3.4.98 \(\int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [398]

Optimal. Leaf size=474 \[ \frac {a^4 e x}{b^5}+\frac {a^2 e x}{2 b^3}+\frac {a^4 f x^2}{2 b^5}+\frac {a^2 f x^2}{4 b^3}-\frac {(e+f x)^2}{16 b f}-\frac {a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac {f \cosh (4 c+4 d x)}{128 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^3 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {a^3 \sqrt {a^2+b^2} f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {a^3 \sqrt {a^2+b^2} f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {a^3 f \sinh (c+d x)}{b^4 d^2}+\frac {a f \sinh (c+d x)}{3 b^2 d^2}+\frac {a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d} \]

[Out]

a^4*e*x/b^5+1/2*a^2*e*x/b^3+1/2*a^4*f*x^2/b^5+1/4*a^2*f*x^2/b^3-1/16*(f*x+e)^2/b/f-a^3*(f*x+e)*cosh(d*x+c)/b^4
/d-1/4*a^2*f*cosh(d*x+c)^2/b^3/d^2-1/3*a*(f*x+e)*cosh(d*x+c)^3/b^2/d-1/128*f*cosh(4*d*x+4*c)/b/d^2+a^3*f*sinh(
d*x+c)/b^4/d^2+1/3*a*f*sinh(d*x+c)/b^2/d^2+1/2*a^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^3/d+1/9*a*f*sinh(d*x+c)^3
/b^2/d^2+1/32*(f*x+e)*sinh(4*d*x+4*c)/b/d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b
^5/d+a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d-a^3*f*polylog(2,-b*exp(d*x+c)/(a
-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^5/d^2+a^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/
b^5/d^2

________________________________________________________________________________________

Rubi [A]
time = 0.62, antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 14, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5698, 5556, 3377, 2718, 5555, 2713, 3391, 5684, 2717, 3403, 2296, 2221, 2317, 2438} \begin {gather*} \frac {a^4 e x}{b^5}+\frac {a^4 f x^2}{2 b^5}+\frac {a^3 f \sinh (c+d x)}{b^4 d^2}-\frac {a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}+\frac {a^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^3 d}+\frac {a^2 e x}{2 b^3}+\frac {a^2 f x^2}{4 b^3}-\frac {a^3 f \sqrt {a^2+b^2} \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {a^3 f \sqrt {a^2+b^2} \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {a^3 \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^5 d}+\frac {a^3 \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^5 d}+\frac {a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac {a f \sinh (c+d x)}{3 b^2 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac {f \cosh (4 c+4 d x)}{128 b d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d}-\frac {(e+f x)^2}{16 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

(a^4*e*x)/b^5 + (a^2*e*x)/(2*b^3) + (a^4*f*x^2)/(2*b^5) + (a^2*f*x^2)/(4*b^3) - (e + f*x)^2/(16*b*f) - (a^3*(e
+ f*x)*Cosh[c + d*x])/(b^4*d) - (a^2*f*Cosh[c + d*x]^2)/(4*b^3*d^2) - (a*(e + f*x)*Cosh[c + d*x]^3)/(3*b^2*d)
- (f*Cosh[4*c + 4*d*x])/(128*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 +
b^2])])/(b^5*d) + (a^3*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (a^
3*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^3*Sqrt[a^2 + b^2]*f*P
olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^3*f*Sinh[c + d*x])/(b^4*d^2) + (a*f*Sinh[c
+ d*x])/(3*b^2*d^2) + (a^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (a*f*Sinh[c + d*x]^3)/(9*b^2*d^
2) + ((e + f*x)*Sinh[4*c + 4*d*x])/(32*b*d)

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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 5556

