3.4.45 \(\int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [345]

Optimal. Leaf size=400 \[ -\frac {a f x}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a \left (a^2+b^2\right ) f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d} \]

[Out]

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

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Rubi [A]
time = 0.35, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5698, 3391, 3377, 2718, 5684, 5554, 2715, 8, 5680, 2221, 2317, 2438} \begin {gather*} -\frac {a^2 f \cosh (c+d x)}{b^3 d^2}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}-\frac {a f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a \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^4 d}-\frac {a \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^4 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}+\frac {a f \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^2}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac {a f x}{4 b^2 d}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a*f*x)/(b^2*d) + (a*(a^2 + b^2)*(e + f*x)^2)/(2*b^4*f) - (a^2*f*Cosh[c + d*x])/(b^3*d^2) - (2*f*Cosh[c +
d*x])/(3*b*d^2) - (f*Cosh[c + d*x]^3)/(9*b*d^2) - (a*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[
a^2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (a*
(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (a*(a^2 + b^2)*f*PolyLog[2, -(
(b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (a^2*(e + f*x)*Sinh[c + d*x])/(b^3*d) + (2*(e + f*x)*Sinh
[c + d*x])/(3*b*d) + (a*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])
/(3*b*d) - (a*(e + f*x)*Sinh[c + d*x]^2)/(2*b^2*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 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*} \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \cosh ^3(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {f \cosh ^3(c+d x)}{9 b d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}+\frac {a^2 \int (e+f x) \cosh (c+d x) \, dx}{b^3}-\frac {a \int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {2 \int (e+f x) \cosh (c+d x) \, dx}{3 b}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}\\ &=\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {f \cosh ^3(c+d x)}{9 b d^2}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac {\left (a \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^3}-\frac {\left (a \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^3}-\frac {\left (a^2 f\right ) \int \sinh (c+d x) \, dx}{b^3 d}+\frac {(a f) \int \sinh ^2(c+d x) \, dx}{2 b^2 d}-\frac {(2 f) \int \sinh (c+d x) \, dx}{3 b d}\\ &=\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {a \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^4 d}-\frac {a \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^4 d}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac {(a f) \int 1 \, dx}{4 b^2 d}+\frac {\left (a \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^4 d}+\frac {\left (a \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^4 d}\\ &=-\frac {a f x}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {a \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^4 d}-\frac {a \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^4 d}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}+\frac {\left (a \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^4 d^2}+\frac {\left (a \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^4 d^2}\\ &=-\frac {a f x}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {a \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^4 d}-\frac {a \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^4 d}-\frac {a \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}\\ \end {align*}

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Mathematica [A]
time = 2.15, size = 551, normalized size = 1.38 \begin {gather*} -\frac {-36 b^2 d e (-a \log (a+b \sinh (c+d x))+b \sinh (c+d x))+36 b^2 f \left (b \cosh (c+d x)+a \left (-\frac {1}{2} (c+d x)^2+(c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-c \log (a+b \sinh (c+d x))+\text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+\text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-b d x \sinh (c+d x)\right )+6 d e \left (3 a b^2 \cosh (2 (c+d x))+6 a \left (2 a^2+b^2\right ) \log (a+b \sinh (c+d x))-3 b \left (4 a^2+b^2\right ) \sinh (c+d x)-b^3 \sinh (3 (c+d x))\right )+f \left (18 b \left (4 a^2+b^2\right ) \cosh (c+d x)+18 a b^2 d x \cosh (2 (c+d x))+2 b^3 \cosh (3 (c+d x))+36 a \left (2 a^2+b^2\right ) \left (-\frac {1}{2} (c+d x)^2+(c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-c \log (a+b \sinh (c+d x))+\text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+\text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-18 b \left (4 a^2+b^2\right ) d x \sinh (c+d x)-9 a b^2 \sinh (2 (c+d x))-6 b^3 d x \sinh (3 (c+d x))\right )}{72 b^4 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(372)=744\).
time = 1.63, size = 1102, normalized size = 2.76

