\(\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [403]
Optimal result
Integrand size = 34, antiderivative size = 641 \[
\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 f x}{4 b^4 d}+\frac {3 a f x}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}-\frac {a^4 f \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac {f \cosh (c+d x)}{8 b d^2}-\frac {a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac {f \cosh (3 c+3 d x)}{144 b d^2}-\frac {f \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {a^3 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x) \sinh (c+d x)}{8 b d}+\frac {a^3 f \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 b d}
\]
[Out]
-1/4*a^3*f*x/b^4/d+3/32*a*f*x/b^2/d+1/2*a^3*(a^2+b^2)*(f*x+e)^2/b^6/f-a^4*f*cosh(d*x+c)/b^5/d^2-2/3*a^2*f*cosh
(d*x+c)/b^3/d^2+1/8*f*cosh(d*x+c)/b/d^2-1/9*a^2*f*cosh(d*x+c)^3/b^3/d^2-1/4*a*(f*x+e)*cosh(d*x+c)^4/b^2/d-1/14
4*f*cosh(3*d*x+3*c)/b/d^2-1/400*f*cosh(5*d*x+5*c)/b/d^2-a^3*(a^2+b^2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(
1/2)))/b^6/d-a^3*(a^2+b^2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d-a^3*(a^2+b^2)*f*polylog(2,-b*e
xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^2-a^3*(a^2+b^2)*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^2+a^
4*(f*x+e)*sinh(d*x+c)/b^5/d+2/3*a^2*(f*x+e)*sinh(d*x+c)/b^3/d-1/8*(f*x+e)*sinh(d*x+c)/b/d+1/4*a^3*f*cosh(d*x+c
)*sinh(d*x+c)/b^4/d^2+3/32*a*f*cosh(d*x+c)*sinh(d*x+c)/b^2/d^2+1/3*a^2*(f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/b^3/d
+1/16*a*f*cosh(d*x+c)^3*sinh(d*x+c)/b^2/d^2-1/2*a^3*(f*x+e)*sinh(d*x+c)^2/b^4/d+1/48*(f*x+e)*sinh(3*d*x+3*c)/b
/d+1/80*(f*x+e)*sinh(5*d*x+5*c)/b/d
Rubi [A] (verified)
Time = 0.62 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.00, number of
steps used = 31, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5698, 5556, 3377, 2718,
5555, 2715, 8, 3391, 5684, 5554, 5680, 2221, 2317, 2438} \[
\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^4 f \cosh (c+d x)}{b^5 d^2}+\frac {a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac {a^3 f \sinh (c+d x) \cosh (c+d x)}{4 b^4 d^2}-\frac {a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}-\frac {a^3 f x}{4 b^4 d}-\frac {a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac {2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}+\frac {a^2 (e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b^3 d}-\frac {a^3 f \left (a^2+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {a^3 f \left (a^2+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^6 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}+\frac {a f \sinh (c+d x) \cosh ^3(c+d x)}{16 b^2 d^2}+\frac {3 a f \sinh (c+d x) \cosh (c+d x)}{32 b^2 d^2}-\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}+\frac {3 a f x}{32 b^2 d}+\frac {f \cosh (c+d x)}{8 b d^2}-\frac {f \cosh (3 c+3 d x)}{144 b d^2}-\frac {f \cosh (5 c+5 d x)}{400 b d^2}-\frac {(e+f x) \sinh (c+d x)}{8 b d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 b d}
\]
[In]
