3.478 \(\int \frac{(e+f x) \coth (c+d x) \text{csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=298 \[ -\frac{b^2 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}-\frac{b^2 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^2}+\frac{b^2 f \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac{b^2 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^3 d}-\frac{b^2 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^3 d}+\frac{b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac{b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}+\frac{b (e+f x) \text{csch}(c+d x)}{a^2 d}-\frac{f \coth (c+d x)}{2 a d^2}-\frac{(e+f x) \text{csch}^2(c+d x)}{2 a d} \]

[Out]

(b*f*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) - (f*Coth[c + d*x])/(2*a*d^2) + (b*(e + f*x)*Csch[c + d*x])/(a^2*d) - (
(e + f*x)*Csch[c + d*x]^2)/(2*a*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) -
(b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (b^2*(e + f*x)*Log[1 - E^(2*(c + d*x)
)])/(a^3*d) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (b^2*f*PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b^2*f*PolyLog[2, E^(2*(c + d*x))])/(2*a^3*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.570451, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {5587, 5452, 3767, 8, 3770, 5569, 3716, 2190, 2279, 2391, 5561} \[ -\frac{b^2 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}-\frac{b^2 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^2}+\frac{b^2 f \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac{b^2 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^3 d}-\frac{b^2 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^3 d}+\frac{b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac{b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}+\frac{b (e+f x) \text{csch}(c+d x)}{a^2 d}-\frac{f \coth (c+d x)}{2 a d^2}-\frac{(e+f x) \text{csch}^2(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*f*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) - (f*Coth[c + d*x])/(2*a*d^2) + (b*(e + f*x)*Csch[c + d*x])/(a^2*d) - (
(e + f*x)*Csch[c + d*x]^2)/(2*a*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) -
(b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (b^2*(e + f*x)*Log[1 - E^(2*(c + d*x)
)])/(a^3*d) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (b^2*f*PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b^2*f*PolyLog[2, E^(2*(c + d*x))])/(2*a^3*d^2)

Rule 5587

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Csch[c + d*x]^(p - 1)*Coth[c + d*x]^n)/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 5452

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 5569

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cosh[c + d*x]*Coth[c +
d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rubi steps

\begin{align*} \int \frac{(e+f x) \coth (c+d x) \text{csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \coth (c+d x) \text{csch}^2(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x) \coth (c+d x) \text{csch}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac{(e+f x) \text{csch}^2(c+d x)}{2 a d}-\frac{b \int (e+f x) \coth (c+d x) \text{csch}(c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac{f \int \text{csch}^2(c+d x) \, dx}{2 a d}\\ &=\frac{b (e+f x) \text{csch}(c+d x)}{a^2 d}-\frac{(e+f x) \text{csch}^2(c+d x)}{2 a d}+\frac{b^2 \int (e+f x) \coth (c+d x) \, dx}{a^3}-\frac{b^3 \int \frac{(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac{(i f) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{2 a d^2}-\frac{(b f) \int \text{csch}(c+d x) \, dx}{a^2 d}\\ &=\frac{b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac{f \coth (c+d x)}{2 a d^2}+\frac{b (e+f x) \text{csch}(c+d x)}{a^2 d}-\frac{(e+f x) \text{csch}^2(c+d x)}{2 a d}-\frac{\left (2 b^2\right ) \int \frac{e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a^3}-\frac{b^3 \int \frac{e^{c+d x} (e+f x)}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a^3}-\frac{b^3 \int \frac{e^{c+d x} (e+f x)}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a^3}\\ &=\frac{b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac{f \coth (c+d x)}{2 a d^2}+\frac{b (e+f x) \text{csch}(c+d x)}{a^2 d}-\frac{(e+f x) \text{csch}^2(c+d x)}{2 a d}-\frac{b^2 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d}-\frac{b^2 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac{\left (b^2 f\right ) \int \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a^3 d}+\frac{\left (b^2 f\right ) \int \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a^3 d}-\frac{\left (b^2 f\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^3 d}\\ &=\frac{b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac{f \coth (c+d x)}{2 a d^2}+\frac{b (e+f x) \text{csch}(c+d x)}{a^2 d}-\frac{(e+f x) \text{csch}^2(c+d x)}{2 a d}-\frac{b^2 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d}-\frac{b^2 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac{\left (b^2 f\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^3 d^2}+\frac{\left (b^2 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a-\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}+\frac{\left (b^2 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}\\ &=\frac{b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac{f \coth (c+d x)}{2 a d^2}+\frac{b (e+f x) \text{csch}(c+d x)}{a^2 d}-\frac{(e+f x) \text{csch}^2(c+d x)}{2 a d}-\frac{b^2 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d}-\frac{b^2 (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac{b^2 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}-\frac{b^2 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{b^2 f \text{Li}_2\left (e^{2 (c+d x)}\right )}{2 a^3 d^2}\\ \end{align*}