method	result	size
risch	\(-\frac {a \left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 b^{2} d^{2}}-\frac {a 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^{2}}+\frac {a f \,c^{2}}{d^{2} b^{2}}+\frac {2 a e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{2}}-\frac {a e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{2}}-\frac {2 a f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{2}}+\frac {a f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{2}}-\frac {a 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^{2}}-\frac {a 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^{2}}-\frac {a 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^{2}}-\frac {a 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^{2}}+\frac {a f \,x^{2}}{2 b^{2}}+\frac {a^{3} f \,x^{2}}{2 b^{4}}+\frac {a^{3} f \,c^{2}}{d^{2} b^{4}}-\frac {a^{3} e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{4}}+\frac {2 a^{3} e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{4}}-\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^{4}}+\frac {\left (3 d x f +3 d e -f \right ) {\mathrm e}^{3 d x +3 c}}{72 d^{2} b}-\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^{4}}-\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^{4}}-\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^{4}}-\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^{4}}-\frac {2 a^{3} f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{4}}-\frac {a \left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 b^{2} d^{2}}+\frac {2 a^{3} f c x}{d \,b^{4}}-\frac {a 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^{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^{4}}+\frac {2 a c f x}{d \,b^{2}}-\frac {a e x}{b^{2}}-\frac {\left (3 d x f +3 d e +f \right ) {\mathrm e}^{-3 d x -3 c}}{72 d^{2} b}-\frac {a^{3} e x}{b^{4}}-\frac {\left (4 a^{2}+3 b^{2}\right ) \left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{8 b^{3} d^{2}}+\frac {\left (4 a^{2} d f x +3 b^{2} d f x +4 a^{2} d e +3 b^{2} d e -4 a^{2} f -3 f \,b^{2}\right ) {\mathrm e}^{d x +c}}{8 b^{3} d^{2}}+\frac {a^{3} f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{4}}\)	\(1102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/144*f*((72*(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) + 3*b^3*e^(4*c) - (4*a^2*b*d*e^(4*c) + 3*b
^3*d*e^(4*c))*x)*e^(d*x) + 18*(4*a^2*b*e^(2*c) + 3*b^3*e^(2*c) + (4*a^2*b*d*e^(2*c) + 3*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) - 9*int
egrate(32*((a^4*e^c + a^2*b^2*e^c)*x*e^(d*x) - (a^3*b + a*b^3)*x)/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) -
b^5), x)) - 1/24*((3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + 3*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) + 2
4*(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 + 3*b^2)*e^(-d*x
- c))/(b^3*d) + 24*(a^3 + a*b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^4*d))*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3011 vs. \(2 (377) = 754\).
time = 0.39, size = 3011, normalized size = 7.53 \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)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/144*(2*(3*b^3*d*f*x + 3*b^3*d*cosh(1) + 3*b^3*d*sinh(1) - b^3*f)*cosh(d*x + c)^6 + 2*(3*b^3*d*f*x + 3*b^3*d*
cosh(1) + 3*b^3*d*sinh(1) - 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*cosh(1) + 2*a*
b^2*d*sinh(1) - a*b^2*f)*cosh(d*x + c)^5 - 3*(6*a*b^2*d*f*x + 6*a*b^2*d*cosh(1) + 6*a*b^2*d*sinh(1) - 3*a*b^2*
f - 4*(3*b^3*d*f*x + 3*b^3*d*cosh(1) + 3*b^3*d*sinh(1) - b^3*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 6*b^3*d*cosh(
1) + 18*((4*a^2*b + 3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*cosh(1) + (4*a^2*b + 3*b^3)*d*sinh(1) - (4*a^2*b + 3*b^
3)*f)*cosh(d*x + c)^4 - 6*b^3*d*sinh(1) + 3*(6*(4*a^2*b + 3*b^3)*d*f*x + 6*(4*a^2*b + 3*b^3)*d*cosh(1) + 10*(3
*b^3*d*f*x + 3*b^3*d*cosh(1) + 3*b^3*d*sinh(1) - b^3*f)*cosh(d*x + c)^2 + 6*(4*a^2*b + 3*b^3)*d*sinh(1) - 6*(4
*a^2*b + 3*b^3)*f - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*cosh(1) + 2*a*b^2*d*sinh(1) - a*b^2*f)*cosh(d*x + c))*sinh(d
*x + c)^4 - 2*b^3*f + 72*((a^3 + a*b^2)*d^2*f*x^2 - 2*(a^3 + a*b^2)*c^2*f + 2*((a^3 + a*b^2)*d^2*x + 2*(a^3 +
a*b^2)*c*d)*cosh(1) + 2*((a^3 + a*b^2)*d^2*x + 2*(a^3 + a*b^2)*c*d)*sinh(1))*cosh(d*x + c)^3 + 2*(36*(a^3 + a*
