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

Optimal. Leaf size=327 \[ \frac {b^2 e x}{a^3}+\frac {b^2 f x^2}{2 a^3}+\frac {b f \sinh (c+d x)}{a^2 d^2}-\frac {b (e+f x) \cosh (c+d x)}{a^2 d}-\frac {b f \sqrt {a^2+b^2} \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b f \sqrt {a^2+b^2} \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^3 d}-\frac {f \cosh ^2(c+d x)}{4 a d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {e x}{2 a}+\frac {f x^2}{4 a} \]

[Out]

1/2*e*x/a+b^2*e*x/a^3+1/4*f*x^2/a+1/2*b^2*f*x^2/a^3-b*(f*x+e)*cosh(d*x+c)/a^2/d-1/4*f*cosh(d*x+c)^2/a/d^2+b*f*
sinh(d*x+c)/a^2/d^2+1/2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/a/d-b*(f*x+e)*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(
a^2+b^2)^(1/2)/a^3/d+b*(f*x+e)*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d-b*f*polylog(2,-a*e
xp(d*x+c)/(b-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a^3/d^2+b*f*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))*(a^2+b
^2)^(1/2)/a^3/d^2

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Rubi [A]  time = 0.64, antiderivative size = 327, normalized size of antiderivative = 1.00, 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 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 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 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 2637

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

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 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 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 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 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 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]

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*}

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Mathematica [C]  time = 4.30, size = 1581, 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*I)*b*Pi*ArcTanh[(-a + b*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(Sqr
t[a^2 + b^2]*d^2) + (2*b*(2*(c + I*ArcCos[((-I)*b)/a])*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sq
rt[-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]])*L
og[((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]*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]])*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*Ar
cTan[((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^(-1/2*c - (d*x)/2))/(Sqrt[2]*S
qrt[(-I)*a]*Sqrt[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])/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)^(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*(Pol
yLog[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*T
anh[(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] + Si
nh[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])] + PolyLog[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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fricas [B]  time = 0.44, size = 1284, normalized size = 3.93 \[ \text {result too large to display} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2}}{b \operatorname {csch}\left (d x + c\right ) + a}\,{d x} \]

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)

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maple [B]  time = 1.00, size = 1012, normalized size = 3.09 \[ \frac {f \,x^{2}}{4 a}+\frac {b^{2} f \,x^{2}}{2 a^{3}}+\frac {e x}{2 a}+\frac {b^{2} e x}{a^{3}}+\frac {\left (2 d f x +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 a \,d^{2}}-\frac {b \left (d f x +d e -f \right ) {\mathrm e}^{d x +c}}{2 a^{2} d^{2}}-\frac {b \left (d f x +d e +f \right ) {\mathrm e}^{-d x -c}}{2 a^{2} d^{2}}-\frac {\left (2 d f x +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{2}}+\frac {2 b^{3} e \arctanh \left (\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{3} \sqrt {a^{2}+b^{2}}}+\frac {2 b e \arctanh \left (\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d a \sqrt {a^{2}+b^{2}}}-\frac {b f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) x}{d a \sqrt {a^{2}+b^{2}}}-\frac {b f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) x}{d a \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {b f \dilog \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {b f \dilog \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {b^{3} f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{3} f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}-\frac {b^{3} f \dilog \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}+\frac {b^{3} f \dilog \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}}-\frac {2 b f c \arctanh \left (\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {2 b^{3} f c \arctanh \left (\frac {2 a \,{\mathrm e}^{d x +c}+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3} \sqrt {a^{2}+b^{2}}} \]

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/d*b^3/a^3*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/d*b/a*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))-1/d*b/a*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-1/d^2*b/a*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+1/d*b/a*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+1/d^2*b/a*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-1/d^2*b/a*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)))+1/d^2*b/a*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)))-1/d*b^3/a^3*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-1/d^2*b^3/a^3*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+1/d*b^3/a^3*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+1/d^2*b^3/a^3*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-1/d^2*b^3/a^3*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)))+1/d^2*b^3/a^3*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/d^2*b/a*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/d^2*b^
3/a^3*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]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{16} \, {\left (32 \, {\left (a^{2} b e^{c} + b^{3} e^{c}\right )} \int \frac {x e^{\left (d x\right )}}{a^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b e^{\left (d x + c\right )} - a^{4}}\,{d x} - \frac {{\left (4 \, {\left (a^{2} d^{2} e^{\left (2 \, c\right )} + 2 \, b^{2} d^{2} e^{\left (2 \, c\right )}\right )} x^{2} + {\left (2 \, a^{2} d x e^{\left (4 \, c\right )} - a^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 8 \, {\left (a b d x e^{\left (3 \, c\right )} - a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} - 8 \, {\left (a b d x e^{c} + a b e^{c}\right )} e^{\left (-d x\right )} - {\left (2 \, a^{2} d x + a^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{a^{3} d^{2}}\right )} f - \frac {1}{8} \, e {\left (\frac {{\left (4 \, b e^{\left (-d x - c\right )} - a\right )} e^{\left (2 \, d x + 2 \, c\right )}}{a^{2} d} - \frac {4 \, {\left (a^{2} + 2 \, b^{2}\right )} {\left (d x + c\right )}}{a^{3} d} + \frac {4 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )}}{a^{2} d} + \frac {8 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {a e^{\left (-d x - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3} d}\right )} \]

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]

-1/16*(32*(a^2*b*e^c + b^3*e^c)*integrate(x*e^(d*x)/(a^4*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^4), x) - (4
*(a^2*d^2*e^(2*c) + 2*b^2*d^2*e^(2*c))*x^2 + (2*a^2*d*x*e^(4*c) - a^2*e^(4*c))*e^(2*d*x) - 8*(a*b*d*x*e^(3*c)
- a*b*e^(3*c))*e^(d*x) - 8*(a*b*d*x*e^c + a*b*e^c)*e^(-d*x) - (2*a^2*d*x + a^2)*e^(-2*d*x))*e^(-2*c)/(a^3*d^2)
)*f - 1/8*e*((4*b*e^(-d*x - c) - a)*e^(2*d*x + 2*c)/(a^2*d) - 4*(a^2 + 2*b^2)*(d*x + c)/(a^3*d) + (4*b*e^(-d*x
- c) + a*e^(-2*d*x - 2*c))/(a^2*d) + 8*(a^2*b + b^3)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^2))/(a*e^(-d*x -
c) - b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right ) \cosh ^{2}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]

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]

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

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