\(\int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [369]

Optimal result

Integrand size = 34, antiderivative size = 403 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 e x}{b^4}-\frac {a e x}{2 b^2}-\frac {a^3 f x^2}{2 b^4}-\frac {a f x^2}{4 b^2}+\frac {a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^2 f \sinh (c+d x)}{b^3 d^2}-\frac {f \sinh (c+d x)}{3 b d^2}-\frac {a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f \sinh ^3(c+d x)}{9 b d^2} \]

[Out]

-a^3*e*x/b^4-1/2*a*e*x/b^2-1/2*a^3*f*x^2/b^4-1/4*a*f*x^2/b^2+a^2*(f*x+e)*cosh(d*x+c)/b^3/d+1/4*a*f*cosh(d*x+c)
^2/b^2/d^2+1/3*(f*x+e)*cosh(d*x+c)^3/b/d-a^2*f*sinh(d*x+c)/b^3/d^2-1/3*f*sinh(d*x+c)/b/d^2-1/2*a*(f*x+e)*cosh(
d*x+c)*sinh(d*x+c)/b^2/d-1/9*f*sinh(d*x+c)^3/b/d^2+a^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2
)^(1/2)/b^4/d-a^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^4/d+a^2*f*polylog(2,-b*exp(
d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^4/d^2-a^2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^
2)^(1/2)/b^4/d^2

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5698, 5555, 2713, 3391, 5684, 3377, 2717, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 e x}{b^4}-\frac {a^3 f x^2}{2 b^4}-\frac {a^2 f \sinh (c+d x)}{b^3 d^2}+\frac {a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac {a^2 f \sqrt {a^2+b^2} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^2 f \sqrt {a^2+b^2} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^4 d}+\frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}-\frac {a (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^2 d}-\frac {a e x}{2 b^2}-\frac {a f x^2}{4 b^2}-\frac {f \sinh ^3(c+d x)}{9 b d^2}-\frac {f \sinh (c+d x)}{3 b d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d} \]

[In]

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

[Out]

-((a^3*e*x)/b^4) - (a*e*x)/(2*b^2) - (a^3*f*x^2)/(2*b^4) - (a*f*x^2)/(4*b^2) + (a^2*(e + f*x)*Cosh[c + d*x])/(
b^3*d) + (a*f*Cosh[c + d*x]^2)/(4*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^3)/(3*b*d) + (a^2*Sqrt[a^2 + b^2]*(e + f
*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^2*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) + (a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b
^2]))])/(b^4*d^2) - (a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) - (
a^2*f*Sinh[c + d*x])/(b^3*d^2) - (f*Sinh[c + d*x])/(3*b*d^2) - (a*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^
2*d) - (f*Sinh[c + d*x]^3)/(9*b*d^2)