b^2)*d^2*f*x^2 - 72*(a^3 + a*b^2)*c^2*f + 20*(3*b^3*d*f*x + 3*b^3*d*cosh(1) + 3*b^3*d*sinh(1) - b^3*f)*cosh(d*
x + c)^3 - 45*(2*a*b^2*d*f*x + 2*a*b^2*d*cosh(1) + 2*a*b^2*d*sinh(1) - a*b^2*f)*cosh(d*x + c)^2 + 72*((a^3 + a
*b^2)*d^2*x + 2*(a^3 + a*b^2)*c*d)*cosh(1) + 36*((4*a^2*b + 3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*cosh(1) + (4*a^
2*b + 3*b^3)*d*sinh(1) - (4*a^2*b + 3*b^3)*f)*cosh(d*x + c) + 72*((a^3 + a*b^2)*d^2*x + 2*(a^3 + a*b^2)*c*d)*s
inh(1))*sinh(d*x + c)^3 - 18*((4*a^2*b + 3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*cosh(1) + (4*a^2*b + 3*b^3)*d*sinh
(1) + (4*a^2*b + 3*b^3)*f)*cosh(d*x + c)^2 + 6*(5*(3*b^3*d*f*x + 3*b^3*d*cosh(1) + 3*b^3*d*sinh(1) - b^3*f)*co
sh(d*x + c)^4 - 3*(4*a^2*b + 3*b^3)*d*f*x - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*cosh(1) + 2*a*b^2*d*sinh(1) - a*b^2*
f)*cosh(d*x + c)^3 - 3*(4*a^2*b + 3*b^3)*d*cosh(1) + 18*((4*a^2*b + 3*b^3)*d*f*x + (4*a^2*b + 3*b^3)*d*cosh(1)
+ (4*a^2*b + 3*b^3)*d*sinh(1) - (4*a^2*b + 3*b^3)*f)*cosh(d*x + c)^2 - 3*(4*a^2*b + 3*b^3)*d*sinh(1) - 3*(4*a
^2*b + 3*b^3)*f + 36*((a^3 + a*b^2)*d^2*f*x^2 - 2*(a^3 + a*b^2)*c^2*f + 2*((a^3 + a*b^2)*d^2*x + 2*(a^3 + a*b^
2)*c*d)*cosh(1) + 2*((a^3 + a*b^2)*d^2*x + 2*(a^3 + a*b^2)*c*d)*sinh(1))*cosh(d*x + c))*sinh(d*x + c)^2 - 9*(2
*a*b^2*d*f*x + 2*a*b^2*d*cosh(1) + 2*a*b^2*d*sinh(1) + a*b^2*f)*cosh(d*x + c) - 144*((a^3 + a*b^2)*f*cosh(d*x
+ c)^3 + 3*(a^3 + a*b^2)*f*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^3 + a*b^2)*f*cosh(d*x + c)*sinh(d*x + c)^2 + (
a^3 + a*b^2)*f*sinh(d*x + c)^3)*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^3 + a*b^2)*f*cosh(d*x + c)^3 + 3*(a^3 + a*b^2)*f*cosh(d*x + c)^2*
sinh(d*x + c) + 3*(a^3 + a*b^2)*f*cosh(d*x + c)*sinh(d*x + c)^2 + (a^3 + a*b^2)*f*sinh(d*x + c)^3)*dilog((a*co
sh(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^3 + a*b^2)*c*f - (a^3 + a*b^2)*d*cosh(1) - (a^3 + a*b^2)*d*sinh(1))*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*c*f
- (a^3 + a*b^2)*d*cosh(1) - (a^3 + a*b^2)*d*sinh(1))*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*c*f - (a
^3 + a*b^2)*d*cosh(1) - (a^3 + a*b^2)*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^3 + a*b^2)*c*f - (a^3 + a
*b^2)*d*cosh(1) - (a^3 + a*b^2)*d*sinh(1))*sinh(d*x + c)^3)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sq
rt((a^2 + b^2)/b^2) + 2*a) + 144*(((a^3 + a*b^2)*c*f - (a^3 + a*b^2)*d*cosh(1) - (a^3 + a*b^2)*d*sinh(1))*cosh
(d*x + c)^3 + 3*((a^3 + a*b^2)*c*f - (a^3 + a*b^2)*d*cosh(1) - (a^3 + a*b^2)*d*sinh(1))*cosh(d*x + c)^2*sinh(d
*x + c) + 3*((a^3 + a*b^2)*c*f - (a^3 + a*b^2)*d*cosh(1) - (a^3 + a*b^2)*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c
)^2 + ((a^3 + a*b^2)*c*f - (a^3 + a*b^2)*d*cosh(1) - (a^3 + a*b^2)*d*sinh(1))*sinh(d*x + c)^3)*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^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)
*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2
)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*sinh(d*
x + c)^3)*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^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)^3 + 3*((a^3 + a*b^2)*d*f*x + (a^3 + a*b
^2)*c*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*cosh(d*x + c)*sinh(d*x +
c)^2 + ((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c*f)*sinh(d*x + c)^3)*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) - 3*(6*a*b^2*d*f*x - 4*(3*b^3*d*f*x + 3*b^3*d*
cosh(1) + 3*b^3*d*sinh(1) - b^3*f)*cosh(d*x + c)^5 + 6*a*b^2*d*cosh(1) + 15*(2*a*b^2*d*f*x + 2*a*b^2*d*cosh(1)
+ 2*a*b^2*d*sinh(1) - a*b^2*f)*cosh(d*x + c)^4 + 6*a*b^2*d*sinh(1) + 3*a*b^2*f - 24*((4*a^2*b + 3*b^3)*d*f*x
+ (4*a^2*b + 3*b^3)*d*cosh(1) + (4*a^2*b + 3*b^...

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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)**3*sinh(d*x+c)/(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)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)*cosh(d*x + c)^3*sinh(d*x + c)/(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 )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\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)^3*sinh(c + d*x)*(e + f*x))/(a + b*sinh(c + d*x)),x)

[Out]

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

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