Int[((e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
[Out]
-1/4*(a^3*f*x)/(b^4*d) + (3*a*f*x)/(32*b^2*d) + (a^3*(a^2 + b^2)*(e + f*x)^2)/(2*b^6*f) - (a^4*f*Cosh[c + d*x]
)/(b^5*d^2) - (2*a^2*f*Cosh[c + d*x])/(3*b^3*d^2) + (f*Cosh[c + d*x])/(8*b*d^2) - (a^2*f*Cosh[c + d*x]^3)/(9*b
^3*d^2) - (a*(e + f*x)*Cosh[c + d*x]^4)/(4*b^2*d) - (f*Cosh[3*c + 3*d*x])/(144*b*d^2) - (f*Cosh[5*c + 5*d*x])/
(400*b*d^2) - (a^3*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 +
b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*f*PolyLog[2, -((b*E
^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2) - (a^3*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2
+ b^2]))])/(b^6*d^2) + (a^4*(e + f*x)*Sinh[c + d*x])/(b^5*d) + (2*a^2*(e + f*x)*Sinh[c + d*x])/(3*b^3*d) - ((
e + f*x)*Sinh[c + d*x])/(8*b*d) + (a^3*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^4*d^2) + (3*a*f*Cosh[c + d*x]*Sinh[
c + d*x])/(32*b^2*d^2) + (a^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^3*d) + (a*f*Cosh[c + d*x]^3*Sinh[c
+ d*x])/(16*b^2*d^2) - (a^3*(e + f*x)*Sinh[c + d*x]^2)/(2*b^4*d) + ((e + f*x)*Sinh[3*c + 3*d*x])/(48*b*d) + (
(e + f*x)*Sinh[5*c + 5*d*x])/(80*b*d)
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
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 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 2715
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]
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 5554
Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(c +
d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1))), x] - Dist[d*(m/(b*(n + 1))), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Rule 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 5680
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
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 ^3(c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {a \int (e+f x) \cosh ^3(c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \cosh ^3(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) \cosh (c+d x)+\frac {1}{16} (e+f x) \cosh (3 c+3 d x)+\frac {1}{16} (e+f x) \cosh (5 c+5 d x)\right ) \, dx}{b} \\ & = -\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}+\frac {a^2 \int (e+f x) \cosh ^3(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {\int (e+f x) \cosh (3 c+3 d x) \, dx}{16 b}+\frac {\int (e+f x) \cosh (5 c+5 d x) \, dx}{16 b}-\frac {\int (e+f x) \cosh (c+d x) \, dx}{8 b}+\frac {(a f) \int \cosh ^4(c+d x) \, dx}{4 b^2 d} \\ & = -\frac {a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac {(e+f x) \sinh (c+d x)}{8 b d}+\frac {a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 b d}+\frac {a^4 \int (e+f x) \cosh (c+d x) \, dx}{b^5}-\frac {a^3 \int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{b^4}+\frac {\left (2 a^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{3 b^3}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^5}+\frac {(3 a f) \int \cosh ^2(c+d x) \, dx}{16 b^2 d}-\frac {f \int \sinh (5 c+5 d x) \, dx}{80 b d}-\frac {f \int \sinh (3 c+3 d x) \, dx}{48 b d}+\frac {f \int \sinh (c+d x) \, dx}{8 b d} \\ & = \frac {a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}+\frac {f \cosh (c+d x)}{8 b d^2}-\frac {a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac {f \cosh (3 c+3 d x)}{144 b d^2}-\frac {f \cosh (5 c+5 d x)}{400 b d^2}+\frac {a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x) \sinh (c+d x)}{8 b d}+\frac {3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 b d}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^5}-\frac {\left (a^3 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^5}-\frac {\left (a^4 f\right ) \int \sinh (c+d x) \, dx}{b^5 d}+\frac {\left (a^3 f\right ) \int \sinh ^2(c+d x) \, dx}{2 b^4 d}-\frac {\left (2 a^2 f\right ) \int \sinh (c+d x) \, dx}{3 b^3 d}+\frac {(3 a f) \int 1 \, dx}{32 b^2 d} \\ & = \frac {3 a f x}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}-\frac {a^4 f \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac {f \cosh (c+d x)}{8 b d^2}-\frac {a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac {f \cosh (3 c+3 d x)}{144 b d^2}-\frac {f \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x) \sinh (c+d x)}{8 b d}+\frac {a^3 f \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 b d}-\frac {\left (a^3 f\right ) \int 1 \, dx}{4 b^4 d}+\frac {\left (a^3 \left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^6 d}+\frac {\left (a^3 \left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^6 d} \\ & = -\frac {a^3 f x}{4 b^4 d}+\frac {3 a f x}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}-\frac {a^4 f \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac {f \cosh (c+d x)}{8 b d^2}-\frac {a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac {f \cosh (3 c+3 d x)}{144 b d^2}-\frac {f \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x) \sinh (c+d x)}{8 b d}+\frac {a^3 f \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 b d}+\frac {\left (a^3 \left (a^2+b^2\right ) f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^6 d^2}+\frac {\left (a^3 \left (a^2+b^2\right ) f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^6 d^2} \\ & = -\frac {a^3 f x}{4 b^4 d}+\frac {3 a f x}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^2}{2 b^6 f}-\frac {a^4 f \cosh (c+d x)}{b^5 d^2}-\frac {2 a^2 f \cosh (c+d x)}{3 b^3 d^2}+\frac {f \cosh (c+d x)}{8 b d^2}-\frac {a^2 f \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a (e+f x) \cosh ^4(c+d x)}{4 b^2 d}-\frac {f \cosh (3 c+3 d x)}{144 b d^2}-\frac {f \cosh (5 c+5 d x)}{400 b d^2}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a^3 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^6 d^2}-\frac {a^3 \left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^6 d^2}+\frac {a^4 (e+f x) \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x) \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x) \sinh (c+d x)}{8 b d}+\frac {a^3 f \cosh (c+d x) \sinh (c+d x)}{4 b^4 d^2}+\frac {3 a f \cosh (c+d x) \sinh (c+d x)}{32 b^2 d^2}+\frac {a^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f \cosh ^3(c+d x) \sinh (c+d x)}{16 b^2 d^2}-\frac {a^3 (e+f x) \sinh ^2(c+d x)}{2 b^4 d}+\frac {(e+f x) \sinh (3 c+3 d x)}{48 b d}+\frac {(e+f x) \sinh (5 c+5 d x)}{80 b d} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(2704\) vs. \(2(641)=1282\).