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2438

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 2713

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

Rule 2717

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

Rule 3377

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 3391

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 3403

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 5555

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

Rule 5684

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 5698

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \cosh ^2(c+d x) \sinh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = \frac {(e+f x) \cosh ^3(c+d x)}{3 b d}-\frac {a \int (e+f x) \cosh ^2(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {f \int \cosh ^3(c+d x) \, dx}{3 b d} \\ & = \frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d}-\frac {a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {a^3 \int (e+f x) \, dx}{b^4}+\frac {a^2 \int (e+f x) \sinh (c+d x) \, dx}{b^3}-\frac {a \int (e+f x) \, dx}{2 b^2}+\frac {\left (a^2 \left (a^2+b^2\right )\right ) \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{b^4}-\frac {(i f) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{3 b d^2} \\ & = -\frac {a^3 e x}{b^4}-\frac {a e x}{2 b^2}-\frac {a^3 f x^2}{2 b^4}-\frac {a f x^2}{4 b^2}+\frac {a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d}-\frac {f \sinh (c+d x)}{3 b d^2}-\frac {a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f \sinh ^3(c+d x)}{9 b d^2}+\frac {\left (2 a^2 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^4}-\frac {\left (a^2 f\right ) \int \cosh (c+d x) \, dx}{b^3 d} \\ & = -\frac {a^3 e x}{b^4}-\frac {a e x}{2 b^2}-\frac {a^3 f x^2}{2 b^4}-\frac {a f x^2}{4 b^2}+\frac {a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d}-\frac {a^2 f \sinh (c+d x)}{b^3 d^2}-\frac {f \sinh (c+d x)}{3 b d^2}-\frac {a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f \sinh ^3(c+d x)}{9 b d^2}+\frac {\left (2 a^2 \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^3}-\frac {\left (2 a^2 \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^3} \\ & = -\frac {a^3 e x}{b^4}-\frac {a e x}{2 b^2}-\frac {a^3 f x^2}{2 b^4}-\frac {a f x^2}{4 b^2}+\frac {a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 f \sinh (c+d x)}{b^3 d^2}-\frac {f \sinh (c+d x)}{3 b d^2}-\frac {a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f \sinh ^3(c+d x)}{9 b d^2}-\frac {\left (a^2 \sqrt {a^2+b^2} f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}+\frac {\left (a^2 \sqrt {a^2+b^2} f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^4 d} \\ & = -\frac {a^3 e x}{b^4}-\frac {a e x}{2 b^2}-\frac {a^3 f x^2}{2 b^4}-\frac {a f x^2}{4 b^2}+\frac {a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 f \sinh (c+d x)}{b^3 d^2}-\frac {f \sinh (c+d x)}{3 b d^2}-\frac {a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f \sinh ^3(c+d x)}{9 b d^2}-\frac {\left (a^2 \sqrt {a^2+b^2} f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^2}+\frac {\left (a^2 \sqrt {a^2+b^2} f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^2} \\ & = -\frac {a^3 e x}{b^4}-\frac {a e x}{2 b^2}-\frac {a^3 f x^2}{2 b^4}-\frac {a f x^2}{4 b^2}+\frac {a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^2 f \sinh (c+d x)}{b^3 d^2}-\frac {f \sinh (c+d x)}{3 b d^2}-\frac {a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f \sinh ^3(c+d x)}{9 b d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.52 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.55 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {72 a^3 c d e+36 a b^2 c d e-36 a^3 c^2 f-18 a b^2 c^2 f+72 a^3 d^2 e x+36 a b^2 d^2 e x+36 a^3 d^2 f x^2+18 a b^2 d^2 f x^2+144 a^2 \sqrt {a^2+b^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-144 a^2 \sqrt {a^2+b^2} c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-72 a^2 b d e \cosh (c+d x)-18 b^3 d e \cosh (c+d x)-72 a^2 b d f x \cosh (c+d x)-18 b^3 d f x \cosh (c+d x)-9 a b^2 f \cosh (2 (c+d x))-6 b^3 d e \cosh (3 (c+d x))-6 b^3 d f x \cosh (3 (c+d x))-72 a^2 \sqrt {a^2+b^2} c f \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-72 a^2 \sqrt {a^2+b^2} d f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+72 a^2 \sqrt {a^2+b^2} c f \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+72 a^2 \sqrt {a^2+b^2} d f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-72 a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+72 a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+72 a^2 b f \sinh (c+d x)+18 b^3 f \sinh (c+d x)+18 a b^2 d e \sinh (2 (c+d x))+18 a b^2 d f x \sinh (2 (c+d x))+2 b^3 f \sinh (3 (c+d x))}{72 b^4 d^2} \]

[In]

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

[Out]

-1/72*(72*a^3*c*d*e + 36*a*b^2*c*d*e - 36*a^3*c^2*f - 18*a*b^2*c^2*f + 72*a^3*d^2*e*x + 36*a*b^2*d^2*e*x + 36*
a^3*d^2*f*x^2 + 18*a*b^2*d^2*f*x^2 + 144*a^2*Sqrt[a^2 + b^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]]
- 144*a^2*Sqrt[a^2 + b^2]*c*f*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 72*a^2*b*d*e*Cosh[c + d*x] - 18*b
^3*d*e*Cosh[c + d*x] - 72*a^2*b*d*f*x*Cosh[c + d*x] - 18*b^3*d*f*x*Cosh[c + d*x] - 9*a*b^2*f*Cosh[2*(c + d*x)]
- 6*b^3*d*e*Cosh[3*(c + d*x)] - 6*b^3*d*f*x*Cosh[3*(c + d*x)] - 72*a^2*Sqrt[a^2 + b^2]*c*f*Log[1 + (b*E^(c +
d*x))/(a - Sqrt[a^2 + b^2])] - 72*a^2*Sqrt[a^2 + b^2]*d*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 7
2*a^2*Sqrt[a^2 + b^2]*c*f*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 72*a^2*Sqrt[a^2 + b^2]*d*f*x*Log[1
+ (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 72*a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2
+ b^2])] + 72*a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 72*a^2*b*f*Sinh[c +
d*x] + 18*b^3*f*Sinh[c + d*x] + 18*a*b^2*d*e*Sinh[2*(c + d*x)] + 18*a*b^2*d*f*x*Sinh[2*(c + d*x)] + 2*b^3*f*S
inh[3*(c + d*x)])/(b^4*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1127\) vs. \(2(367)=734\).

