3.4.92 \(\int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [392]

Optimal. Leaf size=578 \[ -\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac {4 f (e+f x) \cosh (c+d x)}{9 b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{9 b d^2}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d} \]

[Out]

-1/2*a*e*f*x/b^2/d-1/4*a*f^2*x^2/b^2/d+1/3*a^3*(f*x+e)^3/b^4/f-2*a^2*f*(f*x+e)*cosh(d*x+c)/b^3/d^2+4/9*f*(f*x+
e)*cosh(d*x+c)/b/d^2-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c
)/(a+(a^2+b^2)^(1/2)))/b^4/d-2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^2-2*a^3*f*(f*x
+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^2+2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2))
)/b^4/d^3+2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^3+2*a^2*f^2*sinh(d*x+c)/b^3/d^3-4/9*f^2
*sinh(d*x+c)/b/d^3+a^2*(f*x+e)^2*sinh(d*x+c)/b^3/d+1/2*a*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^2/d^2-1/4*a*f^2*s
inh(d*x+c)^2/b^2/d^3-1/2*a*(f*x+e)^2*sinh(d*x+c)^2/b^2/d-2/9*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)^2/b/d^2+2/27*f^
2*sinh(d*x+c)^3/b/d^3+1/3*(f*x+e)^2*sinh(d*x+c)^3/b/d

________________________________________________________________________________________

Rubi [A]
time = 0.66, antiderivative size = 578, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5698, 5554, 3391, 3377, 2717, 5680, 2221, 2611, 2320, 6724} \begin {gather*} \frac {a^3 (e+f x)^3}{3 b^4 f}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^4 d}-\frac {a^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^4 d}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}+\frac {a f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^2 d^2}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}-\frac {4 f^2 \sinh (c+d x)}{9 b d^3}+\frac {4 f (e+f x) \cosh (c+d x)}{9 b d^2}-\frac {2 f (e+f x) \sinh ^2(c+d x) \cosh (c+d x)}{9 b d^2}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*e*f*x)/(b^2*d) - (a*f^2*x^2)/(4*b^2*d) + (a^3*(e + f*x)^3)/(3*b^4*f) - (2*a^2*f*(e + f*x)*Cosh[c + d*x
])/(b^3*d^2) + (4*f*(e + f*x)*Cosh[c + d*x])/(9*b*d^2) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^
2 + b^2])])/(b^4*d) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (2*a^3*f*(e +
f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(
c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))]
)/(b^4*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a^2*f^2*Sinh[c +
d*x])/(b^3*d^3) - (4*f^2*Sinh[c + d*x])/(9*b*d^3) + (a^2*(e + f*x)^2*Sinh[c + d*x])/(b^3*d) + (a*f*(e + f*x)*
Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d^2) - (a*f^2*Sinh[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x]^
2)/(2*b^2*d) - (2*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x]^2)/(9*b*d^2) + (2*f^2*Sinh[c + d*x]^3)/(27*b*d^3) +
((e + f*x)^2*Sinh[c + d*x]^3)/(3*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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 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 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 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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d}-\frac {a \int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {(2 f) \int (e+f x) \sinh ^3(c+d x) \, dx}{3 b d}\\ &=-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{9 b d^2}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d}+\frac {a^2 \int (e+f x)^2 \cosh (c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {(a f) \int (e+f x) \sinh ^2(c+d x) \, dx}{b^2 d}+\frac {(4 f) \int (e+f x) \sinh (c+d x) \, dx}{9 b d}\\ &=\frac {a^3 (e+f x)^3}{3 b^4 f}+\frac {4 f (e+f x) \cosh (c+d x)}{9 b d^2}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{9 b d^2}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d}-\frac {a^3 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac {a^3 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac {\left (2 a^2 f\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^3 d}-\frac {(a f) \int (e+f x) \, dx}{2 b^2 d}-\frac {\left (4 f^2\right ) \int \cosh (c+d x) \, dx}{9 b d^2}\\ &=-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac {4 f (e+f x) \cosh (c+d x)}{9 b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {4 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{9 b d^2}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d}+\frac {\left (2 a^3 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}+\frac {\left (2 a^3 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}+\frac {\left (2 a^2 f^2\right ) \int \cosh (c+d x) \, dx}{b^3 d^2}\\ &=-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac {4 f (e+f x) \cosh (c+d x)}{9 b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{9 b d^2}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d}+\frac {\left (2 a^3 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d^2}+\frac {\left (2 a^3 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d^2}\\ &=-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac {4 f (e+f x) \cosh (c+d x)}{9 b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{9 b d^2}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d}+\frac {\left (2 a^3 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^3}+\frac {\left (2 a^3 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^3}\\ &=-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {a^3 (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac {4 f (e+f x) \cosh (c+d x)}{9 b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {2 f (e+f x) \cosh (c+d x) \sinh ^2(c+d x)}{9 b d^2}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {(e+f x)^2 \sinh ^3(c+d x)}{3 b d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1945\) vs. \(2(578)=1156\).