Time = 8.52 (sec) , antiderivative size = 2704, normalized size of antiderivative = 4.22
\[
\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show}
\]
[In]
Integrate[((e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
[Out]
((-4*a^3*(a^2 + b^2)*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*Sqrt[a^2 + b^2]*d*e*ArcTan[(a
+ b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^(c +
d*x))/Sqrt[a^2 + b^2]])/(-a^2 - b^2)^(3/2) + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2
*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x)
)] + 2*d*e*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b
^2])] + 2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(b^6*d^2) + ((Cosh[5*(c + d*x)]/(7200*b^5*d
) - Sinh[5*(c + d*x)]/(7200*b^5*d))*(-360*b^4*d*e - 72*b^4*f + 360*b^4*c*f - 360*b^4*f*(c + d*x) - 900*a*b^3*d
*e*Cosh[c + d*x] - 225*a*b^3*f*Cosh[c + d*x] + 900*a*b^3*c*f*Cosh[c + d*x] - 900*a*b^3*f*(c + d*x)*Cosh[c + d*
x] - 2400*a^2*b^2*d*e*Cosh[2*(c + d*x)] - 600*b^4*d*e*Cosh[2*(c + d*x)] - 800*a^2*b^2*f*Cosh[2*(c + d*x)] - 20
0*b^4*f*Cosh[2*(c + d*x)] + 2400*a^2*b^2*c*f*Cosh[2*(c + d*x)] + 600*b^4*c*f*Cosh[2*(c + d*x)] - 2400*a^2*b^2*
f*(c + d*x)*Cosh[2*(c + d*x)] - 600*b^4*f*(c + d*x)*Cosh[2*(c + d*x)] - 7200*a^3*b*d*e*Cosh[3*(c + d*x)] - 360
0*a*b^3*d*e*Cosh[3*(c + d*x)] - 3600*a^3*b*f*Cosh[3*(c + d*x)] - 1800*a*b^3*f*Cosh[3*(c + d*x)] + 7200*a^3*b*c
*f*Cosh[3*(c + d*x)] + 3600*a*b^3*c*f*Cosh[3*(c + d*x)] - 7200*a^3*b*f*(c + d*x)*Cosh[3*(c + d*x)] - 3600*a*b^
3*f*(c + d*x)*Cosh[3*(c + d*x)] - 28800*a^4*d*e*Cosh[4*(c + d*x)] - 21600*a^2*b^2*d*e*Cosh[4*(c + d*x)] + 3600
*b^4*d*e*Cosh[4*(c + d*x)] - 28800*a^4*f*Cosh[4*(c + d*x)] - 21600*a^2*b^2*f*Cosh[4*(c + d*x)] + 3600*b^4*f*Co
sh[4*(c + d*x)] + 28800*a^4*c*f*Cosh[4*(c + d*x)] + 21600*a^2*b^2*c*f*Cosh[4*(c + d*x)] - 3600*b^4*c*f*Cosh[4*
(c + d*x)] - 28800*a^4*f*(c + d*x)*Cosh[4*(c + d*x)] - 21600*a^2*b^2*f*(c + d*x)*Cosh[4*(c + d*x)] + 3600*b^4*
f*(c + d*x)*Cosh[4*(c + d*x)] + 28800*a^4*d*e*Cosh[6*(c + d*x)] + 21600*a^2*b^2*d*e*Cosh[6*(c + d*x)] - 3600*b
^4*d*e*Cosh[6*(c + d*x)] - 28800*a^4*f*Cosh[6*(c + d*x)] - 21600*a^2*b^2*f*Cosh[6*(c + d*x)] + 3600*b^4*f*Cosh
[6*(c + d*x)] - 28800*a^4*c*f*Cosh[6*(c + d*x)] - 21600*a^2*b^2*c*f*Cosh[6*(c + d*x)] + 3600*b^4*c*f*Cosh[6*(c
+ d*x)] + 28800*a^4*f*(c + d*x)*Cosh[6*(c + d*x)] + 21600*a^2*b^2*f*(c + d*x)*Cosh[6*(c + d*x)] - 3600*b^4*f*
(c + d*x)*Cosh[6*(c + d*x)] - 7200*a^3*b*d*e*Cosh[7*(c + d*x)] - 3600*a*b^3*d*e*Cosh[7*(c + d*x)] + 3600*a^3*b
*f*Cosh[7*(c + d*x)] + 1800*a*b^3*f*Cosh[7*(c + d*x)] + 7200*a^3*b*c*f*Cosh[7*(c + d*x)] + 3600*a*b^3*c*f*Cosh
[7*(c + d*x)] - 7200*a^3*b*f*(c + d*x)*Cosh[7*(c + d*x)] - 3600*a*b^3*f*(c + d*x)*Cosh[7*(c + d*x)] + 2400*a^2
*b^2*d*e*Cosh[8*(c + d*x)] + 600*b^4*d*e*Cosh[8*(c + d*x)] - 800*a^2*b^2*f*Cosh[8*(c + d*x)] - 200*b^4*f*Cosh[
