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

Optimal. Leaf size=420 \[ \frac{f \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{a^2 d^2}-\frac{f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{a^2 d^2}+\frac{f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{a^2 d^2}-\frac{f \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \cosh \left (4 e-\frac{4 c f}{d}\right )}{a^2 d^2}+\frac{f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a^2 d^2}-\frac{f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{a^2 d^2}-\frac{f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a^2 d^2}+\frac{f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{a^2 d^2}-\frac{\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac{\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac{\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac{\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac{\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{1}{4 a^2 d (c+d x)} \]

[Out]

-1/(4*a^2*d*(c + d*x)) + Cosh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Cosh[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) + (f*
Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*
f)/d + 4*f*x])/(a^2*d^2) + (f*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(a^2*d^2) - (f*CoshIntegr
al[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(a^2*d^2) - Sinh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Sinh[2*e + 2*
f*x]^2/(4*a^2*d*(c + d*x)) + Sinh[4*e + 4*f*x]/(4*a^2*d*(c + d*x)) - (f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*
c*f)/d + 2*f*x])/(a^2*d^2) + (f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (f*Cosh[4*e
- (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) - (f*Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4
*f*x])/(a^2*d^2)

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Rubi [A]  time = 0.732656, antiderivative size = 420, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {3728, 3297, 3303, 3298, 3301, 3313, 12} \[ \frac{f \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{a^2 d^2}-\frac{f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{a^2 d^2}+\frac{f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{a^2 d^2}-\frac{f \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \cosh \left (4 e-\frac{4 c f}{d}\right )}{a^2 d^2}+\frac{f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a^2 d^2}-\frac{f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{a^2 d^2}-\frac{f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{a^2 d^2}+\frac{f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{a^2 d^2}-\frac{\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac{\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac{\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac{\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac{\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{1}{4 a^2 d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*a^2*d*(c + d*x)) + Cosh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Cosh[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) + (f*
Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*
f)/d + 4*f*x])/(a^2*d^2) + (f*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(a^2*d^2) - (f*CoshIntegr
al[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(a^2*d^2) - Sinh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Sinh[2*e + 2*
f*x]^2/(4*a^2*d*(c + d*x)) + Sinh[4*e + 4*f*x]/(4*a^2*d*(c + d*x)) - (f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*
c*f)/d + 2*f*x])/(a^2*d^2) + (f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (f*Cosh[4*e
- (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) - (f*Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4
*f*x])/(a^2*d^2)

Rule 3728

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

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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]

Rule 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x)^2 (a+a \coth (e+f x))^2} \, dx &=\int \left (\frac{1}{4 a^2 (c+d x)^2}-\frac{\cosh (2 e+2 f x)}{2 a^2 (c+d x)^2}+\frac{\cosh ^2(2 e+2 f x)}{4 a^2 (c+d x)^2}+\frac{\sinh (2 e+2 f x)}{2 a^2 (c+d x)^2}+\frac{\sinh ^2(2 e+2 f x)}{4 a^2 (c+d x)^2}-\frac{\sinh (4 e+4 f x)}{4 a^2 (c+d x)^2}\right ) \, dx\\ &=-\frac{1}{4 a^2 d (c+d x)}+\frac{\int \frac{\cosh ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{4 a^2}+\frac{\int \frac{\sinh ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{4 a^2}-\frac{\int \frac{\sinh (4 e+4 f x)}{(c+d x)^2} \, dx}{4 a^2}-\frac{\int \frac{\cosh (2 e+2 f x)}{(c+d x)^2} \, dx}{2 a^2}+\frac{\int \frac{\sinh (2 e+2 f x)}{(c+d x)^2} \, dx}{2 a^2}\\ &=-\frac{1}{4 a^2 d (c+d x)}+\frac{\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac{\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac{\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}+\frac{(i f) \int -\frac{i \sinh (4 e+4 f x)}{2 (c+d x)} \, dx}{a^2 d}-\frac{(i f) \int \frac{i \sinh (4 e+4 f x)}{2 (c+d x)} \, dx}{a^2 d}+\frac{f \int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx}{a^2 d}-\frac{f \int \frac{\cosh (4 e+4 f x)}{c+d x} \, dx}{a^2 d}-\frac{f \int \frac{\sinh (2 e+2 f x)}{c+d x} \, dx}{a^2 d}\\ &=-\frac{1}{4 a^2 d (c+d x)}+\frac{\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac{\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac{\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}+2 \frac{f \int \frac{\sinh (4 e+4 f x)}{c+d x} \, dx}{2 a^2 d}-\frac{\left (f \cosh \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{a^2 d}+\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^2 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^2 d}-\frac{\left (f \sinh \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{a^2 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^2 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^2 d}\\ &=-\frac{1}{4 a^2 d (c+d x)}+\frac{\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\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^2 d^2}-\frac{f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Chi}\left (\frac{4 c f}{d}+4 f x\right )}{a^2 d^2}-\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^2 d^2}-\frac{\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac{\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\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^2 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^2 d^2}-\frac{f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (\frac{4 c f}{d}+4 f x\right )}{a^2 d^2}+2 \left (\frac{\left (f \cosh \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^2 d}+\frac{\left (f \sinh \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^2 d}\right )\\ &=-\frac{1}{4 a^2 d (c+d x)}+\frac{\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\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^2 d^2}-\frac{f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Chi}\left (\frac{4 c f}{d}+4 f x\right )}{a^2 d^2}-\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^2 d^2}-\frac{\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac{\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac{\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\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^2 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^2 d^2}-\frac{f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (\frac{4 c f}{d}+4 f x\right )}{a^2 d^2}+2 \left (\frac{f \text{Chi}\left (\frac{4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{2 a^2 d^2}+\frac{f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^2 d^2}\right )\\ \end{align*}