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

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*} \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {a \int (e+f x) \cosh ^2(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {\int \left (\frac {1}{8} (-e-f x)+\frac {1}{8} (e+f x) \cosh (4 c+4 d x)\right ) \, dx}{b}\\ &=-\frac {(e+f x)^2}{16 b f}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}+\frac {a^2 \int (e+f x) \cosh ^2(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {\int (e+f x) \cosh (4 c+4 d x) \, dx}{8 b}+\frac {(a f) \int \cosh ^3(c+d x) \, dx}{3 b^2 d}\\ &=-\frac {(e+f x)^2}{16 b f}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}+\frac {a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d}+\frac {a^4 \int (e+f x) \, dx}{b^5}-\frac {a^3 \int (e+f x) \sinh (c+d x) \, dx}{b^4}+\frac {a^2 \int (e+f x) \, dx}{2 b^3}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac {(i a f) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{3 b^2 d^2}-\frac {f \int \sinh (4 c+4 d x) \, dx}{32 b d}\\ &=\frac {a^4 e x}{b^5}+\frac {a^2 e x}{2 b^3}+\frac {a^4 f x^2}{2 b^5}+\frac {a^2 f x^2}{4 b^3}-\frac {(e+f x)^2}{16 b f}-\frac {a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac {f \cosh (4 c+4 d x)}{128 b d^2}+\frac {a f \sinh (c+d x)}{3 b^2 d^2}+\frac {a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d}-\frac {\left (2 a^3 \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^5}+\frac {\left (a^3 f\right ) \int \cosh (c+d x) \, dx}{b^4 d}\\ &=\frac {a^4 e x}{b^5}+\frac {a^2 e x}{2 b^3}+\frac {a^4 f x^2}{2 b^5}+\frac {a^2 f x^2}{4 b^3}-\frac {(e+f x)^2}{16 b f}-\frac {a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac {f \cosh (4 c+4 d x)}{128 b d^2}+\frac {a^3 f \sinh (c+d x)}{b^4 d^2}+\frac {a f \sinh (c+d x)}{3 b^2 d^2}+\frac {a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d}-\frac {\left (2 a^3 \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^4}+\frac {\left (2 a^3 \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^4}\\ &=\frac {a^4 e x}{b^5}+\frac {a^2 e x}{2 b^3}+\frac {a^4 f x^2}{2 b^5}+\frac {a^2 f x^2}{4 b^3}-\frac {(e+f x)^2}{16 b f}-\frac {a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac {f \cosh (4 c+4 d x)}{128 b d^2}-\frac {a^3 \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^5 d}+\frac {a^3 \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^5 d}+\frac {a^3 f \sinh (c+d x)}{b^4 d^2}+\frac {a f \sinh (c+d x)}{3 b^2 d^2}+\frac {a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d}+\frac {\left (a^3 \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^5 d}-\frac {\left (a^3 \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^5 d}\\ &=\frac {a^4 e x}{b^5}+\frac {a^2 e x}{2 b^3}+\frac {a^4 f x^2}{2 b^5}+\frac {a^2 f x^2}{4 b^3}-\frac {(e+f x)^2}{16 b f}-\frac {a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac {f \cosh (4 c+4 d x)}{128 b d^2}-\frac {a^3 \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^5 d}+\frac {a^3 \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^5 d}+\frac {a^3 f \sinh (c+d x)}{b^4 d^2}+\frac {a f \sinh (c+d x)}{3 b^2 d^2}+\frac {a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d}+\frac {\left (a^3 \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^5 d^2}-\frac {\left (a^3 \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^5 d^2}\\ &=\frac {a^4 e x}{b^5}+\frac {a^2 e x}{2 b^3}+\frac {a^4 f x^2}{2 b^5}+\frac {a^2 f x^2}{4 b^3}-\frac {(e+f x)^2}{16 b f}-\frac {a^3 (e+f x) \cosh (c+d x)}{b^4 d}-\frac {a^2 f \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {a (e+f x) \cosh ^3(c+d x)}{3 b^2 d}-\frac {f \cosh (4 c+4 d x)}{128 b d^2}-\frac {a^3 \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^5 d}+\frac {a^3 \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^5 d}-\frac {a^3 \sqrt {a^2+b^2} f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {a^3 \sqrt {a^2+b^2} f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {a^3 f \sinh (c+d x)}{b^4 d^2}+\frac {a f \sinh (c+d x)}{3 b^2 d^2}+\frac {a^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {a f \sinh ^3(c+d x)}{9 b^2 d^2}+\frac {(e+f x) \sinh (4 c+4 d x)}{32 b d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.13, size = 2275, normalized size = 4.80 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