time = 3.30, size = 1945, normalized size = 3.37 \begin {gather*} \frac {e^{-3 c} \left (432 a^3 c^2 d e e^{3 c} f+864 a^3 c d^2 e e^{3 c} f x+432 a^3 d^3 e e^{3 c} f x^2+144 a^3 d^3 e^{3 c} f^2 x^3-432 a^2 b e^{2 c} f^2 \cosh (d x)+108 b^3 e^{2 c} f^2 \cosh (d x)+432 a^2 b e^{4 c} f^2 \cosh (d x)-108 b^3 e^{4 c} f^2 \cosh (d x)-432 a^2 b d e^{2 c} f^2 x \cosh (d x)+108 b^3 d e^{2 c} f^2 x \cosh (d x)-432 a^2 b d e^{4 c} f^2 x \cosh (d x)+108 b^3 d e^{4 c} f^2 x \cosh (d x)-216 a^2 b d^2 e^{2 c} f^2 x^2 \cosh (d x)+54 b^3 d^2 e^{2 c} f^2 x^2 \cosh (d x)+216 a^2 b d^2 e^{4 c} f^2 x^2 \cosh (d x)-54 b^3 d^2 e^{4 c} f^2 x^2 \cosh (d x)-27 a b^2 e^c f^2 \cosh (2 d x)-27 a b^2 e^{5 c} f^2 \cosh (2 d x)-54 a b^2 d e^c f^2 x \cosh (2 d x)+54 a b^2 d e^{5 c} f^2 x \cosh (2 d x)-54 a b^2 d^2 e^c f^2 x^2 \cosh (2 d x)-54 a b^2 d^2 e^{5 c} f^2 x^2 \cosh (2 d x)-4 b^3 f^2 \cosh (3 d x)+4 b^3 e^{6 c} f^2 \cosh (3 d x)-12 b^3 d f^2 x \cosh (3 d x)-12 b^3 d e^{6 c} f^2 x \cosh (3 d x)-18 b^3 d^2 f^2 x^2 \cosh (3 d x)+18 b^3 d^2 e^{6 c} f^2 x^2 \cosh (3 d x)-864 a^2 b d e e^{3 c} f \cosh (c+d x)+216 b^3 d e e^{3 c} f \cosh (c+d x)-108 a b^2 d^2 e^2 e^{3 c} \cosh (2 (c+d x))-216 a b^2 d^2 e e^{3 c} f x \cosh (2 (c+d x))-24 b^3 d e e^{3 c} f \cosh (3 (c+d x))-864 a^3 c d e e^{3 c} f \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-864 a^3 d^2 e e^{3 c} f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-864 a^3 c d e e^{3 c} f \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-864 a^3 d^2 e e^{3 c} f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-432 a^3 d^2 e^{3 c} f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-432 a^3 d^2 e^{3 c} f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-432 a^3 d^2 e^2 e^{3 c} \log (a+b \sinh (c+d x))+864 a^3 c d e e^{3 c} f \log (a+b \sinh (c+d x))-864 a^3 d e e^{3 c} f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-864 a^3 d e e^{3 c} f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-864 a^3 d e^{3 c} f^2 x \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-864 a^3 d e^{3 c} f^2 x \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+864 a^3 e^{3 c} f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+864 