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

Optimal. Leaf size=327 \[ -\frac{b f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{b f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^3 d^2}-\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}+1\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}+1\right )}{a^3 d}+\frac{b^2 e x}{a^3}+\frac{b^2 f x^2}{2 a^3}+\frac{b f \sinh (c+d x)}{a^2 d^2}-\frac{b (e+f x) \cosh (c+d x)}{a^2 d}-\frac{f \cosh ^2(c+d x)}{4 a d^2}+\frac{(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac{e x}{2 a}+\frac{f x^2}{4 a} \]

[Out]

(e*x)/(2*a) + (b^2*e*x)/a^3 + (f*x^2)/(4*a) + (b^2*f*x^2)/(2*a^3) - (b*(e + f*x)*Cosh[c + d*x])/(a^2*d) - (f*C
osh[c + d*x]^2)/(4*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^3*
d) + (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3*d) - (b*Sqrt[a^2 + b^2]
*f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*Sqrt[a^2 + b^2]*f*PolyLog[2, -((a*E^(c
+ d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*f*Sinh[c + d*x])/(a^2*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c
+ d*x])/(2*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.639711, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {5594, 5579, 3310, 5565, 3296, 2637, 3322, 2264, 2190, 2279, 2391} \[ -\frac{b f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{b f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}\right )}{a^3 d^2}-\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}+1\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (\frac{a e^{c+d x}}{\sqrt{a^2+b^2}+b}+1\right )}{a^3 d}+\frac{b^2 e x}{a^3}+\frac{b^2 f x^2}{2 a^3}+\frac{b f \sinh (c+d x)}{a^2 d^2}-\frac{b (e+f x) \cosh (c+d x)}{a^2 d}-\frac{f \cosh ^2(c+d x)}{4 a d^2}+\frac{(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac{e x}{2 a}+\frac{f x^2}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x)/(2*a) + (b^2*e*x)/a^3 + (f*x^2)/(4*a) + (b^2*f*x^2)/(2*a^3) - (b*(e + f*x)*Cosh[c + d*x])/(a^2*d) - (f*C
osh[c + d*x]^2)/(4*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^3*
d) + (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3*d) - (b*Sqrt[a^2 + b^2]
*f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*Sqrt[a^2 + b^2]*f*PolyLog[2, -((a*E^(c
+ d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*f*Sinh[c + d*x])/(a^2*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c
+ d*x])/(2*a*d)

Rule 5594

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n)/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]

Rule 5579

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[((e + f*x)^m*Cosh[c + d*x]^p*Sinh[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 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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]

Rubi steps

\begin{align*} \int \frac{(e+f x) \cosh ^2(c+d x)}{a+b \text{csch}(c+d x)} \, dx &=\int \frac{(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx\\ &=\frac{\int (e+f x) \cosh ^2(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x) \cosh ^2(c+d x)}{b+a \sinh (c+d x)} \, dx}{a}\\ &=-\frac{f \cosh ^2(c+d x)}{4 a d^2}+\frac{(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{\int (e+f x) \, dx}{2 a}-\frac{b \int (e+f x) \sinh (c+d x) \, dx}{a^2}+\frac{b^2 \int (e+f x) \, dx}{a^3}-\frac{\left (b \left (a^2+b^2\right )\right ) \int \frac{e+f x}{b+a \sinh (c+d x)} \, dx}{a^3}\\ &=\frac{e x}{2 a}+\frac{b^2 e x}{a^3}+\frac{f x^2}{4 a}+\frac{b^2 f x^2}{2 a^3}-\frac{b (e+f x) \cosh (c+d x)}{a^2 d}-\frac{f \cosh ^2(c+d x)}{4 a d^2}+\frac{(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac{\left (2 b \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx}{a^3}+\frac{(b f) \int \cosh (c+d x) \, dx}{a^2 d}\\ &=\frac{e x}{2 a}+\frac{b^2 e x}{a^3}+\frac{f x^2}{4 a}+\frac{b^2 f x^2}{2 a^3}-\frac{b (e+f x) \cosh (c+d x)}{a^2 d}-\frac{f \cosh ^2(c+d x)}{4 a d^2}+\frac{b f \sinh (c+d x)}{a^2 d^2}+\frac{(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac{\left (2 b \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)}{2 b-2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx}{a^2}+\frac{\left (2 b \sqrt{a^2+b^2}\right ) \int \frac{e^{c+d x} (e+f x)}{2 b+2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx}{a^2}\\ &=\frac{e x}{2 a}+\frac{b^2 e x}{a^3}+\frac{f x^2}{4 a}+\frac{b^2 f x^2}{2 a^3}-\frac{b (e+f x) \cosh (c+d x)}{a^2 d}-\frac{f \cosh ^2(c+d x)}{4 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b f \sinh (c+d x)}{a^2 d^2}+\frac{(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{\left (b \sqrt{a^2+b^2} f\right ) \int \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 d}-\frac{\left (b \sqrt{a^2+b^2} f\right ) \int \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 d}\\ &=\frac{e x}{2 a}+\frac{b^2 e x}{a^3}+\frac{f x^2}{4 a}+\frac{b^2 f x^2}{2 a^3}-\frac{b (e+f x) \cosh (c+d x)}{a^2 d}-\frac{f \cosh ^2(c+d x)}{4 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b f \sinh (c+d x)}{a^2 d^2}+\frac{(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac{\left (b \sqrt{a^2+b^2} f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b-2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}-\frac{\left (b \sqrt{a^2+b^2} f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b+2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}\\ &=\frac{e x}{2 a}+\frac{b^2 e x}{a^3}+\frac{f x^2}{4 a}+\frac{b^2 f x^2}{2 a^3}-\frac{b (e+f x) \cosh (c+d x)}{a^2 d}-\frac{f \cosh ^2(c+d x)}{4 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d}-\frac{b \sqrt{a^2+b^2} f \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{b \sqrt{a^2+b^2} f \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{b f \sinh (c+d x)}{a^2 d^2}+\frac{(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}\\ \end{align*}

