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

Optimal. Leaf size=243 \[ \frac{b f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{b f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2}-\frac{b f \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}+\frac{b (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d}+\frac{b (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d}-\frac{b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{(e+f x) \text{csch}(c+d x)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.46413, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5587, 5452, 3770, 5569, 3716, 2190, 2279, 2391, 5561} \[ \frac{b f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{b f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2}-\frac{b f \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}+\frac{b (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d}+\frac{b (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d}-\frac{b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{(e+f x) \text{csch}(c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((f*ArcTanh[Cosh[c + d*x]])/(a*d^2)) - ((e + f*x)*Csch[c + d*x])/(a*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))
/(a - Sqrt[a^2 + b^2])])/(a^2*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*d) - (b*(
e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d
^2) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) - (b*f*PolyLog[2, E^(2*(c + d*x))])
/(2*a^2*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 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}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \coth (c+d x) \text{csch}(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac{(e+f x) \text{csch}(c+d x)}{a d}-\frac{b \int (e+f x) \coth (c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac{f \int \text{csch}(c+d x) \, dx}{a d}\\ &=-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{(e+f x) \text{csch}(c+d x)}{a d}+\frac{(2 b) \int \frac{e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a^2}+\frac{b^2 \int \frac{e^{c+d x} (e+f x)}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a^2}+\frac{b^2 \int \frac{e^{c+d x} (e+f x)}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a^2}\\ &=-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{(e+f x) \text{csch}(c+d x)}{a d}+\frac{b (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d}+\frac{b (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d}-\frac{b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{(b f) \int \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a^2 d}-\frac{(b f) \int \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a^2 d}+\frac{(b f) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d}\\ &=-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{(e+f x) \text{csch}(c+d x)}{a d}+\frac{b (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d}+\frac{b (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d}-\frac{b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac{(b f) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac{(b f) \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^2 d^2}-\frac{(b f) \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^2 d^2}\\ &=-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{(e+f x) \text{csch}(c+d x)}{a d}+\frac{b (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d}+\frac{b (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d}-\frac{b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac{b f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{b f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^2}-\frac{b f \text{Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}\\ \end{align*}

Mathematica [A]  time = 1.89411, size = 416, normalized size = 1.71 \[ \frac{2 b f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+2 b f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+b f \text{PolyLog}\left (2,e^{-2 (c+d x)}\right )+2 b c f \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+2 b c f \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+2 b d f x \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+2 b d f x \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+2 b d e \log (a+b \sinh (c+d x))-2 b c f \log (a+b \sinh (c+d x))+a d e \tanh \left (\frac{1}{2} (c+d x)\right )-a d e \coth \left (\frac{1}{2} (c+d x)\right )+a d f x \tanh \left (\frac{1}{2} (c+d x)\right )-a d f x \coth \left (\frac{1}{2} (c+d x)\right )+2 a f \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-2 b c^2 f-2 b d e \log (\sinh (c+d x))-4 b c d f x-2 b c f \log \left (1-e^{-2 (c+d x)}\right )-2 b d f x \log \left (1-e^{-2 (c+d x)}\right )+2 b c f \log (\sinh (c+d x))-2 b d^2 f x^2}{2 a^2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.138, size = 528, normalized size = 2.2 \begin{align*} -2\,{\frac{ \left ( fx+e \right ){{\rm e}^{dx+c}}}{da \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) }}+{\frac{bfc\ln \left ({{\rm e}^{dx+c}}-1 \right ) }{{a}^{2}{d}^{2}}}-{\frac{bfc\ln \left ( b{{\rm e}^{2\,dx+2\,c}}+2\,a{{\rm e}^{dx+c}}-b \right ) }{{a}^{2}{d}^{2}}}-{\frac{f\ln \left ({{\rm e}^{dx+c}}+1 \right ) }{a{d}^{2}}}+{\frac{f\ln \left ({{\rm e}^{dx+c}}-1 \right ) }{a{d}^{2}}}+{\frac{bf{\it dilog} \left ({{\rm e}^{dx+c}} \right ) }{{a}^{2}{d}^{2}}}+{\frac{bf}{{a}^{2}{d}^{2}}{\it dilog} \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }+{\frac{bf}{{a}^{2}{d}^{2}}{\it dilog} \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{bf{\it dilog} \left ({{\rm e}^{dx+c}}+1 \right ) }{{a}^{2}{d}^{2}}}-{\frac{be\ln \left ({{\rm e}^{dx+c}}-1 \right ) }{{a}^{2}d}}+{\frac{be\ln \left ( b{{\rm e}^{2\,dx+2\,c}}+2\,a{{\rm e}^{dx+c}}-b \right ) }{{a}^{2}d}}-{\frac{be\ln \left ({{\rm e}^{dx+c}}+1 \right ) }{{a}^{2}d}}+{\frac{bfx}{{a}^{2}d}\ln \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }+{\frac{bfc}{{a}^{2}{d}^{2}}\ln \left ({ \left ( -b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}-a \right ) \left ( -a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }+{\frac{bfx}{{a}^{2}d}\ln \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }+{\frac{bfc}{{a}^{2}{d}^{2}}\ln \left ({ \left ( b{{\rm e}^{dx+c}}+\sqrt{{a}^{2}+{b}^{2}}+a \right ) \left ( a+\sqrt{{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{bf\ln \left ({{\rm e}^{dx+c}}+1 \right ) x}{{a}^{2}d}} \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)/(a+b*sinh(d*x+c)),x)

