh(-2*(d*e - c*f)/d) + (d
^2*f*x + c*d*f + 2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d))*cosh(f
*x + e) - (d^2*f*x + c*d*f - 2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f
)/d) - 2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*sinh(-2*(d*e - c*f)/d) - d^2)*sinh(f*x +
e))/((a*d^5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)*cosh(f*x + e) + (a*d^5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)*sinh(f*x + e
))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c^{3} \coth{\left (e + f x \right )} + c^{3} + 3 c^{2} d x \coth{\left (e + f x \right )} + 3 c^{2} d x + 3 c d^{2} x^{2} \coth{\left (e + f x \right )} + 3 c d^{2} x^{2} + d^{3} x^{3} \coth{\left (e + f x \right )} + d^{3} x^{3}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**3/(a+a*coth(f*x+e)),x)

[Out]

Integral(1/(c**3*coth(e + f*x) + c**3 + 3*c**2*d*x*coth(e + f*x) + 3*c**2*d*x + 3*c*d**2*x**2*coth(e + f*x) +
3*c*d**2*x**2 + d**3*x**3*coth(e + f*x) + d**3*x**3), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.14919, size = 242, normalized size = 1.15 \begin{align*} -\frac{4 \, d^{2} f^{2} x^{2}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d}\right )} + 8 \, c d f^{2} x{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d}\right )} + 4 \, c^{2} f^{2}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d}\right )} + 2 \, d^{2} f x e^{\left (-2 \, f x\right )} + 2 \, c d f e^{\left (-2 \, f x\right )} - d^{2} e^{\left (-2 \, f x\right )} + d^{2} e^{\left (2 \, e\right )}}{4 \,{\left (a d^{5} x^{2} e^{\left (2 \, e\right )} + 2 \, a c d^{4} x e^{\left (2 \, e\right )} + a c^{2} d^{3} e^{\left (2 \, e\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3/(a+a*coth(f*x+e)),x, algorithm="giac")

[Out]

-1/4*(4*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d) + 8*c*d*f^2*x*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d) + 4*c^
2*f^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d) + 2*d^2*f*x*e^(-2*f*x) + 2*c*d*f*e^(-2*f*x) - d^2*e^(-2*f*x) + d^2*e^
(2*e))/(a*d^5*x^2*e^(2*e) + 2*a*c*d^4*x*e^(2*e) + a*c^2*d^3*e^(2*e))