8*(c + d*x)] - 2400*a^2*b^2*c*f*Cosh[8*(c + d*x)] - 600*b^4*c*f*Cosh[8*(c + d*x)] + 2400*a^2*b^2*f*(c + d*x)*C
osh[8*(c + d*x)] + 600*b^4*f*(c + d*x)*Cosh[8*(c + d*x)] - 900*a*b^3*d*e*Cosh[9*(c + d*x)] + 225*a*b^3*f*Cosh[
9*(c + d*x)] + 900*a*b^3*c*f*Cosh[9*(c + d*x)] - 900*a*b^3*f*(c + d*x)*Cosh[9*(c + d*x)] + 360*b^4*d*e*Cosh[10
*(c + d*x)] - 72*b^4*f*Cosh[10*(c + d*x)] - 360*b^4*c*f*Cosh[10*(c + d*x)] + 360*b^4*f*(c + d*x)*Cosh[10*(c +
d*x)] - 900*a*b^3*d*e*Sinh[c + d*x] - 225*a*b^3*f*Sinh[c + d*x] + 900*a*b^3*c*f*Sinh[c + d*x] - 900*a*b^3*f*(c
+ d*x)*Sinh[c + d*x] - 2400*a^2*b^2*d*e*Sinh[2*(c + d*x)] - 600*b^4*d*e*Sinh[2*(c + d*x)] - 800*a^2*b^2*f*Sin
h[2*(c + d*x)] - 200*b^4*f*Sinh[2*(c + d*x)] + 2400*a^2*b^2*c*f*Sinh[2*(c + d*x)] + 600*b^4*c*f*Sinh[2*(c + d*
x)] - 2400*a^2*b^2*f*(c + d*x)*Sinh[2*(c + d*x)] - 600*b^4*f*(c + d*x)*Sinh[2*(c + d*x)] - 7200*a^3*b*d*e*Sinh
[3*(c + d*x)] - 3600*a*b^3*d*e*Sinh[3*(c + d*x)] - 3600*a^3*b*f*Sinh[3*(c + d*x)] - 1800*a*b^3*f*Sinh[3*(c + d
*x)] + 7200*a^3*b*c*f*Sinh[3*(c + d*x)] + 3600*a*b^3*c*f*Sinh[3*(c + d*x)] - 7200*a^3*b*f*(c + d*x)*Sinh[3*(c
+ d*x)] - 3600*a*b^3*f*(c + d*x)*Sinh[3*(c + d*x)] - 28800*a^4*d*e*Sinh[4*(c + d*x)] - 21600*a^2*b^2*d*e*Sinh[
4*(c + d*x)] + 3600*b^4*d*e*Sinh[4*(c + d*x)] - 28800*a^4*f*Sinh[4*(c + d*x)] - 21600*a^2*b^2*f*Sinh[4*(c + d*
x)] + 3600*b^4*f*Sinh[4*(c + d*x)] + 28800*a^4*c*f*Sinh[4*(c + d*x)] + 21600*a^2*b^2*c*f*Sinh[4*(c + d*x)] - 3
600*b^4*c*f*Sinh[4*(c + d*x)] - 28800*a^4*f*(c + d*x)*Sinh[4*(c + d*x)] - 21600*a^2*b^2*f*(c + d*x)*Sinh[4*(c
+ d*x)] + 3600*b^4*f*(c + d*x)*Sinh[4*(c + d*x)] + 28800*a^4*d*e*Sinh[6*(c + d*x)] + 21600*a^2*b^2*d*e*Sinh[6*
(c + d*x)] - 3600*b^4*d*e*Sinh[6*(c + d*x)] - 28800*a^4*f*Sinh[6*(c + d*x)] - 21600*a^2*b^2*f*Sinh[6*(c + d*x)
] + 3600*b^4*f*Sinh[6*(c + d*x)] - 28800*a^4*c*f*Sinh[6*(c + d*x)] - 21600*a^2*b^2*c*f*Sinh[6*(c + d*x)] + 360
0*b^4*c*f*Sinh[6*(c + d*x)] + 28800*a^4*f*(c + d*x)*Sinh[6*(c + d*x)] + 21600*a^2*b^2*f*(c + d*x)*Sinh[6*(c +
d*x)] - 3600*b^4*f*(c + d*x)*Sinh[6*(c + d*x)] - 7200*a^3*b*d*e*Sinh[7*(c + d*x)] - 3600*a*b^3*d*e*Sinh[7*(c +
d*x)] + 3600*a^3*b*f*Sinh[7*(c + d*x)] + 1800*a*b^3*f*Sinh[7*(c + d*x)] + 7200*a^3*b*c*f*Sinh[7*(c + d*x)] +
3600*a*b^3*c*f*Sinh[7*(c + d*x)] - 7200*a^3*b*f*(c + d*x)*Sinh[7*(c + d*x)] - 3600*a*b^3*f*(c + d*x)*Sinh[7*(c
+ d*x)] + 2400*a^2*b^2*d*e*Sinh[8*(c + d*x)] + 600*b^4*d*e*Sinh[8*(c + d*x)] - 800*a^2*b^2*f*Sinh[8*(c + d*x)
] - 200*b^4*f*Sinh[8*(c + d*x)] - 2400*a^2*b^2*c*f*Sinh[8*(c + d*x)] - 600*b^4*c*f*Sinh[8*(c + d*x)] + 2400*a^
2*b^2*f*(c + d*x)*Sinh[8*(c + d*x)] + 600*b^4*f*(c + d*x)*Sinh[8*(c + d*x)] - 900*a*b^3*d*e*Sinh[9*(c + d*x)]
+ 225*a*b^3*f*Sinh[9*(c + d*x)] + 900*a*b^3*c*f*Sinh[9*(c + d*x)] - 900*a*b^3*f*(c + d*x)*Sinh[9*(c + d*x)] +
360*b^4*d*e*Sinh[10*(c + d*x)] - 72*b^4*f*Sinh[10*(c + d*x)] - 360*b^4*c*f*Sinh[10*(c + d*x)] + 360*b^4*f*(c +
d*x)*Sinh[10*(c + d*x)]))/d)/8
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1362\) vs. \(2(593)=1186\).