[Out]

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

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

[Out]

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

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Fricas [B]  time = 2.40876, size = 3152, normalized size = 12.97 \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)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(2*(a*d*f*x + a*d*e)*cosh(d*x + c) - (b*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x
+ c)^2 - b*f)*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*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x + c)^2 - b*f)*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*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x + c)^2 - b*f)*dilog(cosh(d*x + c) + si
nh(d*x + c)) + (b*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x + c)^2 - b*f)*dilog(-co
sh(d*x + c) - sinh(d*x + c)) + (b*d*e - b*c*f - (b*d*e - b*c*f)*cosh(d*x + c)^2 - 2*(b*d*e - b*c*f)*cosh(d*x +
c)*sinh(d*x + c) - (b*d*e - b*c*f)*sinh(d*x + c)^2)*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*d*e - b*c*f - (b*d*e - b*c*f)*cosh(d*x + c)^2 - 2*(b*d*e - b*c*f)*cosh(d*x + c)*sinh(
d*x + c) - (b*d*e - b*c*f)*sinh(d*x + c)^2)*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*d*f*x + b*c*f - (b*d*f*x + b*c*f)*cosh(d*x + c)^2 - 2*(b*d*f*x + b*c*f)*cosh(d*x + c)*sinh(d*x
+ c) - (b*d*f*x + b*c*f)*sinh(d*x + c)^2)*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*d*f*x + b*c*f - (b*d*f*x + b*c*f)*cosh(d*x + c)^2 - 2*(b*d*f*x +
b*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f*x + b*c*f)*sinh(d*x + c)^2)*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*d*f*x + b*d*e - (b*d*f*x + b*d*e +
a*f)*cosh(d*x + c)^2 - 2*(b*d*f*x + b*d*e + a*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f*x + b*d*e + a*f)*sinh(d
*x + c)^2 + a*f)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (b*d*e - (b*d*e - (b*c + a)*f)*cosh(d*x + c)^2 - 2*(
b*d*e - (b*c + a)*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*e - (b*c + a)*f)*sinh(d*x + c)^2 - (b*c + a)*f)*log(co
sh(d*x + c) + sinh(d*x + c) - 1) - (b*d*f*x + b*c*f - (b*d*f*x + b*c*f)*cosh(d*x + c)^2 - 2*(b*d*f*x + b*c*f)*
cosh(d*x + c)*sinh(d*x + c) - (b*d*f*x + b*c*f)*sinh(d*x + c)^2)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) + 2*(
a*d*f*x + a*d*e)*sinh(d*x + c))/(a^2*d^2*cosh(d*x + c)^2 + 2*a^2*d^2*cosh(d*x + c)*sinh(d*x + c) + a^2*d^2*sin
h(d*x + c)^2 - a^2*d^2)

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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)/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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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 )}{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)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

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