(-144*b^4*e*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d)) - 72*b^4
*f*(x^2 + ((2*I)*a*Pi*ArcTanh[(-b + a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d^2) + (2*a*(2*((-
I)*c + ArcCos[((-I)*a)/b])*ArcTanh[((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[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (ArcCos[((-I)*a)/
b] + (2*I)*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[((I*a + b)*(a + I*(b +
Sqrt[-a^2 - b^2]))*(-I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c +
Pi + (2*I)*d*x)/4]))] - (ArcCos[((-I)*a)/b] - (2*I)*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt
[-a^2 - b^2]])*Log[((I*a + b)*(I*a - b + Sqrt[-a^2 - b^2])*(I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(a - I*
b + Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] + (ArcCos[((-I)*a)/b] - (2*I)*ArcTanh[((a + I*b)*Cot
[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (2*I)*ArcTanh[((a - I*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]*Sqrt[(-I)*b]*Sqrt[a + b
*Sinh[c + d*x]]))] + (ArcCos[((-I)*a)/b] + (2*I)*(ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-
a^2 - b^2]] + ArcTanh[((a - I*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)*b]*Sqrt[a + b*Sinh[c + d*x]])] + I*(PolyLog[2, ((I*a + Sqrt[-a^
2 - b^2])*(I*a + b - I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*C
ot[((2*I)*c + Pi + (2*I)*d*x)/4]))] - PolyLog[2, ((a + I*Sqrt[-a^2 - b^2])*(-a + I*b + Sqrt[-a^2 - b^2]*Cot[((
2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))])))/(Sqrt[-
a^2 - b^2]*d^2)) - (72*b^2*e*((4*a^2 + b^2)*(c + d*x) - (2*a*(4*a^2 + 3*b^2)*ArcTan[(b - a*Tanh[(c + d*x)/2])/
Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + b^2*Sinh[2*(c + d*x)]))/d - (36*b^2*f*((4*a^2 + b^
2)*(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d*x] - b^2*Cosh[2*(c + d*x)] - (2*a*(4*a^2 + 3*b^2)*(2*c*ArcTanh[
(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]
))/(a - Sqrt[a^2 + b^2])] - (c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + Pol
yLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, -((b*(Cosh[c + d*x] + Sinh[c
+ d*x]))/(a + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*b^2*d*x*Sinh[2*(c + d*x)]))/d^2 +
(12*e*(6*(16*a^4 + 12*a^2*b^2 + b^4)*(c + d*x) - (12*a*(16*a^4 + 20*a^2*b^2 + 5*b^4)*ArcTan[(b - a*Tanh[(c +
d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 48*a*b*(2*a^2 + b^2)*Cosh[c + d*x] - 8*a*b^3*Cosh[3*(c + d*x)]
+ 6*b^2*(4*a^2 + b^2)*Sinh[2*(c + d*x)] + 3*b^4*Sinh[4*(c + d*x)]))/d + (f*(-576*a^4*c^2 - 432*a^2*b^2*c^2 - 3
6*b^4*c^2 + 576*a^4*d^2*x^2 + 432*a^2*b^2*d^2*x^2 + 36*b^4*d^2*x^2 - 576*a*b*(2*a^2 + b^2)*d*x*Cosh[c + d*x] -
36*(4*a^2*b^2 + b^4)*Cosh[2*(c + d*x)] - 96*a*b^3*d*x*Cosh[3*(c + d*x)] - 9*b^4*Cosh[4*(c + d*x)] + (72*a*(16
*a^4 + 20*a^2*b^2 + 5*b^4)*(2*Sqrt[-(a^2 + b^2)^2]*c*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]] + a^2*(c + d
*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + b^2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2
])] - a^2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - b^2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])] + (a^2 + b^2)*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - (a^2 + b^2)*PolyLog[2,
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[-a^2 - b^2]*Sqrt[-(a^2 + b^2)^2]) + 1152*a^3*b*Sinh[c + d*x]
+ 576*a*b^3*Sinh[c + d*x] + 288*a^2*b^2*d*x*Sinh[2*(c + d*x)] + 72*b^4*d*x*Sinh[2*(c + d*x)] + 32*a*b^3*Sinh[
3*(c + d*x)] + 36*b^4*d*x*Sinh[4*(c + d*x)]))/d^2)/(1152*b^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1212\) vs. \(2(432)=864\).
time = 1.62, size = 1213, normalized size = 2.56

