\(\int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx\) [59]

Optimal result

Integrand size = 18, antiderivative size = 196 \[ \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx=-\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {b (b d-2 a c f-2 a d f x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {a b d \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2} \]

[Out]

-1/2*(d*x+c)^2/(a^2-b^2)/d+1/4*(-2*a*d*f*x-2*a*c*f+b*d)^2/a/(a-b)/(a+b)^2/d/f^2+b*(d*x+c)/(a^2-b^2)/f/(a+b*cot
h(f*x+e))+b*(-2*a*d*f*x-2*a*c*f+b*d)*ln(1+(-a+b)/(a+b)/exp(2*f*x+2*e))/(a^2-b^2)^2/f^2+a*b*d*polylog(2,(a-b)/(
a+b)/exp(2*f*x+2*e))/(a^2-b^2)^2/f^2

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3814, 3812, 2221, 2317, 2438} \[ \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx=\frac {b (-2 a c f-2 a d f x+b d) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {b (c+d x)}{f \left (a^2-b^2\right ) (a+b \coth (e+f x))}-\frac {(c+d x)^2}{2 d \left (a^2-b^2\right )}+\frac {a b d \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{f^2 \left (a^2-b^2\right )^2}+\frac {(-2 a c f-2 a d f x+b d)^2}{4 a d f^2 (a-b) (a+b)^2} \]

[In]

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

[Out]

-1/2*(c + d*x)^2/((a^2 - b^2)*d) + (b*d - 2*a*c*f - 2*a*d*f*x)^2/(4*a*(a - b)*(a + b)^2*d*f^2) + (b*(c + d*x))
/((a^2 - b^2)*f*(a + b*Coth[e + f*x])) + (b*(b*d - 2*a*c*f - 2*a*d*f*x)*Log[1 - (a - b)/((a + b)*E^(2*(e + f*x
)))])/((a^2 - b^2)^2*f^2) + (a*b*d*PolyLog[2, (a - b)/((a + b)*E^(2*(e + f*x)))])/((a^2 - b^2)^2*f^2)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3812

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^
(m + 1)/(d*(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^Simp[2*I*(e + f*x), x]/((a + I
*b)^2 + (a^2 + b^2)*E^(2*I*k*Pi)*E^Simp[2*I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && Integer
Q[4*k] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 3814

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}-\frac {i \int \frac {-i b d+2 i a c f+2 i a d f x}{a+b \coth (e+f x)} \, dx}{\left (a^2-b^2\right ) f} \\ & = -\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {(2 i b) \int \frac {e^{-2 (e+f x)} (-i b d+2 i a c f+2 i a d f x)}{(a+b)^2+\left (-a^2+b^2\right ) e^{-2 (e+f x)}} \, dx}{\left (a^2-b^2\right ) f} \\ & = -\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {b (b d-2 a c f-2 a d f x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {(2 a b d) \int \log \left (1+\frac {\left (-a^2+b^2\right ) e^{-2 (e+f x)}}{(a+b)^2}\right ) \, dx}{\left (a^2-b^2\right )^2 f} \\ & = -\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {b (b d-2 a c f-2 a d f x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {(a b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {\left (-a^2+b^2\right ) x}{(a+b)^2}\right )}{x} \, dx,x,e^{-2 (e+f x)}\right )}{\left (a^2-b^2\right )^2 f^2} \\ & = -\frac {(c+d x)^2}{2 \left (a^2-b^2\right ) d}+\frac {(b d-2 a c f-2 a d f x)^2}{4 a (a-b) (a+b)^2 d f^2}+\frac {b (c+d x)}{\left (a^2-b^2\right ) f (a+b \coth (e+f x))}+\frac {b (b d-2 a c f-2 a d f x) \log \left (1-\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {a b d \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )}{\left (a^2-b^2\right )^2 f^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.87 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.24 \[ \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx=\frac {\text {csch}^2(e+f x) (b \cosh (e+f x)+a \sinh (e+f x)) \left (4 (a-b) b f (c+d x) \sinh (e+f x)+2 (a-b) (e+f x) (-2 c f+d (e-f x)) (b \cosh (e+f x)+a \sinh (e+f x))+\frac {\left ((-b d+2 a f (c+d x)) \left ((a-b) (-b d+2 a f (c+d x))-4 a b d \log \left (1+\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )\right )+4 a^2 b d^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )\right ) (b \cosh (e+f x)+a \sinh (e+f x))}{a (a+b) d}\right )}{4 (a-b)^2 (a+b) f^2 (a+b \coth (e+f x))^2} \]