Mathematica [A]  time = 1.41906, size = 442, normalized size = 1.05 \[ \frac{\left (\sinh \left (2 \left (f \left (x-\frac{c}{d}\right )+e\right )\right )-\cosh \left (2 \left (f \left (x-\frac{c}{d}\right )+e\right )\right )\right ) \left (4 f (c+d x) \text{Chi}\left (\frac{4 f (c+d x)}{d}\right ) \left (\cosh \left (2 e-\frac{2 f (c+d x)}{d}\right )-\sinh \left (2 e-\frac{2 f (c+d x)}{d}\right )\right )-4 f (c+d x) (\sinh (2 f x)+\cosh (2 f x)) \text{Chi}\left (\frac{2 f (c+d x)}{d}\right )+4 c f \text{Shi}\left (\frac{4 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac{2 f (c+d x)}{d}\right )+4 d f x \text{Shi}\left (\frac{4 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac{2 f (c+d x)}{d}\right )-4 c f \text{Shi}\left (\frac{4 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac{2 f (c+d x)}{d}\right )-4 d f x \text{Shi}\left (\frac{4 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac{2 f (c+d x)}{d}\right )+d \sinh \left (2 \left (f \left (x-\frac{c}{d}\right )+e\right )\right )-d \sinh \left (2 \left (f \left (\frac{c}{d}+x\right )+e\right )\right )+d \cosh \left (2 \left (f \left (x-\frac{c}{d}\right )+e\right )\right )+d \cosh \left (2 \left (f \left (\frac{c}{d}+x\right )+e\right )\right )+4 c f \sinh (2 f x) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+4 d f x \sinh (2 f x) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+4 c f \cosh (2 f x) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+4 d f x \cosh (2 f x) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+2 d \sinh \left (\frac{2 c f}{d}\right )-2 d \cosh \left (\frac{2 c f}{d}\right )\right )}{4 a^2 d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-Cosh[2*(e + f*(-(c/d) + x))] + Sinh[2*(e + f*(-(c/d) + x))])*(-2*d*Cosh[(2*c*f)/d] + d*Cosh[2*(e + f*(-(c/d
) + x))] + d*Cosh[2*(e + f*(c/d + x))] + 2*d*Sinh[(2*c*f)/d] - 4*f*(c + d*x)*CoshIntegral[(2*f*(c + d*x))/d]*(
Cosh[2*f*x] + Sinh[2*f*x]) + d*Sinh[2*(e + f*(-(c/d) + x))] - d*Sinh[2*(e + f*(c/d + x))] + 4*f*(c + d*x)*Cosh
Integral[(4*f*(c + d*x))/d]*(Cosh[2*e - (2*f*(c + d*x))/d] - Sinh[2*e - (2*f*(c + d*x))/d]) + 4*c*f*Cosh[2*f*x
]*SinhIntegral[(2*f*(c + d*x))/d] + 4*d*f*x*Cosh[2*f*x]*SinhIntegral[(2*f*(c + d*x))/d] + 4*c*f*Sinh[2*f*x]*Si
nhIntegral[(2*f*(c + d*x))/d] + 4*d*f*x*Sinh[2*f*x]*SinhIntegral[(2*f*(c + d*x))/d] - 4*c*f*Cosh[2*e - (2*f*(c
+ d*x))/d]*SinhIntegral[(4*f*(c + d*x))/d] - 4*d*f*x*Cosh[2*e - (2*f*(c + d*x))/d]*SinhIntegral[(4*f*(c + d*x
))/d] + 4*c*f*Sinh[2*e - (2*f*(c + d*x))/d]*SinhIntegral[(4*f*(c + d*x))/d] + 4*d*f*x*Sinh[2*e - (2*f*(c + d*x
))/d]*SinhIntegral[(4*f*(c + d*x))/d]))/(4*a^2*d^2*(c + d*x))