method	result	size
risch	\(-\frac {2 a^{5} f c \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{5} \sqrt {a^{2}+b^{2}}}-\frac {2 a^{3} f c \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a^{5} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{5} \sqrt {a^{2}+b^{2}}}-\frac {a^{5} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{5} \sqrt {a^{2}+b^{2}}}+\frac {a^{5} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{5} \sqrt {a^{2}+b^{2}}}+\frac {a^{5} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{5} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}+\frac {\left (4 d x f +4 d e -f \right ) {\mathrm e}^{4 d x +4 c}}{256 d^{2} b}-\frac {a \left (4 a^{2} d f x +b^{2} d f x +4 a^{2} d e +b^{2} d e -4 a^{2} f -f \,b^{2}\right ) {\mathrm e}^{d x +c}}{8 b^{4} d^{2}}+\frac {a^{2} f \,x^{2}}{4 b^{3}}+\frac {a^{5} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{5} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}-\frac {f \,x^{2}}{16 b}-\frac {a^{5} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{5} \sqrt {a^{2}+b^{2}}}-\frac {\left (4 d x f +4 d e +f \right ) {\mathrm e}^{-4 d x -4 c}}{256 d^{2} b}+\frac {a^{4} f \,x^{2}}{2 b^{5}}-\frac {a \left (4 a^{2}+b^{2}\right ) \left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{8 b^{4} d^{2}}+\frac {a^{2} \left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 b^{3} d^{2}}-\frac {a^{2} \left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 b^{3} d^{2}}+\frac {a^{2} e x}{2 b^{3}}+\frac {2 a^{5} e \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{5} \sqrt {a^{2}+b^{2}}}+\frac {2 a^{3} e \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a \left (3 d x f +3 d e -f \right ) {\mathrm e}^{3 d x +3 c}}{72 b^{2} d^{2}}-\frac {e x}{8 b}+\frac {a^{4} e x}{b^{5}}-\frac {a \left (3 d x f +3 d e +f \right ) {\mathrm e}^{-3 d x -3 c}}{72 b^{2} d^{2}}\)	\(1213\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-2/d^2*a^5/b^5*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))-2/d^2*a^3/b^3*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*a^3/b^3*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*a^3/b^3*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*a^5/b^5*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*a^5/b^5*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*a^5/b^5*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*a^5/b^5*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*a^3/b^3*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*a^3/b^3*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+1/256*(4*d*f*x+4*d*e-f)/d^2/b*exp(4*d*x+4*c)-1/8*a*(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^4/d^2*exp(d*x+c)+1/4*a^2*f*x^2/b^3-1/16*f*x^2/b-1/d^2*a^5/b^5*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^5/b^5*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^3/b^3*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^3/b^3*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)))-1/25