[In]

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

[Out]

(Csch[e + f*x]^2*(b*Cosh[e + f*x] + a*Sinh[e + f*x])*(4*(a - b)*b*f*(c + d*x)*Sinh[e + f*x] + 2*(a - b)*(e + f
*x)*(-2*c*f + d*(e - f*x))*(b*Cosh[e + f*x] + a*Sinh[e + f*x]) + (((-(b*d) + 2*a*f*(c + d*x))*((a - b)*(-(b*d)
+ 2*a*f*(c + d*x)) - 4*a*b*d*Log[1 + (-a + b)/((a + b)*E^(2*(e + f*x)))]) + 4*a^2*b*d^2*PolyLog[2, (a - b)/((
a + b)*E^(2*(e + f*x)))])*(b*Cosh[e + f*x] + a*Sinh[e + f*x]))/(a*(a + b)*d)))/(4*(a - b)^2*(a + b)*f^2*(a + b
*Coth[e + f*x])^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(523\) vs. \(2(195)=390\).

Time = 0.44 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.67

method	result	size
risch	\(\frac {d \,x^{2}}{2 a^{2}+4 a b +2 b^{2}}+\frac {c x}{a^{2}+2 a b +b^{2}}-\frac {2 \left (d x +c \right ) b^{2}}{\left (a -b \right ) f \left (a^{2}+2 a b +b^{2}\right ) \left ({\mathrm e}^{2 f x +2 e} a +b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}-\frac {2 b^{2} d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2} \left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {b^{2} d \ln \left ({\mathrm e}^{2 f x +2 e} a +b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}{f^{2} \left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {4 b a c \ln \left ({\mathrm e}^{f x +e}\right )}{f \left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {2 b a c \ln \left ({\mathrm e}^{2 f x +2 e} a +b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}{f \left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {4 b e d a \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2} \left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {2 b e d a \ln \left ({\mathrm e}^{2 f x +2 e} a +b \,{\mathrm e}^{2 f x +2 e}-a +b \right )}{f^{2} \left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {2 b d a \,x^{2}}{\left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {2 b d a \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right ) x}{f \left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {4 b d a e x}{f \left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {2 b d a \ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right ) e}{f^{2} \left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {2 b d a \,e^{2}}{f^{2} \left (a -b \right )^{2} \left (a +b \right )^{2}}-\frac {b d a \operatorname {polylog}\left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 f x +2 e}}{a -b}\right )}{f^{2} \left (a -b \right )^{2} \left (a +b \right )^{2}}\)	\(524\)

[In]

int((d*x+c)/(a+b*coth(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/2/(a^2+2*a*b+b^2)*d*x^2+1/(a^2+2*a*b+b^2)*c*x-2/(a-b)/f/(a^2+2*a*b+b^2)*(d*x+c)*b^2/(exp(2*f*x+2*e)*a+b*exp(
2*f*x+2*e)-a+b)-2/f^2/(a-b)^2*b^2/(a+b)^2*d*ln(exp(f*x+e))+1/f^2/(a-b)^2*b^2/(a+b)^2*d*ln(exp(2*f*x+2*e)*a+b*e
xp(2*f*x+2*e)-a+b)+4/f/(a-b)^2*b/(a+b)^2*a*c*ln(exp(f*x+e))-2/f/(a-b)^2*b/(a+b)^2*a*c*ln(exp(2*f*x+2*e)*a+b*ex
p(2*f*x+2*e)-a+b)-4/f^2/(a-b)^2*b/(a+b)^2*e*d*a*ln(exp(f*x+e))+2/f^2/(a-b)^2*b/(a+b)^2*e*d*a*ln(exp(2*f*x+2*e)
*a+b*exp(2*f*x+2*e)-a+b)+2/(a-b)^2*b/(a+b)^2*d*a*x^2-2/f/(a-b)^2*b/(a+b)^2*d*a*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b)
)*x+4/f/(a-b)^2*b/(a+b)^2*d*a*e*x-2/f^2/(a-b)^2*b/(a+b)^2*d*a*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*e+2/f^2/(a-b)^2
*b/(a+b)^2*d*a*e^2-1/f^2/(a-b)^2*b/(a+b)^2*d*a*polylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1797 vs. \(2 (194) = 388\).