Time = 187.96 (sec) , antiderivative size = 1363, normalized size of antiderivative =
2.13
| | |
method | result | size |
| | |
risch |
\(\text {Expression too large to display}\) |
\(1363\) |
| | |
|
|
|
[In]
int((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
-1/d^2*a^3/b^4*f*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^3/b^4*f*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^4*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1
/2)))*x-1/d*a^3/b^4*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/256*a*(4*d*f*x+4*d*e-f)/b^2
/d^2*exp(4*d*x+4*c)-1/32*a*(4*a^2*d*f*x+2*b^2*d*f*x+4*a^2*d*e+2*b^2*d*e-2*a^2*f-b^2*f)/b^4/d^2*exp(2*d*x+2*c)+
1/d^2*a^5/b^6*c*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/d*a^5/b^6*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a
+(a^2+b^2)^(1/2)))*x-1/d*a^5/b^6*f*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^6*f*
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^5/b^6*f*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^5/b^6*f*c^2+1/16*(8*a^4*d*f*x+6*a^2*b^2*d*f*x-b^4*d*f*x+8*a^4*d*e+6*a^2*b^2*
d*e-b^4*d*e-8*a^4*f-6*a^2*b^2*f+b^4*f)/b^5/d^2*exp(d*x+c)-1/16*(8*a^4+6*a^2*b^2-b^4)*(d*f*x+d*e+f)/b^5/d^2*exp
(-d*x-c)-1/288*(4*a^2+b^2)*(3*d*f*x+3*d*e+f)/b^3/d^2*exp(-3*d*x-3*c)-1/256*a*(4*d*f*x+4*d*e+f)/b^2/d^2*exp(-4*
d*x-4*c)-1/d^2*a^3/b^4*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d*a^3/b^4*e*ln(b*exp(2*
d*x+2*c)+2*a*exp(d*x+c)-b)-a^3*e*x/b^4+1/d^2*a^3/b^4*f*c^2+2/d*a^3/b^4*e*ln(exp(d*x+c))-1/d^2*a^3/b^4*f*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^6*f*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^6*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/d*a^5/b^
6*e*ln(exp(d*x+c))-1/d*a^5/b^6*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/32*a*(2*a^2+b^2)*(2*d*f*x+2*d*e+f)/b^
4/d^2*exp(-2*d*x-2*c)+2/d*a^3/b^4*c*f*x-2/d^2*a^3/b^4*c*f*ln(exp(d*x+c))+1/d^2*a^3/b^4*c*f*ln(b*exp(2*d*x+2*c)
+2*a*exp(d*x+c)-b)-2/d^2*a^5/b^6*c*f*ln(exp(d*x+c))+2/d*a^5/b^6*f*c*x+1/2*a^3*f*x^2/b^4+1/800*(5*d*f*x+5*d*e-f
)/b/d^2*exp(5*d*x+5*c)+1/288*(12*a^2*d*f*x+3*b^2*d*f*x+12*a^2*d*e+3*b^2*d*e-4*a^2*f-b^2*f)/b^3/d^2*exp(3*d*x+3
*c)-1/800*(5*d*f*x+5*d*e+f)/b/d^2*exp(-5*d*x-5*c)+1/2*a^5/b^6*f*x^2-a^5/b^6*e*x
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5548 vs. \(2 (591) = 1182\).
