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

Optimal. Leaf size=322 \[ -\frac{f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a b^2 d^2}-\frac{f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a b^2 d^2}+\frac{f \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}-\frac{\left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a b^2 d}-\frac{\left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a b^2 d}+\frac{\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}+\frac{(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac{(e+f x)^2}{2 a f}-\frac{f \cosh (c+d x)}{b d^2}+\frac{(e+f x) \sinh (c+d x)}{b d} \]

[Out]

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

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Rubi [A]  time = 0.591985, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 13, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.406, Rules used = {5585, 5450, 5446, 2635, 8, 3716, 2190, 2279, 2391, 5565, 3296, 2638, 5561} \[ -\frac{f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a b^2 d^2}-\frac{f \left (a^2+b^2\right ) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a b^2 d^2}+\frac{f \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}-\frac{\left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a b^2 d}-\frac{\left (a^2+b^2\right ) (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a b^2 d}+\frac{\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}+\frac{(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac{(e+f x)^2}{2 a f}-\frac{f \cosh (c+d x)}{b d^2}+\frac{(e+f x) \sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5585

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

Rule 5450

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

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 5565

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

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

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

Mathematica [A]  time = 1.92519, size = 296, normalized size = 0.92 \[ \frac{\left (a^2+b^2\right ) \left (-f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )-f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-d e \log (a+b \sinh (c+d x))+c f \log (a+b \sinh (c+d x))+\frac{1}{2} f (c+d x)^2\right )+\frac{1}{2} b^2 f \left ((c+d x) \left (2 \log \left (1-e^{-2 (c+d x)}\right )+c+d x\right )-\text{PolyLog}\left (2,e^{-2 (c+d x)}\right )\right )+a b d (e+f x) \sinh (c+d x)-a b f \cosh (c+d x)+b^2 d e \log (\sinh (c+d x))-b^2 c f \log (\sinh (c+d x))}{a b^2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.255, size = 932, normalized size = 2.9 \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)*cosh(d*x+c)^2*coth(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, e{\left (\frac{2 \,{\left (d x + c\right )} a}{b^{2} d} - \frac{e^{\left (d x + c\right )}}{b d} + \frac{e^{\left (-d x - c\right )}}{b d} - \frac{2 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac{2 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} + \frac{2 \,{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a b^{2} d}\right )} - \frac{1}{4} \, f{\left (\frac{2 \,{\left (a d^{2} x^{2} e^{c} -{\left (b d x e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} +{\left (b d x + b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{b^{2} d^{2}} - \int \frac{8 \,{\left ({\left (a^{3} e^{c} + a b^{2} e^{c}\right )} x e^{\left (d x\right )} -{\left (a^{2} b + b^{3}\right )} x\right )}}{a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} b^{2} e^{\left (d x + c\right )} - a b^{3}}\,{d x} + 4 \, \int \frac{x}{a e^{\left (d x + c\right )} + a}\,{d x} - 4 \, \int \frac{x}{a e^{\left (d x + c\right )} - a}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.99753, size = 2799, normalized size = 8.69 \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)*cosh(d*x+c)^2*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(a*b*d*f*x + a*b*d*e + a*b*f - (a*b*d*f*x + a*b*d*e - a*b*f)*cosh(d*x + c)^2 - (a*b*d*f*x + a*b*d*e - a*b
*f)*sinh(d*x + c)^2 - (a^2*d^2*f*x^2 + 2*a^2*d^2*e*x + 4*a^2*c*d*e - 2*a^2*c^2*f)*cosh(d*x + c) + 2*((a^2 + b^
2)*f*cosh(d*x + c) + (a^2 + b^2)*f*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) + 2*((a^2 + b^2)*f*cosh(d*x + c) + (a^2 + b^2)*f*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) - 2*(b^2*f*cosh(d*x + c) + b^2*f*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) - 2*(b^2*f*cosh
(d*x + c) + b^2*f*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) + 2*(((a^2 + b^2)*d*e - (a^2 + b^2)*c*f
)*cosh(d*x + c) + ((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*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) + 2*(((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*cosh(d*x + c) + ((a^2 + b^2)*d*e
- (a^2 + b^2)*c*f)*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)
+ 2*(((a^2 + b^2)*d*f*x + (a^2 + b^2)*c*f)*cosh(d*x + c) + ((a^2 + b^2)*d*f*x + (a^2 + b^2)*c*f)*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)
+ 2*(((a^2 + b^2)*d*f*x + (a^2 + b^2)*c*f)*cosh(d*x + c) + ((a^2 + b^2)*d*f*x + (a^2 + b^2)*c*f)*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)
- 2*((b^2*d*f*x + b^2*d*e)*cosh(d*x + c) + (b^2*d*f*x + b^2*d*e)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x
+ c) + 1) - 2*((b^2*d*e - b^2*c*f)*cosh(d*x + c) + (b^2*d*e - b^2*c*f)*sinh(d*x + c))*log(cosh(d*x + c) + sinh
(d*x + c) - 1) - 2*((b^2*d*f*x + b^2*c*f)*cosh(d*x + c) + (b^2*d*f*x + b^2*c*f)*sinh(d*x + c))*log(-cosh(d*x +
c) - sinh(d*x + c) + 1) - (a^2*d^2*f*x^2 + 2*a^2*d^2*e*x + 4*a^2*c*d*e - 2*a^2*c^2*f + 2*(a*b*d*f*x + a*b*d*e
- a*b*f)*cosh(d*x + c))*sinh(d*x + c))/(a*b^2*d^2*cosh(d*x + c) + a*b^2*d^2*sinh(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [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)*cosh(d*x+c)^2*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out