Time = 0.31 (sec) , antiderivative size = 1797, normalized size of antiderivative = 9.17 \[ \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*((a^3 + a^2*b - a*b^2 - b^3)*d*f^2*x^2 + 2*(a^3 + a^2*b - a*b^2 - b^3)*c*f^2*x - 4*(a^2*b - a*b^2)*d*e^2
- 4*(a*b^2 - b^3)*d*e - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*f^2*x^2 - 4*(a^2*b + a*b^2)*d*e^2 + 8*(a^2*b + a*b^
2)*c*e*f - 4*(a*b^2 + b^3)*d*e + 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*f^2 - 2*(a*b^2 + b^3)*d*f)*x)*cosh(f*x +
e)^2 - 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*f^2*x^2 - 4*(a^2*b + a*b^2)*d*e^2 + 8*(a^2*b + a*b^2)*c*e*f - 4*(
a*b^2 + b^3)*d*e + 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*f^2 - 2*(a*b^2 + b^3)*d*f)*x)*cosh(f*x + e)*sinh(f*x +
e) - ((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*f^2*x^2 - 4*(a^2*b + a*b^2)*d*e^2 + 8*(a^2*b + a*b^2)*c*e*f - 4*(a*b^
2 + b^3)*d*e + 2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c*f^2 - 2*(a*b^2 + b^3)*d*f)*x)*sinh(f*x + e)^2 + 4*(2*(a^2*
b - a*b^2)*c*e + (a*b^2 - b^3)*c)*f + 4*((a^2*b + a*b^2)*d*cosh(f*x + e)^2 + 2*(a^2*b + a*b^2)*d*cosh(f*x + e)
*sinh(f*x + e) + (a^2*b + a*b^2)*d*sinh(f*x + e)^2 - (a^2*b - a*b^2)*d)*dilog(sqrt((a + b)/(a - b))*(cosh(f*x
+ e) + sinh(f*x + e))) + 4*((a^2*b + a*b^2)*d*cosh(f*x + e)^2 + 2*(a^2*b + a*b^2)*d*cosh(f*x + e)*sinh(f*x + e
) + (a^2*b + a*b^2)*d*sinh(f*x + e)^2 - (a^2*b - a*b^2)*d)*dilog(-sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(
f*x + e))) + 2*(2*(a^2*b - a*b^2)*d*e - 2*(a^2*b - a*b^2)*c*f - (2*(a^2*b + a*b^2)*d*e - 2*(a^2*b + a*b^2)*c*f
+ (a*b^2 + b^3)*d)*cosh(f*x + e)^2 - 2*(2*(a^2*b + a*b^2)*d*e - 2*(a^2*b + a*b^2)*c*f + (a*b^2 + b^3)*d)*cosh
(f*x + e)*sinh(f*x + e) - (2*(a^2*b + a*b^2)*d*e - 2*(a^2*b + a*b^2)*c*f + (a*b^2 + b^3)*d)*sinh(f*x + e)^2 +
(a*b^2 - b^3)*d)*log(2*(a + b)*cosh(f*x + e) + 2*(a + b)*sinh(f*x + e) + 2*(a - b)*sqrt((a + b)/(a - b))) + 2*
(2*(a^2*b - a*b^2)*d*e - 2*(a^2*b - a*b^2)*c*f - (2*(a^2*b + a*b^2)*d*e - 2*(a^2*b + a*b^2)*c*f + (a*b^2 + b^3
)*d)*cosh(f*x + e)^2 - 2*(2*(a^2*b + a*b^2)*d*e - 2*(a^2*b + a*b^2)*c*f + (a*b^2 + b^3)*d)*cosh(f*x + e)*sinh(
f*x + e) - (2*(a^2*b + a*b^2)*d*e - 2*(a^2*b + a*b^2)*c*f + (a*b^2 + b^3)*d)*sinh(f*x + e)^2 + (a*b^2 - b^3)*d
)*log(2*(a + b)*cosh(f*x + e) + 2*(a + b)*sinh(f*x + e) - 2*(a - b)*sqrt((a + b)/(a - b))) - 4*((a^2*b - a*b^2
)*d*f*x + (a^2*b - a*b^2)*d*e - ((a^2*b + a*b^2)*d*f*x + (a^2*b + a*b^2)*d*e)*cosh(f*x + e)^2 - 2*((a^2*b + a*
b^2)*d*f*x + (a^2*b + a*b^2)*d*e)*cosh(f*x + e)*sinh(f*x + e) - ((a^2*b + a*b^2)*d*f*x + (a^2*b + a*b^2)*d*e)*
sinh(f*x + e)^2)*log(sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e)) + 1) - 4*((a^2*b - a*b^2)*d*f*x + (
a^2*b - a*b^2)*d*e - ((a^2*b + a*b^2)*d*f*x + (a^2*b + a*b^2)*d*e)*cosh(f*x + e)^2 - 2*((a^2*b + a*b^2)*d*f*x
+ (a^2*b + a*b^2)*d*e)*cosh(f*x + e)*sinh(f*x + e) - ((a^2*b + a*b^2)*d*f*x + (a^2*b + a*b^2)*d*e)*sinh(f*x +
e)^2)*log(-sqrt((a + b)/(a - b))*(cosh(f*x + e) + sinh(f*x + e)) + 1))/((a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 +
a*b^4 + b^5)*f^2*cosh(f*x + e)^2 + 2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*f^2*cosh(f*x + e)*si
nh(f*x + e) + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*f^2*sinh(f*x + e)^2 - (a^5 - a^4*b - 2*a^3*b
^2 + 2*a^2*b^3 + a*b^4 - b^5)*f^2)