Time = 0.33 (sec) , antiderivative size = 5548, normalized size of antiderivative = 8.66
\[
\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
[Out]
Too large to include
Sympy [F(-1)]
Timed out. \[
\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out}
\]
[In]
integrate((f*x+e)*cosh(d*x+c)**3*sinh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x }
\]
[In]
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-1/960*e*((15*a*b^3*e^(-d*x - c) - 6*b^4 - 10*(4*a^2*b^2 + b^4)*e^(-2*d*x - 2*c) + 60*(2*a^3*b + a*b^3)*e^(-3*
d*x - 3*c) - 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-4*d*x - 4*c))*e^(5*d*x + 5*c)/(b^5*d) + 960*(a^5 + a^3*b^2)*(d*x
+ c)/(b^6*d) + (15*a*b^3*e^(-4*d*x - 4*c) + 6*b^4*e^(-5*d*x - 5*c) + 60*(8*a^4 + 6*a^2*b^2 - b^4)*e^(-d*x - c
) + 60*(2*a^3*b + a*b^3)*e^(-2*d*x - 2*c) + 10*(4*a^2*b^2 + b^4)*e^(-3*d*x - 3*c))/(b^5*d) + 960*(a^5 + a^3*b^
2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^6*d)) - 1/57600*f*((28800*(a^5*d^2*e^(5*c) + a^3*b^2*d^2
*e^(5*c))*x^2 - 72*(5*b^5*d*x*e^(10*c) - b^5*e^(10*c))*e^(5*d*x) + 225*(4*a*b^4*d*x*e^(9*c) - a*b^4*e^(9*c))*e
^(4*d*x) + 200*(4*a^2*b^3*e^(8*c) + b^5*e^(8*c) - 3*(4*a^2*b^3*d*e^(8*c) + b^5*d*e^(8*c))*x)*e^(3*d*x) - 1800*
(2*a^3*b^2*e^(7*c) + a*b^4*e^(7*c) - 2*(2*a^3*b^2*d*e^(7*c) + a*b^4*d*e^(7*c))*x)*e^(2*d*x) + 3600*(8*a^4*b*e^
(6*c) + 6*a^2*b^3*e^(6*c) - b^5*e^(6*c) - (8*a^4*b*d*e^(6*c) + 6*a^2*b^3*d*e^(6*c) - b^5*d*e^(6*c))*x)*e^(d*x)
+ 3600*(8*a^4*b*e^(4*c) + 6*a^2*b^3*e^(4*c) - b^5*e^(4*c) + (8*a^4*b*d*e^(4*c) + 6*a^2*b^3*d*e^(4*c) - b^5*d*
e^(4*c))*x)*e^(-d*x) + 1800*(2*a^3*b^2*e^(3*c) + a*b^4*e^(3*c) + 2*(2*a^3*b^2*d*e^(3*c) + a*b^4*d*e^(3*c))*x)*
e^(-2*d*x) + 200*(4*a^2*b^3*e^(2*c) + b^5*e^(2*c) + 3*(4*a^2*b^3*d*e^(2*c) + b^5*d*e^(2*c))*x)*e^(-3*d*x) + 22
5*(4*a*b^4*d*x*e^c + a*b^4*e^c)*e^(-4*d*x) + 72*(5*b^5*d*x + b^5)*e^(-5*d*x))*e^(-5*c)/(b^6*d^2) - 900*integra
te(128*((a^6*e^c + a^4*b^2*e^c)*x*e^(d*x) - (a^5*b + a^3*b^3)*x)/(b^7*e^(2*d*x + 2*c) + 2*a*b^6*e^(d*x + c) -
b^7), x))
Giac [F]
\[
\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x }
\]
[In]
integrate((f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
[Out]
integrate((f*x + e)*cosh(d*x + c)^3*sinh(d*x + c)^3/(b*sinh(d*x + c) + a), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(e+f x) \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x
\]
[In]
int((cosh(c + d*x)^3*sinh(c + d*x)^3*(e + f*x))/(a + b*sinh(c + d*x)),x)
[Out]
int((cosh(c + d*x)^3*sinh(c + d*x)^3*(e + f*x))/(a + b*sinh(c + d*x)), x)