6*(4*d*f*x+4*d*e+f)/d^2/b*exp(-4*d*x-4*c)+1/2*a^4*f*x^2/b^5-1/8*a*(4*a^2+b^2)*(d*f*x+d*e+f)/b^4/d^2*exp(-d*x-c
)+1/16*a^2*(2*d*f*x+2*d*e-f)/b^3/d^2*exp(2*d*x+2*c)-1/16*a^2*(2*d*f*x+2*d*e+f)/b^3/d^2*exp(-2*d*x-2*c)+2/d*a^5
/b^5*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*a^3/b^3*e/(a^2+b^2)^(1/2)*arctanh
(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/2*a^2*e*x/b^3-1/72*a*(3*d*f*x+3*d*e-f)/b^2/d^2*exp(3*d*x+3*c)-1/8
*e*x/b+a^4*e*x/b^5-1/72*a*(3*d*f*x+3*d*e+f)/b^2/d^2*exp(-3*d*x-3*c)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2304*(4608*(a^5*e^c + a^3*b^2*e^c)*integrate(x*e^(d*x)/(b^6*e^(2*d*x + 2*c) + 2*a*b^5*e^(d*x + c) - b^6), x
) - (144*(8*a^4*d^2*e^(4*c) + 4*a^2*b^2*d^2*e^(4*c) - b^4*d^2*e^(4*c))*x^2 + 9*(4*b^4*d*x*e^(8*c) - b^4*e^(8*c
))*e^(4*d*x) - 32*(3*a*b^3*d*x*e^(7*c) - a*b^3*e^(7*c))*e^(3*d*x) + 144*(2*a^2*b^2*d*x*e^(6*c) - a^2*b^2*e^(6*
c))*e^(2*d*x) + 288*(4*a^3*b*e^(5*c) + a*b^3*e^(5*c) - (4*a^3*b*d*e^(5*c) + a*b^3*d*e^(5*c))*x)*e^(d*x) - 288*
(4*a^3*b*e^(3*c) + a*b^3*e^(3*c) + (4*a^3*b*d*e^(3*c) + a*b^3*d*e^(3*c))*x)*e^(-d*x) - 144*(2*a^2*b^2*d*x*e^(2
*c) + a^2*b^2*e^(2*c))*e^(-2*d*x) - 32*(3*a*b^3*d*x*e^c + a*b^3*e^c)*e^(-3*d*x) - 9*(4*b^4*d*x + b^4)*e^(-4*d*
x))*e^(-4*c)/(b^5*d^2))*f - 1/192*(192*sqrt(a^2 + b^2)*a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d
*x - c) - a + sqrt(a^2 + b^2)))/(b^5*d) + (8*a*b^2*e^(-d*x - c) - 24*a^2*b*e^(-2*d*x - 2*c) - 3*b^3 + 24*(4*a^
3 + a*b^2)*e^(-3*d*x - 3*c))*e^(4*d*x + 4*c)/(b^4*d) - 24*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/(b^5*d) + (24*a^
2*b*e^(-2*d*x - 2*c) + 8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e^(-4*d*x - 4*c) + 24*(4*a^3 + a*b^2)*e^(-d*x - c))/(b
^4*d))*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3950 vs. \(2 (439) = 878\).
time = 0.45, size = 3950, normalized size = 8.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/2304*(9*(4*b^4*d*f*x + 4*b^4*d*cosh(1) + 4*b^4*d*sinh(1) - b^4*f)*cosh(d*x + c)^8 + 9*(4*b^4*d*f*x + 4*b^4*d
*cosh(1) + 4*b^4*d*sinh(1) - b^4*f)*sinh(d*x + c)^8 - 32*(3*a*b^3*d*f*x + 3*a*b^3*d*cosh(1) + 3*a*b^3*d*sinh(1
) - a*b^3*f)*cosh(d*x + c)^7 - 8*(12*a*b^3*d*f*x + 12*a*b^3*d*cosh(1) + 12*a*b^3*d*sinh(1) - 4*a*b^3*f - 9*(4*