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Maple [A]  time = 0.282, size = 164, normalized size = 0.4 \begin{align*} -{\frac{1}{4\,{a}^{2}d \left ( dx+c \right ) }}-{\frac{f{{\rm e}^{-4\,fx-4\,e}}}{4\,{a}^{2}d \left ( dfx+cf \right ) }}+{\frac{f}{{a}^{2}{d}^{2}}{{\rm e}^{4\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,4\,fx+4\,e+4\,{\frac{cf-de}{d}} \right ) }+{\frac{f{{\rm e}^{-2\,fx-2\,e}}}{2\,{a}^{2}d \left ( dfx+cf \right ) }}-{\frac{f}{{a}^{2}{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))^2,x)

[Out]

-1/4/a^2/d/(d*x+c)-1/4/a^2*f*exp(-4*f*x-4*e)/d/(d*f*x+c*f)+1/a^2*f/d^2*exp(4*(c*f-d*e)/d)*Ei(1,4*f*x+4*e+4*(c*
f-d*e)/d)+1/2/a^2*f*exp(-2*f*x-2*e)/d/(d*f*x+c*f)-1/a^2*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 = 6.0564, size = 135, normalized size = 0.32 \begin{align*} -\frac{1}{4 \,{\left (a^{2} d^{2} x + a^{2} c d\right )}} - \frac{e^{\left (-4 \, e + \frac{4 \, c f}{d}\right )} E_{2}\left (\frac{4 \,{\left (d x + c\right )} f}{d}\right )}{4 \,{\left (d x + c\right )} a^{2} d} + \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^{2} 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.24221, size = 1413, normalized size = 3.36 \begin{align*} \text{result too large to display} \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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c^{2} \coth ^{2}{\left (e + f x \right )} + 2 c^{2} \coth{\left (e + f x \right )} + c^{2} + 2 c d x \coth ^{2}{\left (e + f x \right )} + 4 c d x \coth{\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \coth{\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \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))**2,x)

[Out]

Integral(1/(c**2*coth(e + f*x)**2 + 2*c**2*coth(e + f*x) + c**2 + 2*c*d*x*coth(e + f*x)**2 + 4*c*d*x*coth(e +
f*x) + 2*c*d*x + d**2*x**2*coth(e + f*x)**2 + 2*d**2*x**2*coth(e + f*x) + d**2*x**2), x)/a**2

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Giac [A]  time = 1.92818, size = 255, normalized size = 0.61 \begin{align*} -\frac{4 \, d f x{\rm Ei}\left (-\frac{4 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{4 \, c f}{d} - 4 \, e\right )} - 4 \, 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 )} + 4 \, c f{\rm Ei}\left (-\frac{4 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{4 \, c f}{d} - 4 \, e\right )} - 4 \, 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 )} - 2 \, d e^{\left (-2 \, f x - 2 \, e\right )} + d e^{\left (-4 \, f x - 4 \, e\right )}}{4 \,{\left (a^{2} d^{3} x + a^{2} c d^{2}\right )}} - \frac{1}{4 \,{\left (d x + c\right )} a^{2} 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))^2,x, algorithm="giac")

[Out]

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