a^3 e^{3 c} f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+432 a^2 b e^{2 c} f^2 \sinh (d x)-108 b^3 e^{2 c} f^2 \sinh (d x)+432 a^2 b e^{4 c} f^2 \sinh (d x)-108 b^3 e^{4 c} f^2 \sinh (d x)+432 a^2 b d e^{2 c} f^2 x \sinh (d x)-108 b^3 d e^{2 c} f^2 x \sinh (d x)-432 a^2 b d e^{4 c} f^2 x \sinh (d x)+108 b^3 d e^{4 c} f^2 x \sinh (d x)+216 a^2 b d^2 e^{2 c} f^2 x^2 \sinh (d x)-54 b^3 d^2 e^{2 c} f^2 x^2 \sinh (d x)+216 a^2 b d^2 e^{4 c} f^2 x^2 \sinh (d x)-54 b^3 d^2 e^{4 c} f^2 x^2 \sinh (d x)+27 a b^2 e^c f^2 \sinh (2 d x)-27 a b^2 e^{5 c} f^2 \sinh (2 d x)+54 a b^2 d e^c f^2 x \sinh (2 d x)+54 a b^2 d e^{5 c} f^2 x \sinh (2 d x)+54 a b^2 d^2 e^c f^2 x^2 \sinh (2 d x)-54 a b^2 d^2 e^{5 c} f^2 x^2 \sinh (2 d x)+4 b^3 f^2 \sinh (3 d x)+4 b^3 e^{6 c} f^2 \sinh (3 d x)+12 b^3 d f^2 x \sinh (3 d x)-12 b^3 d e^{6 c} f^2 x \sinh (3 d x)+18 b^3 d^2 f^2 x^2 \sinh (3 d x)+18 b^3 d^2 e^{6 c} f^2 x^2 \sinh (3 d x)+432 a^2 b d^2 e^2 e^{3 c} \sinh (c+d x)-108 b^3 d^2 e^2 e^{3 c} \sinh (c+d x)+864 a^2 b d^2 e e^{3 c} f x \sinh (c+d x)-216 b^3 d^2 e e^{3 c} f x \sinh (c+d x)+108 a b^2 d e e^{3 c} f \sinh (2 (c+d x))+36 b^3 d^2 e^2 e^{3 c} \sinh (3 (c+d x))+72 b^3 d^2 e e^{3 c} f x \sinh (3 (c+d x))\right )}{432 b^4 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(432*a^3*c^2*d*e*E^(3*c)*f + 864*a^3*c*d^2*e*E^(3*c)*f*x + 432*a^3*d^3*e*E^(3*c)*f*x^2 + 144*a^3*d^3*E^(3*c)*f
^2*x^3 - 432*a^2*b*E^(2*c)*f^2*Cosh[d*x] + 108*b^3*E^(2*c)*f^2*Cosh[d*x] + 432*a^2*b*E^(4*c)*f^2*Cosh[d*x] - 1
08*b^3*E^(4*c)*f^2*Cosh[d*x] - 432*a^2*b*d*E^(2*c)*f^2*x*Cosh[d*x] + 108*b^3*d*E^(2*c)*f^2*x*Cosh[d*x] - 432*a
^2*b*d*E^(4*c)*f^2*x*Cosh[d*x] + 108*b^3*d*E^(4*c)*f^2*x*Cosh[d*x] - 216*a^2*b*d^2*E^(2*c)*f^2*x^2*Cosh[d*x] +
54*b^3*d^2*E^(2*c)*f^2*x^2*Cosh[d*x] + 216*a^2*b*d^2*E^(4*c)*f^2*x^2*Cosh[d*x] - 54*b^3*d^2*E^(4*c)*f^2*x^2*C
osh[d*x] - 27*a*b^2*E^c*f^2*Cosh[2*d*x] - 27*a*b^2*E^(5*c)*f^2*Cosh[2*d*x] - 54*a*b^2*d*E^c*f^2*x*Cosh[2*d*x]
+ 54*a*b^2*d*E^(5*c)*f^2*x*Cosh[2*d*x] - 54*a*b^2*d^2*E^c*f^2*x^2*Cosh[2*d*x] - 54*a*b^2*d^2*E^(5*c)*f^2*x^2*C
osh[2*d*x] - 4*b^3*f^2*Cosh[3*d*x] + 4*b^3*E^(6*c)*f^2*Cosh[3*d*x] - 12*b^3*d*f^2*x*Cosh[3*d*x] - 12*b^3*d*E^(