Mathematica [C]  time = 5.65869, size = 1579, normalized size = 4.83 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Csch[c + d*x]*(b + a*Sinh[c + d*x])*(2*a^2*e*(c/d + x - (2*b*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2
]])/(Sqrt[-a^2 - b^2]*d)) + a^2*f*(x^2 + (2*b*((I*Pi*ArcTanh[(-a + b*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/Sqrt
[a^2 + b^2] + (2*(c + I*ArcCos[((-I)*b)/a])*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b
^2]] + ((-2*I)*c + Pi - (2*I)*d*x)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]]
- (ArcCos[((-I)*b)/a] - 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[((a + I*
b)*(a - I*b + Sqrt[-a^2 - b^2])*(1 + I*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Co
t[((2*I)*c + Pi + (2*I)*d*x)/4]))] - (ArcCos[((-I)*b)/a] + 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/
4])/Sqrt[-a^2 - b^2]])*Log[(I*(a + I*b)*(-a + I*b + Sqrt[-a^2 - b^2])*(I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))
/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] + (ArcCos[((-I)*b)/a] + 2*ArcTan[((a -
I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (2*I)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2
*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[-a^2 - b^2]*E^(-c/2 - (d*x)/2))/(Sqrt[2]*Sqrt[(-I)*a]*S
qrt[b + a*Sinh[c + d*x]]))] + (ArcCos[((-I)*b)/a] - 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqr
t[-a^2 - b^2]] + (2*I)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[((-1)^(
1/4)*Sqrt[-a^2 - b^2]*E^((c + d*x)/2))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b + a*Sinh[c + d*x]])] + I*(PolyLog[2, ((I*b
+ Sqrt[-a^2 - b^2])*(a + I*b - I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a
^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] - PolyLog[2, ((b + I*Sqrt[-a^2 - b^2])*(I*a - b + Sqrt[-a^2 - b
^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))])
)/Sqrt[-a^2 - b^2]))/d^2) + (2*e*((a^2 + 4*b^2)*(c + d*x) - (2*b*(3*a^2 + 4*b^2)*ArcTan[(a - b*Tanh[(c + d*x)/
2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + a^2*Sinh[2*(c + d*x)]))/d + (f*((a^2 + 4*b^2)*
(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d*x] - a^2*Cosh[2*(c + d*x)] - (2*b*(3*a^2 + 4*b^2)*(2*c*ArcTanh[(b
+ a*Cosh[c + d*x] + a*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/
(b - Sqrt[a^2 + b^2])] - (c + d*x)*Log[1 + (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(b + Sqrt[a^2 + b^2])] + PolyLo
g[2, (a*(Cosh[c + d*x] + Sinh[c + d*x]))/(-b + Sqrt[a^2 + b^2])] - PolyLog[2, -((a*(Cosh[c + d*x] + Sinh[c + d
*x]))/(b + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*a^2*d*x*Sinh[2*(c + d*x)]))/d^2))/(8
*a^3*(a + b*Csch[c + d*x]))

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Maple [B]  time = 0.184, size = 1012, normalized size = 3.1 \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/(a+b*csch(d*x+c)),x)

[Out]