Sympy [F(-2)]

Exception generated. \[ \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> Invalid NaN comparison

Maxima [F]

\[ \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx=\int { \frac {d x + c}{{\left (b \coth \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-1/2*(8*a*b*f*integrate(x/(a^4*f*e^(2*f*x + 2*e) + 2*a^3*b*f*e^(2*f*x + 2*e) - 2*a*b^3*f*e^(2*f*x + 2*e) - b^4
*f*e^(2*f*x + 2*e) - a^4*f + 2*a^2*b^2*f - b^4*f), x) + 2*b^2*(2*(f*x + e)/((a^4 - 2*a^2*b^2 + b^4)*f^2) - log
((a + b)*e^(2*f*x + 2*e) - a + b)/((a^4 - 2*a^2*b^2 + b^4)*f^2)) + ((a^2*f*e^(2*e) - b^2*f*e^(2*e))*x^2*e^(2*f
*x) - 4*b^2*x - (a^2*f - 2*a*b*f + b^2*f)*x^2)/(a^4*f - 2*a^2*b^2*f + b^4*f - (a^4*f*e^(2*e) + 2*a^3*b*f*e^(2*
e) - 2*a*b^3*f*e^(2*e) - b^4*f*e^(2*e))*e^(2*f*x)))*d - c*(2*a*b*log(-(a - b)*e^(-2*f*x - 2*e) + a + b)/((a^4
- 2*a^2*b^2 + b^4)*f) + 2*b^2/((a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^(-2*f*x - 2*e))*f) -
(f*x + e)/((a^2 + 2*a*b + b^2)*f))

Giac [F]

\[ \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx=\int { \frac {d x + c}{{\left (b \coth \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x}{(a+b \coth (e+f x))^2} \, dx=\int \frac {c+d\,x}{{\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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