6*c)*f^2*x*Cosh[3*d*x] - 18*b^3*d^2*f^2*x^2*Cosh[3*d*x] + 18*b^3*d^2*E^(6*c)*f^2*x^2*Cosh[3*d*x] - 864*a^2*b*d
*e*E^(3*c)*f*Cosh[c + d*x] + 216*b^3*d*e*E^(3*c)*f*Cosh[c + d*x] - 108*a*b^2*d^2*e^2*E^(3*c)*Cosh[2*(c + d*x)]
- 216*a*b^2*d^2*e*E^(3*c)*f*x*Cosh[2*(c + d*x)] - 24*b^3*d*e*E^(3*c)*f*Cosh[3*(c + d*x)] - 864*a^3*c*d*e*E^(3
*c)*f*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 864*a^3*d^2*e*E^(3*c)*f*x*Log[1 + (b*E^(c + d*x))/(a -
Sqrt[a^2 + b^2])] - 864*a^3*c*d*e*E^(3*c)*f*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 864*a^3*d^2*e*E^(
3*c)*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 432*a^3*d^2*E^(3*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x)
)/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 432*a^3*d^2*E^(3*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(
a^2 + b^2)*E^(2*c)])] - 432*a^3*d^2*e^2*E^(3*c)*Log[a + b*Sinh[c + d*x]] + 864*a^3*c*d*e*E^(3*c)*f*Log[a + b*S
inh[c + d*x]] - 864*a^3*d*e*E^(3*c)*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 864*a^3*d*e*E^(3*c)
*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 864*a^3*d*E^(3*c)*f^2*x*PolyLog[2, -((b*E^(2*c + d*x
))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 864*a^3*d*E^(3*c)*f^2*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqr
t[(a^2 + b^2)*E^(2*c)]))] + 864*a^3*E^(3*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*
c)]))] + 864*a^3*E^(3*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 432*a^2*b*
E^(2*c)*f^2*Sinh[d*x] - 108*b^3*E^(2*c)*f^2*Sinh[d*x] + 432*a^2*b*E^(4*c)*f^2*Sinh[d*x] - 108*b^3*E^(4*c)*f^2*
Sinh[d*x] + 432*a^2*b*d*E^(2*c)*f^2*x*Sinh[d*x] - 108*b^3*d*E^(2*c)*f^2*x*Sinh[d*x] - 432*a^2*b*d*E^(4*c)*f^2*
x*Sinh[d*x] + 108*b^3*d*E^(4*c)*f^2*x*Sinh[d*x] + 216*a^2*b*d^2*E^(2*c)*f^2*x^2*Sinh[d*x] - 54*b^3*d^2*E^(2*c)
*f^2*x^2*Sinh[d*x] + 216*a^2*b*d^2*E^(4*c)*f^2*x^2*Sinh[d*x] - 54*b^3*d^2*E^(4*c)*f^2*x^2*Sinh[d*x] + 27*a*b^2
*E^c*f^2*Sinh[2*d*x] - 27*a*b^2*E^(5*c)*f^2*Sinh[2*d*x] + 54*a*b^2*d*E^c*f^2*x*Sinh[2*d*x] + 54*a*b^2*d*E^(5*c
)*f^2*x*Sinh[2*d*x] + 54*a*b^2*d^2*E^c*f^2*x^2*Sinh[2*d*x] - 54*a*b^2*d^2*E^(5*c)*f^2*x^2*Sinh[2*d*x] + 4*b^3*
f^2*Sinh[3*d*x] + 4*b^3*E^(6*c)*f^2*Sinh[3*d*x] + 12*b^3*d*f^2*x*Sinh[3*d*x] - 12*b^3*d*E^(6*c)*f^2*x*Sinh[3*d