Time = 21.24 (sec) , antiderivative size = 1128, normalized size of antiderivative = 2.80

method	result	size
risch	\(\text {Expression too large to display}\)	\(1128\)

[In]

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

[Out]

-1/d^2/b^2*a^2*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/8*(4*a^2+b^2)*(
d*f*x+d*e+f)/b^3/d^2*exp(-d*x-c)+1/d/b^2*a^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b
^2)^(1/2)))*x-1/d/b^2*a^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/d^2/b
^2*a^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d^2/b^2*a^2*f/(a^2+b^2
)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+2/d^2/b^2*a^2*f*c/(a^2+b^2)^(1/2)*arctanh(1
/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d^2*a^4/b^4*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/
(-a+(a^2+b^2)^(1/2)))*c-1/d^2*a^4/b^4*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)
))*c+2/d^2*a^4/b^4*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/4*a*f*x^2/b^2+1/8*(
4*a^2*d*f*x+b^2*d*f*x+4*a^2*d*e+b^2*d*e-4*a^2*f-b^2*f)/b^3/d^2*exp(d*x+c)+1/d*a^4/b^4*f/(a^2+b^2)^(1/2)*ln((-b
*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d*a^4/b^4*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2
)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-a^3*e*x/b^4-2/d*a^4/b^4*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(
a^2+b^2)^(1/2))-2/d/b^2*a^2*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d^2/b^2*a^2*
f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/2*a*e*x/b^2+1/72*(3*d*f*x+3*
d*e+f)/b/d^2*exp(-3*d*x-3*c)+1/d^2*a^4/b^4*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+
b^2)^(1/2)))-1/d^2*a^4/b^4*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/16*
a*(2*d*f*x+2*d*e-f)/b^2/d^2*exp(2*d*x+2*c)+1/16*a*(2*d*f*x+2*d*e+f)/b^2/d^2*exp(-2*d*x-2*c)-1/2*a^3*f*x^2/b^4+
1/72*(3*d*f*x+3*d*e-f)/b/d^2*exp(3*d*x+3*c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2195 vs. \(2 (365) = 730\).