Mathematica [A]  time = 6.8722, size = 376, normalized size = 1.26 \[ \frac{-8 b^2 \left (f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac{1}{2} f (c+d x)^2\right )+4 b^2 f \left ((c+d x) \left (2 \log \left (1-e^{-2 (c+d x)}\right )+c+d x\right )-\text{PolyLog}\left (2,e^{-2 (c+d x)}\right )\right )-a^2 d (e+f x) \text{csch}^2\left (\frac{1}{2} (c+d x)\right )+a^2 d (e+f x) \text{sech}^2\left (\frac{1}{2} (c+d x)\right )-2 a \tanh \left (\frac{1}{2} (c+d x)\right ) (a f+2 b d (e+f x))+2 a \coth \left (\frac{1}{2} (c+d x)\right ) (2 b d (e+f x)-a f)-8 a b f \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+8 b^2 d e \log (\sinh (c+d x))-8 b^2 c f \log (\sinh (c+d x))}{8 a^3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*(-(a*f) + 2*b*d*(e + f*x))*Coth[(c + d*x)/2] - a^2*d*(e + f*x)*Csch[(c + d*x)/2]^2 + 8*b^2*d*e*Log[Sinh[c
+ d*x]] - 8*b^2*c*f*Log[Sinh[c + d*x]] - 8*a*b*f*Log[Tanh[(c + d*x)/2]] + 4*b^2*f*((c + d*x)*(c + d*x + 2*Log
[1 - E^(-2*(c + d*x))]) - PolyLog[2, E^(-2*(c + d*x))]) - 8*b^2*(-(f*(c + d*x)^2)/2 + f*(c + d*x)*Log[1 + (b*E
^(c + d*x))/(a - Sqrt[a^2 + b^2])] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d*e*Log[a +
b*Sinh[c + d*x]] - c*f*Log[a + b*Sinh[c + d*x]] + f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*Pol
yLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) + a^2*d*(e + f*x)*Sech[(c + d*x)/2]^2 - 2*a*(a*f + 2*b*d*(e
+ f*x))*Tanh[(c + d*x)/2])/(8*a^3*d^2)

________________________________________________________________________________________

Maple [B]  time = 0.143, size = 649, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-2*b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c)-2*b*d*e*exp(3*d*x+3*c)+2*a*d*e*exp(2*d*x+2*c)+2*b*d*f*x*e
xp(d*x+c)+a*f*exp(2*d*x+2*c)+2*b*d*e*exp(d*x+c)-a*f)/d^2/a^2/(exp(2*d*x+2*c)-1)^2-1/a^3/d^2*b^2*f*c*ln(exp(d*x
+c)-1)+1/a^3*b^2/d^2*f*c*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/a^2/d^2*b*f*ln(exp(d*x+c)+1)-1/a^2/d^2*b*f*ln
(exp(d*x+c)-1)-1/a^3/d^2*b^2*f*dilog(exp(d*x+c))-1/a^3*b^2/d^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(
a^2+b^2)^(1/2)))-1/a^3*b^2/d^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/a^3/d^2*b^2*f*d
ilog(exp(d*x+c)+1)+1/a^3/d*b^2*e*ln(exp(d*x+c)-1)-1/a^3*b^2/d*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/a^3/d*
b^2*e*ln(exp(d*x+c)+1)-1/a^3*b^2/d*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/a^3*b^2/d^
2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/a^3*b^2/d*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2
)+a)/(a+(a^2+b^2)^(1/2)))*x-1/a^3*b^2/d^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/a^3/d
*b^2*f*ln(exp(d*x+c)+1)*x