*x] + 18*b^3*d^2*f^2*x^2*Sinh[3*d*x] + 18*b^3*d^2*E^(6*c)*f^2*x^2*Sinh[3*d*x] + 432*a^2*b*d^2*e^2*E^(3*c)*Sinh
[c + d*x] - 108*b^3*d^2*e^2*E^(3*c)*Sinh[c + d*x] + 864*a^2*b*d^2*e*E^(3*c)*f*x*Sinh[c + d*x] - 216*b^3*d^2*e*
E^(3*c)*f*x*Sinh[c + d*x] + 108*a*b^2*d*e*E^(3*c)*f*Sinh[2*(c + d*x)] + 36*b^3*d^2*e^2*E^(3*c)*Sinh[3*(c + d*x
)] + 72*b^3*d^2*e*E^(3*c)*f*x*Sinh[3*(c + d*x)])/(432*b^4*d^3*E^(3*c))

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Maple [F]
time = 2.30, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-1/24*(24*(d*x + c)*a^3/(b^4*d) + 24*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(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) + (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))*e^2 - 1/432*(144*a^3*d^3*f^2*x^3*e^(3*c) + 432*a^3*d^3*
f*x^2*e^(3*c + 1) - 2*(9*b^3*d^2*f^2*x^2*e^(6*c) + 2*b^3*f^2*e^(6*c) - 6*b^3*d*f*e^(6*c + 1) - 6*(b^3*d*f^2*e^
(6*c) - 3*b^3*d^2*f*e^(6*c + 1))*x)*e^(3*d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^(5*c) + a*b^2*f^2*e^(5*c) - 2*a*b^2*
d*f*e^(5*c + 1) - 2*(a*b^2*d*f^2*e^(5*c) - 2*a*b^2*d^2*f*e^(5*c + 1))*x)*e^(2*d*x) - 54*(8*a^2*b*f^2*e^(4*c) -
2*b^3*f^2*e^(4*c) + (4*a^2*b*d^2*f^2*e^(4*c) - b^3*d^2*f^2*e^(4*c))*x^2 - 2*(4*a^2*b*d*f^2*e^(4*c) - b^3*d*f^
2*e^(4*c) - (4*a^2*b*d^2*f*e^(4*c) - b^3*d^2*f*e^(4*c))*e)*x - 2*(4*a^2*b*d*f*e^(4*c) - b^3*d*f*e^(4*c))*e)*e^
(d*x) + 54*(8*a^2*b*f^2*e^(2*c) - 2*b^3*f^2*e^(2*c) + (4*a^2*b*d^2*f^2*e^(2*c) - b^3*d^2*f^2*e^(2*c))*x^2 + 2*
(4*a^2*b*d*f^2*e^(2*c) - b^3*d*f^2*e^(2*c) + (4*a^2*b*d^2*f*e^(2*c) - b^3*d^2*f*e^(2*c))*e)*x + 2*(4*a^2*b*d*f
*e^(2*c) - b^3*d*f*e^(2*c))*e)*e^(-d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^c + 2*a*b^2*d*f*e^(c + 1) + a*b^2*f^2*e^c
+ 2*(2*a*b^2*d^2*f*e^(c + 1) + a*b^2*d*f^2*e^c)*x)*e^(-2*d*x) + 2*(9*b^3*d^2*f^2*x^2 + 6*b^3*d*f*e + 2*b^3*f^2
+ 6*(3*b^3*d^2*f*e + b^3*d*f^2)*x)*e^(-3*d*x))*e^(-3*c)/(b^4*d^3) + integrate(-2*(a^3*b*f^2*x^2 + 2*a^3*b*f*x
*e - (a^4*f^2*x^2*e^c + 2*a^4*f*x*e^(c + 1))*e^(d*x))/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d*x + c) - b^5), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6483 vs. \(2 (549) = 1098\).