Time = 0.31 (sec) , antiderivative size = 2195, normalized size of antiderivative = 5.45 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/144*(2*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^6 + 2*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*sinh(d*x + c)
^6 + 6*b^3*d*f*x - 9*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^5 - 3*(6*a*b^2*d*f*x + 6*a*b^2*d*e
- 3*a*b^2*f - 4*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*b^3*d*e + 18*((4*a^2*b +
b^3)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c)^4 + 3*(6*(4*a^2*b + b^3)*d*f*x + 6*(4*a^2*
b + b^3)*d*e + 10*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^2 - 6*(4*a^2*b + b^3)*f - 15*(2*a*b^2*d*f*x
+ 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c))*sinh(d*x + c)^4 + 2*b^3*f - 36*((2*a^3 + a*b^2)*d^2*f*x^2 + 2*(2*a^3 +
a*b^2)*d^2*e*x)*cosh(d*x + c)^3 - 2*(18*(2*a^3 + a*b^2)*d^2*f*x^2 + 36*(2*a^3 + a*b^2)*d^2*e*x - 20*(3*b^3*d*
f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^3 + 45*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^2 - 36*((4
*a^2*b + b^3)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^3 + 18*((4*a^2*b +
b^3)*d*f*x + (4*a^2*b + b^3)*d*e + (4*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 6*(5*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f
)*cosh(d*x + c)^4 + 3*(4*a^2*b + b^3)*d*f*x - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^3 + 3*(
4*a^2*b + b^3)*d*e + 18*((4*a^2*b + b^3)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 3*
(4*a^2*b + b^3)*f - 18*((2*a^3 + a*b^2)*d^2*f*x^2 + 2*(2*a^3 + a*b^2)*d^2*e*x)*cosh(d*x + c))*sinh(d*x + c)^2
+ 144*(a^2*b*f*cosh(d*x + c)^3 + 3*a^2*b*f*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*b*f*cosh(d*x + c)*sinh(d*x +
c)^2 + a^2*b*f*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)*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) - 144*(a^2*b*f*cosh(d*x + c)^3 + 3*a^2*b*f*cosh(d*x +
c)^2*sinh(d*x + c) + 3*a^2*b*f*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*b*f*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^2)
*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) - 144*((a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)^3 + 3*(a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c) +
3*(a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d*e - a^2*b*c*f)*sinh(d*x + c)^3)*sqrt((a^2
+ b^2)/b^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) + 144*((a^2*b*d*e - a
^2*b*c*f)*cosh(d*x + c)^3 + 3*(a^2*b*d*e - a^2*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d*e - a^2*b*c*f
)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d*e - a^2*b*c*f)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^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) + 144*((a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)
^3 + 3*(a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)*sin
h(d*x + c)^2 + (a^2*b*d*f*x + a^2*b*c*f)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^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) - 144*((a^2*b*d*f*x + a^2*b*c*f)
*cosh(d*x + c)^3 + 3*(a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b*d*f*x + a^2*b*c*f)*cos
h(d*x + c)*sinh(d*x + c)^2 + (a^2*b*d*f*x + a^2*b*c*f)*sinh(d*x + c)^3)*sqrt((a^2 + b^2)/b^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) + 9*(2*a*b^2*d*f*x
+ 2*a*b^2*d*e + a*b^2*f)*cosh(d*x + c) + 3*(6*a*b^2*d*f*x + 4*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)
^5 + 6*a*b^2*d*e - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^4 + 3*a*b^2*f + 24*((4*a^2*b + b^3
)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c)^3 - 36*((2*a^3 + a*b^2)*d^2*f*x^2 + 2*(2*a^3
+ a*b^2)*d^2*e*x)*cosh(d*x + c)^2 + 12*((4*a^2*b + b^3)*d*f*x + (4*a^2*b + b^3)*d*e + (4*a^2*b + b^3)*f)*cosh(
d*x + c))*sinh(d*x + c))/(b^4*d^2*cosh(d*x + c)^3 + 3*b^4*d^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^4*d^2*cosh(d
*x + c)*sinh(d*x + c)^2 + b^4*d^2*sinh(d*x + c)^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

1/144*(288*(a^4*e^c + a^2*b^2*e^c)*integrate(x*e^(d*x)/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x) -
(36*(2*a^3*d^2*e^(3*c) + a*b^2*d^2*e^(3*c))*x^2 - 2*(3*b^3*d*x*e^(6*c) - b^3*e^(6*c))*e^(3*d*x) + 9*(2*a*b^2*
d*x*e^(5*c) - a*b^2*e^(5*c))*e^(2*d*x) + 18*(4*a^2*b*e^(4*c) + b^3*e^(4*c) - (4*a^2*b*d*e^(4*c) + b^3*d*e^(4*c
))*x)*e^(d*x) - 18*(4*a^2*b*e^(2*c) + b^3*e^(2*c) + (4*a^2*b*d*e^(2*c) + b^3*d*e^(2*c))*x)*e^(-d*x) - 9*(2*a*b
^2*d*x*e^c + a*b^2*e^c)*e^(-2*d*x) - 2*(3*b^3*d*x + b^3)*e^(-3*d*x))*e^(-3*c)/(b^4*d^2))*f + 1/24*e*(24*sqrt(a
^2 + b^2)*a^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^4*d) - (3*
a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) - 12*(2*a^3 + a*b^2)*(d*x +
c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c) + 3*(4*a^2 + b^2)*e^(-d*x - c))/(b^3*d))

Giac [F]

\[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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