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -{\left (4 \, b^{2} d \int \frac{x}{4 \,{\left (a^{3} d e^{\left (d x + c\right )} + a^{3} d\right )}}\,{d x} - 4 \, b^{2} d \int \frac{x}{4 \,{\left (a^{3} d e^{\left (d x + c\right )} - a^{3} d\right )}}\,{d x} + a b{\left (\frac{d x + c}{a^{3} d^{2}} - \frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3} d^{2}}\right )} - a b{\left (\frac{d x + c}{a^{3} d^{2}} - \frac{\log \left (e^{\left (d x + c\right )} - 1\right )}{a^{3} d^{2}}\right )} - \frac{2 \, b d x e^{\left (3 \, d x + 3 \, c\right )} - 2 \, b d x e^{\left (d x + c\right )} -{\left (2 \, a d x e^{\left (2 \, c\right )} + a e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + a}{a^{2} d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d^{2}} - 4 \, \int \frac{a b^{2} x e^{\left (d x + c\right )} - b^{3} x}{2 \,{\left (a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} e^{\left (d x + c\right )} - a^{3} b\right )}}\,{d x}\right )} f - e{\left (\frac{2 \,{\left (b e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} + \frac{b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{3} d} - \frac{b^{2} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} - \frac{b^{2} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.60422, size = 7004, normalized size = 23.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c)^3 + 2*(a*b*d*f*x + a*b*d*e)*sinh(d*x + c)^3 + a^2*f - (2*a^2*d*f*x + 2*
a^2*d*e + a^2*f)*cosh(d*x + c)^2 - (2*a^2*d*f*x + 2*a^2*d*e + a^2*f - 6*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c))*s
inh(d*x + c)^2 - 2*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c) - (b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d
*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh
(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d
*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (b^2*f*cosh(d*x + c)^4 + 4*b
^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh
(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*dilog((a
*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + (b^
2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2
+ b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c)
)*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + (b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d*
x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(
d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x +
c)) - ((b^2*d*e - b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*e - b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*e -
b^2*c*f)*sinh(d*x + c)^4 + b^2*d*e - b^2*c*f - 2*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*e - b^2*c*f -
3*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*e - b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*e - b
^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) +
2*a) - ((b^2*d*e - b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*e - b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*e -
b^2*c*f)*sinh(d*x + c)^4 + b^2*d*e - b^2*c*f - 2*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*e - b^2*c*f -
3*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*e - b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*e -
b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) +
2*a) - (b^2*d*f*x + (b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)*sinh(d*x +
c)^3 + (b^2*d*f*x + b^2*c*f)*sinh(d*x + c)^4 + b^2*c*f - 2*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*f*
x + b^2*c*f - 3*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f*x + b^2*c*f)*cosh(d*x + c
)^3 - (b^2*d*f*x + b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*
x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - (b^2*d*f*x + (b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^4 +
4*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f*x + b^2*c*f)*sinh(d*x + c)^4 + b^2*c*f - 2*(
b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*f*x + b^2*c*f - 3*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2)*sinh(
d*x + c)^2 + 4*((b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*f*x + b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*lo
g(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + (b
^2*d*f*x + (b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c)^4 + 4*(b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c)*sinh(
d*x + c)^3 + (b^2*d*f*x + b^2*d*e + a*b*f)*sinh(d*x + c)^4 + b^2*d*e + a*b*f - 2*(b^2*d*f*x + b^2*d*e + a*b*f)
*cosh(d*x + c)^2 - 2*(b^2*d*f*x + b^2*d*e + a*b*f - 3*(b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 4*((b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c)^3 - (b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c))*sinh(
d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((b^2*d*e - (b^2*c + a*b)*f)*cosh(d*x + c)^4 + 4*(b^2*d*e -
(b^2*c + a*b)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*e - (b^2*c + a*b)*f)*sinh(d*x + c)^4 + b^2*d*e - 2*(b
^2*d*e - (b^2*c + a*b)*f)*cosh(d*x + c)^2 - 2*(b^2*d*e - 3*(b^2*d*e - (b^2*c + a*b)*f)*cosh(d*x + c)^2 - (b^2*
c + a*b)*f)*sinh(d*x + c)^2 - (b^2*c + a*b)*f + 4*((b^2*d*e - (b^2*c + a*b)*f)*cosh(d*x + c)^3 - (b^2*d*e - (b
^2*c + a*b)*f)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (b^2*d*f*x + (b^2*d*f*x
+ b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f*x + b^2*c*f)*sin
h(d*x + c)^4 + b^2*c*f - 2*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*f*x + b^2*c*f - 3*(b^2*d*f*x + b^2
*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*f*x + b^2*c*f)*cosh
(d*x + c))*sinh(d*x + c))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 2*(a*b*d*f*x + a*b*d*e - 3*(a*b*d*f*x + a*
b*d*e)*cosh(d*x + c)^2 + (2*a^2*d*f*x + 2*a^2*d*e + a^2*f)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d^2*cosh(d*x + c
)^4 + 4*a^3*d^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d^2*sinh(d*x + c)^4 - 2*a^3*d^2*cosh(d*x + c)^2 + a^3*d^2
+ 2*(3*a^3*d^2*cosh(d*x + c)^2 - a^3*d^2)*sinh(d*x + c)^2 + 4*(a^3*d^2*cosh(d*x + c)^3 - a^3*d^2*cosh(d*x + c)
)*sinh(d*x + c))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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