time = 0.42, size = 6483, normalized size = 11.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/432*(18*b^3*d^2*f^2*x^2 + 12*b^3*d*f^2*x + 18*b^3*d^2*cosh(1)^2 - 2*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*
b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x +
3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*cosh(d*x + c)^6 + 18*b^3*d^2*sinh(1)^2 - 2*(9*b^3*d^2*f^2*x^2 - 6*b^3*d
*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^
3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*sinh(d*x + c)^6 + 27*(2*a*b^2*d^2*f^2*x^2 - 2*a*b^2*d*f^2*x
+ 2*a*b^2*d^2*cosh(1)^2 + 2*a*b^2*d^2*sinh(1)^2 + a*b^2*f^2 + 2*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cosh(1) + 2*(2*a
*b^2*d^2*f*x + 2*a*b^2*d^2*cosh(1) - a*b^2*d*f)*sinh(1))*cosh(d*x + c)^5 + 3*(18*a*b^2*d^2*f^2*x^2 - 18*a*b^2*
d*f^2*x + 18*a*b^2*d^2*cosh(1)^2 + 18*a*b^2*d^2*sinh(1)^2 + 9*a*b^2*f^2 + 18*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cos
h(1) - 4*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3
*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*cosh(d*x + c) + 18*(2*a
*b^2*d^2*f*x + 2*a*b^2*d^2*cosh(1) - a*b^2*d*f)*sinh(1))*sinh(d*x + c)^5 + 4*b^3*f^2 - 54*((4*a^2*b - b^3)*d^2
*f^2*x^2 - 2*(4*a^2*b - b^3)*d*f^2*x + (4*a^2*b - b^3)*d^2*cosh(1)^2 + (4*a^2*b - b^3)*d^2*sinh(1)^2 + 2*(4*a^
2*b - b^3)*f^2 + 2*((4*a^2*b - b^3)*d^2*f*x - (4*a^2*b - b^3)*d*f)*cosh(1) + 2*((4*a^2*b - b^3)*d^2*f*x + (4*a
^2*b - b^3)*d^2*cosh(1) - (4*a^2*b - b^3)*d*f)*sinh(1))*cosh(d*x + c)^4 - 3*(18*(4*a^2*b - b^3)*d^2*f^2*x^2 -
36*(4*a^2*b - b^3)*d*f^2*x + 18*(4*a^2*b - b^3)*d^2*cosh(1)^2 + 18*(4*a^2*b - b^3)*d^2*sinh(1)^2 + 36*(4*a^2*b
- b^3)*f^2 + 10*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 +
6*(3*b^3*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*cosh(d*x + c)^2
+ 36*((4*a^2*b - b^3)*d^2*f*x - (4*a^2*b - b^3)*d*f)*cosh(1) - 45*(2*a*b^2*d^2*f^2*x^2 - 2*a*b^2*d*f^2*x + 2*
a*b^2*d^2*cosh(1)^2 + 2*a*b^2*d^2*sinh(1)^2 + a*b^2*f^2 + 2*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cosh(1) + 2*(2*a*b^2
*d^2*f*x + 2*a*b^2*d^2*cosh(1) - a*b^2*d*f)*sinh(1))*cosh(d*x + c) + 36*((4*a^2*b - b^3)*d^2*f*x + (4*a^2*b -
b^3)*d^2*cosh(1) - (4*a^2*b - b^3)*d*f)*sinh(1))*sinh(d*x + c)^4 - 144*(a^3*d^3*f^2*x^3 + 2*a^3*c^3*f^2 + 3*(a
^3*d^3*x + 2*a^3*c*d^2)*cosh(1)^2 + 3*(a^3*d^3*x + 2*a^3*c*d^2)*sinh(1)^2 + 3*(a^3*d^3*f*x^2 - 2*a^3*c^2*d*f)*
cosh(1) + 3*(a^3*d^3*f*x^2 - 2*a^3*c^2*d*f + 2*(a^3*d^3*x + 2*a^3*c*d^2)*cosh(1))*sinh(1))*cosh(d*x + c)^3 - 2
*(72*a^3*d^3*f^2*x^3 + 144*a^3*c^3*f^2 + 20*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d
^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*