b^4*d*f*x + 4*b^4*d*cosh(1) + 4*b^4*d*sinh(1) - b^4*f)*cosh(d*x + c))*sinh(d*x + c)^7 - 36*b^4*d*f*x + 144*(2*
a^2*b^2*d*f*x + 2*a^2*b^2*d*cosh(1) + 2*a^2*b^2*d*sinh(1) - a^2*b^2*f)*cosh(d*x + c)^6 + 4*(72*a^2*b^2*d*f*x +
72*a^2*b^2*d*cosh(1) + 72*a^2*b^2*d*sinh(1) - 36*a^2*b^2*f + 63*(4*b^4*d*f*x + 4*b^4*d*cosh(1) + 4*b^4*d*sinh
(1) - b^4*f)*cosh(d*x + c)^2 - 56*(3*a*b^3*d*f*x + 3*a*b^3*d*cosh(1) + 3*a*b^3*d*sinh(1) - a*b^3*f)*cosh(d*x +
c))*sinh(d*x + c)^6 - 36*b^4*d*cosh(1) - 288*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*cosh(1) + (4*a^3*
b + a*b^3)*d*sinh(1) - (4*a^3*b + a*b^3)*f)*cosh(d*x + c)^5 - 36*b^4*d*sinh(1) - 24*(12*(4*a^3*b + a*b^3)*d*f*
x - 21*(4*b^4*d*f*x + 4*b^4*d*cosh(1) + 4*b^4*d*sinh(1) - b^4*f)*cosh(d*x + c)^3 + 12*(4*a^3*b + a*b^3)*d*cosh
(1) + 28*(3*a*b^3*d*f*x + 3*a*b^3*d*cosh(1) + 3*a*b^3*d*sinh(1) - a*b^3*f)*cosh(d*x + c)^2 + 12*(4*a^3*b + a*b
^3)*d*sinh(1) - 12*(4*a^3*b + a*b^3)*f - 36*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*cosh(1) + 2*a^2*b^2*d*sinh(1) - a^2
*b^2*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 9*b^4*f + 144*((8*a^4 + 4*a^2*b^2 - b^4)*d^2*f*x^2 + 2*(8*a^4 + 4*a^2
*b^2 - b^4)*d^2*x*cosh(1) + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*x*sinh(1))*cosh(d*x + c)^4 + 2*(72*(8*a^4 + 4*a^2*
b^2 - b^4)*d^2*f*x^2 + 144*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*x*cosh(1) + 315*(4*b^4*d*f*x + 4*b^4*d*cosh(1) + 4*b^
4*d*sinh(1) - b^4*f)*cosh(d*x + c)^4 + 144*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*x*sinh(1) - 560*(3*a*b^3*d*f*x + 3*a*
b^3*d*cosh(1) + 3*a*b^3*d*sinh(1) - a*b^3*f)*cosh(d*x + c)^3 + 1080*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*cosh(1) + 2
*a^2*b^2*d*sinh(1) - a^2*b^2*f)*cosh(d*x + c)^2 - 720*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*cosh(1) +
(4*a^3*b + a*b^3)*d*sinh(1) - (4*a^3*b + a*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^4 - 288*((4*a^3*b + a*b^3)*d*
f*x + (4*a^3*b + a*b^3)*d*cosh(1) + (4*a^3*b + a*b^3)*d*sinh(1) + (4*a^3*b + a*b^3)*f)*cosh(d*x + c)^3 + 8*(63
*(4*b^4*d*f*x + 4*b^4*d*cosh(1) + 4*b^4*d*sinh(1) - b^4*f)*cosh(d*x + c)^5 - 140*(3*a*b^3*d*f*x + 3*a*b^3*d*co
sh(1) + 3*a*b^3*d*sinh(1) - a*b^3*f)*cosh(d*x + c)^4 - 36*(4*a^3*b + a*b^3)*d*f*x + 360*(2*a^2*b^2*d*f*x + 2*a
^2*b^2*d*cosh(1) + 2*a^2*b^2*d*sinh(1) - a^2*b^2*f)*cosh(d*x + c)^3 - 36*(4*a^3*b + a*b^3)*d*cosh(1) - 360*((4
*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*cosh(1) + (4*a^3*b + a*b^3)*d*sinh(1) - (4*a^3*b + a*b^3)*f)*cosh(
d*x + c)^2 - 36*(4*a^3*b + a*b^3)*d*sinh(1) - 36*(4*a^3*b + a*b^3)*f + 72*((8*a^4 + 4*a^2*b^2 - b^4)*d^2*f*x^2
