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/(d*x+c)-f*Chi(4*c*f/d+4*f*x)*cosh(-4*e+4*c*f/d)/a^2/d^2+f*Chi(2*c*f/d+2*f*x)*cosh(-2*e+2*c*f/d)/a^2
/d^2+1/2*cosh(2*f*x+2*e)/a^2/d/(d*x+c)-1/4*cosh(2*f*x+2*e)^2/a^2/d/(d*x+c)-f*cosh(-2*e+2*c*f/d)*Shi(2*c*f/d+2*
f*x)/a^2/d^2+f*cosh(-4*e+4*c*f/d)*Shi(4*c*f/d+4*f*x)/a^2/d^2-f*Chi(4*c*f/d+4*f*x)*sinh(-4*e+4*c*f/d)/a^2/d^2+f
*Shi(4*c*f/d+4*f*x)*sinh(-4*e+4*c*f/d)/a^2/d^2+f*Chi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/a^2/d^2-f*Shi(2*c*f/d+2
*f*x)*sinh(-2*e+2*c*f/d)/a^2/d^2-1/2*sinh(2*f*x+2*e)/a^2/d/(d*x+c)-1/4*sinh(2*f*x+2*e)^2/a^2/d/(d*x+c)+1/4*sin
h(4*f*x+4*e)/a^2/d/(d*x+c)

________________________________________________________________________________________

Rubi [A]  time = 0.73, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, 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 12

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

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 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 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 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 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]

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 = 1.48, 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))

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 615, normalized size = 1.46 \[ \frac {2 \, {\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 )^{2} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right )^{2} \sinh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + {\left (2 \, {\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 ) - 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) - d\right )} \cosh \left (f x + e\right )^{2} + {\left (2 \, {\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 ) - 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + 2 \, {\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 ) - 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) - d\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\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 (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \sinh \left (-\frac {4 \, {\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 {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right )\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + d}{2 \, {\left ({\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \sinh \left (f x + e\right )^{2}\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 633, normalized size = 1.51 \[ -\frac {{\left (4 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {4 \, {\left (c f - d e\right )}}{d}\right )} - 4 \, c f^{3} {\rm Ei}\left (\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {4 \, {\left (c f - d e\right )}}{d}\right )} - 4 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (c f - d e\right )}}{d}\right )} + 4 \, c f^{3} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (c f - d e\right )}}{d}\right )} + 4 \, d f^{2} {\rm Ei}\left (\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {4 \, {\left (c f - d e\right )}}{d} + 1\right )} - 4 \, d f^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (c f - d e\right )}}{d} + 1\right )} - d f^{2} e^{\left (\frac {4 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )} + 2 \, d f^{2} e^{\left (\frac {2 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )} - d f^{2}\right )} d^{2}}{4 \, {\left ({\left (d x + c\right )} a^{2} d^{4} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - a^{2} c d^{4} f + a^{2} d^{5} e\right )} f} \]

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

________________________________________________________________________________________

maple [A]  time = 1.17, size = 164, normalized size = 0.39 \[ -\frac {1}{4 a^{2} d \left (d x +c \right )}-\frac {f \,{\mathrm e}^{-4 f x -4 e}}{4 a^{2} d \left (d f x +c f \right )}+\frac {f \,{\mathrm e}^{\frac {4 c f -4 d e}{d}} \Ei \left (1, 4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{a^{2} d^{2}}+\frac {f \,{\mathrm e}^{-2 f x -2 e}}{2 a^{2} d \left (d f x +c f \right )}-\frac {f \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \Ei \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a^{2} d^{2}} \]

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

________________________________________________________________________________________

maxima [A]  time = 2.16, size = 100, normalized size = 0.24 \[ -\frac {1}{4 \, {\left (a^{2} d^{2} x + a^{2} c d\right )}} - \frac {e^{\left (-4 \, e + \frac {4 \, c f}{d}\right )} E_{2}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{4 \, {\left (d x + c\right )} a^{2} d} + \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, {\left (d x + c\right )} a^{2} d} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{c^{2} \coth ^{2}{\left (e + f x \right )} + 2 c^{2} \coth {\left (e + f x \right )} + c^{2} + 2 c d x \coth ^{2}{\left (e + f x \right )} + 4 c d x \coth {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \coth {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \]

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

________________________________________________________________________________________