1/4*f*x^2/a+1/2*b^2*f*x^2/a^3+1/2*e*x/a+b^2*e*x/a^3+1/16*(2*d*f*x+2*d*e-f)/a/d^2*exp(2*d*x+2*c)-1/2*b*(d*f*x+d
*e-f)/a^2/d^2*exp(d*x+c)-1/2*b*(d*f*x+d*e+f)/a^2/d^2*exp(-d*x-c)-1/16*(2*d*f*x+2*d*e+f)/a/d^2*exp(-2*d*x-2*c)+
2*b^3/a^3/d*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))+2*b/a/d*e/(a^2+b^2)^(1/2)*arct
anh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))-b/a/d*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b
+(a^2+b^2)^(1/2)))*x-b/a/d^2*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*c+b/
a/d*f/(a^2+b^2)^(1/2)*ln((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*x+b/a/d^2*f/(a^2+b^2)^(1/2)*ln(
(a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*c-b/a/d^2*f/(a^2+b^2)^(1/2)*dilog((-a*exp(d*x+c)+(a^2+b^
2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))+b/a/d^2*f/(a^2+b^2)^(1/2)*dilog((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2
)^(1/2)))-b^3/a^3/d*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*x-b^3/a^3/d^2
*f/(a^2+b^2)^(1/2)*ln((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+(a^2+b^2)^(1/2)))*c+b^3/a^3/d*f/(a^2+b^2)^(1/2)*ln
((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*x+b^3/a^3/d^2*f/(a^2+b^2)^(1/2)*ln((a*exp(d*x+c)+(a^2+b
^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))*c-b^3/a^3/d^2*f/(a^2+b^2)^(1/2)*dilog((-a*exp(d*x+c)+(a^2+b^2)^(1/2)-b)/(-b+
(a^2+b^2)^(1/2)))+b^3/a^3/d^2*f/(a^2+b^2)^(1/2)*dilog((a*exp(d*x+c)+(a^2+b^2)^(1/2)+b)/(b+(a^2+b^2)^(1/2)))-2*
b/a/d^2*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))-2*b^3/a^3/d^2*f*c/(a^2+b^2)^(1/2
)*arctanh(1/2*(2*a*exp(d*x+c)+2*b)/(a^2+b^2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.88607, size = 3200, normalized size = 9.79 \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/(a+b*csch(d*x+c)),x, algorithm="fricas")

[Out]

-1/16*(2*a^2*d*f*x - (2*a^2*d*f*x + 2*a^2*d*e - a^2*f)*cosh(d*x + c)^4 - (2*a^2*d*f*x + 2*a^2*d*e - a^2*f)*sin
h(d*x + c)^4 + 2*a^2*d*e + 8*(a*b*d*f*x + a*b*d*e - a*b*f)*cosh(d*x + c)^3 + 4*(2*a*b*d*f*x + 2*a*b*d*e - 2*a*
b*f - (2*a^2*d*f*x + 2*a^2*d*e - a^2*f)*cosh(d*x + c))*sinh(d*x + c)^3 + a^2*f - 4*((a^2 + 2*b^2)*d^2*f*x^2 +
2*(a^2 + 2*b^2)*d^2*e*x)*cosh(d*x + c)^2 - 2*(2*(a^2 + 2*b^2)*d^2*f*x^2 + 4*(a^2 + 2*b^2)*d^2*e*x + 3*(2*a^2*d
*f*x + 2*a^2*d*e - a^2*f)*cosh(d*x + c)^2 - 12*(a*b*d*f*x + a*b*d*e - a*b*f)*cosh(d*x + c))*sinh(d*x + c)^2 +
16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*cosh(d*x + c)*sinh(d*x + c) + a*b*f*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)
*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a +
1) - 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*cosh(d*x + c)*sinh(d*x + c) + a*b*f*sinh(d*x + c)^2)*sqrt((a^2 + b^2
)/a^2)*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) -
a)/a + 1) - 16*((a*b*d*e - a*b*c*f)*cosh(d*x + c)^2 + 2*(a*b*d*e - a*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b
*d*e - a*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a
^2 + b^2)/a^2) + 2*b) + 16*((a*b*d*e - a*b*c*f)*cosh(d*x + c)^2 + 2*(a*b*d*e - a*b*c*f)*cosh(d*x + c)*sinh(d*x
+ c) + (a*b*d*e - a*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) -
2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 16*((a*b*d*f*x + a*b*c*f)*cosh(d*x + c)^2 + 2*(a*b*d*f*x + a*b*c*f)*cosh(d
*x + c)*sinh(d*x + c) + (a*b*d*f*x + a*b*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/a^2)*log(-(b*cosh(d*x + c) + b
*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) - 16*((a*b*d*f*x + a*b*c*f)
*cosh(d*x + c)^2 + 2*(a*b*d*f*x + a*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f*x + a*b*c*f)*sinh(d*x + c)^2
)*sqrt((a^2 + b^2)/a^2)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^
2 + b^2)/a^2) - a)/a) + 8*(a*b*d*f*x + a*b*d*e + a*b*f)*cosh(d*x + c) + 4*(2*a*b*d*f*x + 2*a*b*d*e - (2*a^2*d*
f*x + 2*a^2*d*e - a^2*f)*cosh(d*x + c)^3 + 2*a*b*f + 6*(a*b*d*f*x + a*b*d*e - a*b*f)*cosh(d*x + c)^2 - 2*((a^2
+ 2*b^2)*d^2*f*x^2 + 2*(a^2 + 2*b^2)*d^2*e*x)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d^2*cosh(d*x + c)^2 + 2*a^3*
d^2*cosh(d*x + c)*sinh(d*x + c) + a^3*d^2*sinh(d*x + c)^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)*cosh(d*x+c)**2/(a+b*csch(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 )} \cosh \left (d x + c\right )^{2}}{b \operatorname{csch}\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)*cosh(d*x+c)^2/(a+b*csch(d*x+c)),x, algorithm="giac")

[Out]

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