+ 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*x*cosh(1) + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*x*sinh(1))*cosh(d*x + c))*sinh(
d*x + c)^3 - 144*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*cosh(1) + 2*a^2*b^2*d*sinh(1) + a^2*b^2*f)*cosh(d*x + c)^2 - 1
2*(24*a^2*b^2*d*f*x - 21*(4*b^4*d*f*x + 4*b^4*d*cosh(1) + 4*b^4*d*sinh(1) - b^4*f)*cosh(d*x + c)^6 + 24*a^2*b^
2*d*cosh(1) + 56*(3*a*b^3*d*f*x + 3*a*b^3*d*cosh(1) + 3*a*b^3*d*sinh(1) - a*b^3*f)*cosh(d*x + c)^5 + 24*a^2*b^
2*d*sinh(1) + 12*a^2*b^2*f - 180*(2*a^2*b^2*d*f*x + 2*a^2*b^2*d*cosh(1) + 2*a^2*b^2*d*sinh(1) - a^2*b^2*f)*cos
h(d*x + c)^4 + 240*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b + a*b^3)*d*cosh(1) + (4*a^3*b + a*b^3)*d*sinh(1) - (4*a
^3*b + a*b^3)*f)*cosh(d*x + c)^3 - 72*((8*a^4 + 4*a^2*b^2 - b^4)*d^2*f*x^2 + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*x
*cosh(1) + 2*(8*a^4 + 4*a^2*b^2 - b^4)*d^2*x*sinh(1))*cosh(d*x + c)^2 + 72*((4*a^3*b + a*b^3)*d*f*x + (4*a^3*b
+ a*b^3)*d*cosh(1) + (4*a^3*b + a*b^3)*d*sinh(1) + (4*a^3*b + a*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 2304
*(a^3*b*f*cosh(d*x + c)^4 + 4*a^3*b*f*cosh(d*x + c)^3*sinh(d*x + c) + 6*a^3*b*f*cosh(d*x + c)^2*sinh(d*x + c)^
2 + 4*a^3*b*f*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*b*f*sinh(d*x + c)^4)*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) + 2304*(a^3*b*
f*cosh(d*x + c)^4 + 4*a^3*b*f*cosh(d*x + c)^3*sinh(d*x + c) + 6*a^3*b*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*a^
3*b*f*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*b*f*sinh(d*x + c)^4)*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) - 2304*((a^3*b*c*f - a
^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x + c)^4 + 4*(a^3*b*c*f - a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x
+ c)^3*sinh(d*x + c) + 6*(a^3*b*c*f - a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*
(a^3*b*c*f - a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b*c*f - a^3*b*d*cosh(1) -
a^3*b*d*sinh(1))*sinh(d*x + c)^4)*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) + 2304*((a^3*b*c*f - a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x + c)^4 + 4*(a^3*b*c*f
- a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)**2*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)*cosh(d*x + c)^2*sinh(d*x + c)^3/(b*sinh(d*x + c) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)^2*sinh(c + d*x)^3*(e + f*x))/(a + b*sinh(c + d*x)),x)

[Out]

int((cosh(c + d*x)^2*sinh(c + d*x)^3*(e + f*x))/(a + b*sinh(c + d*x)), x)

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