f)*sinh(1))*cosh(d*x + c)^3 + 216*(a^3*d^3*x + 2*a^3*c*d^2)*cosh(1)^2 - 135*(2*a*b^2*d^2*f^2*x^2 - 2*a*b^2*d*f
^2*x + 2*a*b^2*d^2*cosh(1)^2 + 2*a*b^2*d^2*sinh(1)^2 + a*b^2*f^2 + 2*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cosh(1) + 2
*(2*a*b^2*d^2*f*x + 2*a*b^2*d^2*cosh(1) - a*b^2*d*f)*sinh(1))*cosh(d*x + c)^2 + 216*(a^3*d^3*x + 2*a^3*c*d^2)*
sinh(1)^2 + 216*(a^3*d^3*f*x^2 - 2*a^3*c^2*d*f)*cosh(1) + 108*((4*a^2*b - b^3)*d^2*f^2*x^2 - 2*(4*a^2*b - b^3)
*d*f^2*x + (4*a^2*b - b^3)*d^2*cosh(1)^2 + (4*a^2*b - b^3)*d^2*sinh(1)^2 + 2*(4*a^2*b - b^3)*f^2 + 2*((4*a^2*b
- b^3)*d^2*f*x - (4*a^2*b - b^3)*d*f)*cosh(1) + 2*((4*a^2*b - b^3)*d^2*f*x + (4*a^2*b - b^3)*d^2*cosh(1) - (4
*a^2*b - b^3)*d*f)*sinh(1))*cosh(d*x + c) + 216*(a^3*d^3*f*x^2 - 2*a^3*c^2*d*f + 2*(a^3*d^3*x + 2*a^3*c*d^2)*c
osh(1))*sinh(1))*sinh(d*x + c)^3 + 54*((4*a^2*b - b^3)*d^2*f^2*x^2 + 2*(4*a^2*b - b^3)*d*f^2*x + (4*a^2*b - b^
3)*d^2*cosh(1)^2 + (4*a^2*b - b^3)*d^2*sinh(1)^2 + 2*(4*a^2*b - b^3)*f^2 + 2*((4*a^2*b - b^3)*d^2*f*x + (4*a^2
*b - b^3)*d*f)*cosh(1) + 2*((4*a^2*b - b^3)*d^2*f*x + (4*a^2*b - b^3)*d^2*cosh(1) + (4*a^2*b - b^3)*d*f)*sinh(
1))*cosh(d*x + c)^2 + 6*(9*(4*a^2*b - b^3)*d^2*f^2*x^2 + 18*(4*a^2*b - b^3)*d*f^2*x + 9*(4*a^2*b - b^3)*d^2*co
sh(1)^2 - 5*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*
b^3*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*cosh(d*x + c)^4 + 9*
(4*a^2*b - b^3)*d^2*sinh(1)^2 + 45*(2*a*b^2*d^2*f^2*x^2 - 2*a*b^2*d*f^2*x + 2*a*b^2*d^2*cosh(1)^2 + 2*a*b^2*d^
2*sinh(1)^2 + a*b^2*f^2 + 2*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cosh(1) + 2*(2*a*b^2*d^2*f*x + 2*a*b^2*d^2*cosh(1) -
a*b^2*d*f)*sinh(1))*cosh(d*x + c)^3 + 18*(4*a^2*b - b^3)*f^2 - 54*((4*a^2*b - b^3)*d^2*f^2*x^2 - 2*(4*a^2*b -
b^3)*d*f^2*x + (4*a^2*b - b^3)*d^2*cosh(1)^2 + (4*a^2*b - b^3)*d^2*sinh(1)^2 + 2*(4*a^2*b - b^3)*f^2 + 2*((4*
a^2*b - b^3)*d^2*f*x - (4*a^2*b - b^3)*d*f)*cosh(1) + 2*((4*a^2*b - b^3)*d^2*f*x + (4*a^2*b - b^3)*d^2*cosh(1)
- (4*a^2*b - b^3)*d*f)*sinh(1))*cosh(d*x + c)^2 + 18*((4*a^2*b - b^3)*d^2*f*x + (4*a^2*b - b^3)*d*f)*cosh(1)
- 72*(a^3*d^3*f^2*x^3 + 2*a^3*c^3*f^2 + 3*(a^3*d^3*x + 2*a^3*c*d^2)*cosh(1)^2 + 3*(a^3*d^3*x + 2*a^3*c*d^2)*si
nh(1)^2 + 3*(a^3*d^3*f*x^2 - 2*a^3*c^2*d*f)*cos...

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

[Out]

integrate((f*x + e)^2*cosh(d*x + c)*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 )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{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)*sinh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)

[Out]

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

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