3.3.34 \(\int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [234]

Optimal. Leaf size=522 \[ -\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {2 a^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}-\frac {2 a^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^3}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2} \]

[Out]

-1/4*f^2*x/b/d^2+1/3*a^2*(f*x+e)^3/b^3/f-1/6*(f*x+e)^3/b/f-2*a*f^2*cosh(d*x+c)/b^2/d^3-a*(f*x+e)^2*cosh(d*x+c)
/b^2/d+2*a*f*(f*x+e)*sinh(d*x+c)/b^2/d^2+1/4*f^2*cosh(d*x+c)*sinh(d*x+c)/b/d^3+1/2*(f*x+e)^2*cosh(d*x+c)*sinh(
d*x+c)/b/d-1/2*f*(f*x+e)*sinh(d*x+c)^2/b/d^2-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d/(a^2+b
^2)^(1/2)+a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d/(a^2+b^2)^(1/2)-2*a^3*f*(f*x+e)*polylog(2
,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^2/(a^2+b^2)^(1/2)+2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^
2)^(1/2)))/b^3/d^2/(a^2+b^2)^(1/2)+2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^3/(a^2+b^2)^(1
/2)-2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^3/(a^2+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.73, antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5676, 3392, 32, 2715, 8, 3377, 2718, 3403, 2296, 2221, 2611, 2320, 6724} \begin {gather*} \frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3 \sqrt {a^2+b^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^3 d^3 \sqrt {a^2+b^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^3 d^2 \sqrt {a^2+b^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^3 d^2 \sqrt {a^2+b^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^3 d \sqrt {a^2+b^2}}+\frac {a^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d \sqrt {a^2+b^2}}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac {f^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^3}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {f^2 x}{4 b d^2}-\frac {(e+f x)^3}{6 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(f^2*x)/(b*d^2) + (a^2*(e + f*x)^3)/(3*b^3*f) - (e + f*x)^3/(6*b*f) - (2*a*f^2*Cosh[c + d*x])/(b^2*d^3) -
(a*(e + f*x)^2*Cosh[c + d*x])/(b^2*d) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3
*Sqrt[a^2 + b^2]*d) + (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d)
- (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (2*a^3
*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (2*a^3*f^2*Poly
Log[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) - (2*a^3*f^2*PolyLog[3, -((b*E^(c
+ d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) + (2*a*f*(e + f*x)*Sinh[c + d*x])/(b^2*d^2) + (f^2*
Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^3) + ((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d) - (f*(e + f*x)*Sinh
[c + d*x]^2)/(2*b*d^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 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 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 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 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 5676

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[(e + f*x)^m*(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]

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 \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac {a \int (e+f x)^2 \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {\int (e+f x)^2 \, dx}{2 b}+\frac {f^2 \int \sinh ^2(c+d x) \, dx}{2 b d^2}\\ &=-\frac {(e+f x)^3}{6 b f}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {a^2 \int (e+f x)^2 \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {(2 a f) \int (e+f x) \cosh (c+d x) \, dx}{b^2 d}-\frac {f^2 \int 1 \, dx}{4 b d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^3}-\frac {\left (2 a f^2\right ) \int \sinh (c+d x) \, dx}{b^2 d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 d}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}-\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^2 \sqrt {a^2+b^2}}+\frac {\left (2 a^3\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^2 \sqrt {a^2+b^2}}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} d}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {\left (2 a^3 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 \sqrt {a^2+b^2} d}-\frac {\left (2 a^3 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 \sqrt {a^2+b^2} d}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} d^2}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\frac {\left (2 a^3 f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 \sqrt {a^2+b^2} d^2}-\frac {\left (2 a^3 f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 \sqrt {a^2+b^2} d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} d^2}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}+\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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} d^3}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \cosh (c+d x)}{b^2 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} 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^3 \sqrt {a^2+b^2} d^3}+\frac {2 a f (e+f x) \sinh (c+d x)}{b^2 d^2}+\frac {f^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f (e+f x) \sinh ^2(c+d x)}{2 b d^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1589\) vs. \(2(522)=1044\).
time = 7.46, size = 1589, normalized size = 3.04 \begin {gather*} \frac {a^3 \left (2 d^2 e^2 \sqrt {\left (a^2+b^2\right ) e^{2 c}} \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 \sqrt {a^2+b^2} d^2 e e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\sqrt {a^2+b^2} d^2 e^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 )+2 \sqrt {a^2+b^2} d^2 e e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+\sqrt {a^2+b^2} d^2 e^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 )-2 \sqrt {a^2+b^2} d e^c f (e+f 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 )+2 \sqrt {a^2+b^2} d e^c f (e+f 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 )+2 \sqrt {a^2+b^2} e^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 )-2 \sqrt {a^2+b^2} e^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 )\right )}{b^3 \sqrt {a^2+b^2} d^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\left (\frac {\cosh (2 c+2 d x)}{48 b^3 d^3}-\frac {\sinh (2 c+2 d x)}{48 b^3 d^3}\right ) \left (-6 b^2 d^2 e^2-6 b^2 d e f-3 b^2 f^2-12 b^2 d^2 e f x-6 b^2 d f^2 x-6 b^2 d^2 f^2 x^2-24 a b d^2 e^2 \cosh (c+d x)-48 a b d e f \cosh (c+d x)-48 a b f^2 \cosh (c+d x)-48 a b d^2 e f x \cosh (c+d x)-48 a b d f^2 x \cosh (c+d x)-24 a b d^2 f^2 x^2 \cosh (c+d x)+48 a^2 d^3 e^2 x \cosh (2 c+2 d x)-24 b^2 d^3 e^2 x \cosh (2 c+2 d x)+48 a^2 d^3 e f x^2 \cosh (2 c+2 d x)-24 b^2 d^3 e f x^2 \cosh (2 c+2 d x)+16 a^2 d^3 f^2 x^3 \cosh (2 c+2 d x)-8 b^2 d^3 f^2 x^3 \cosh (2 c+2 d x)-24 a b d^2 e^2 \cosh (3 c+3 d x)+48 a b d e f \cosh (3 c+3 d x)-48 a b f^2 \cosh (3 c+3 d x)-48 a b d^2 e f x \cosh (3 c+3 d x)+48 a b d f^2 x \cosh (3 c+3 d x)-24 a b d^2 f^2 x^2 \cosh (3 c+3 d x)+6 b^2 d^2 e^2 \cosh (4 c+4 d x)-6 b^2 d e f \cosh (4 c+4 d x)+3 b^2 f^2 \cosh (4 c+4 d x)+12 b^2 d^2 e f x \cosh (4 c+4 d x)-6 b^2 d f^2 x \cosh (4 c+4 d x)+6 b^2 d^2 f^2 x^2 \cosh (4 c+4 d x)-24 a b d^2 e^2 \sinh (c+d x)-48 a b d e f \sinh (c+d x)-48 a b f^2 \sinh (c+d x)-48 a b d^2 e f x \sinh (c+d x)-48 a b d f^2 x \sinh (c+d x)-24 a b d^2 f^2 x^2 \sinh (c+d x)+48 a^2 d^3 e^2 x \sinh (2 c+2 d x)-24 b^2 d^3 e^2 x \sinh (2 c+2 d x)+48 a^2 d^3 e f x^2 \sinh (2 c+2 d x)-24 b^2 d^3 e f x^2 \sinh (2 c+2 d x)+16 a^2 d^3 f^2 x^3 \sinh (2 c+2 d x)-8 b^2 d^3 f^2 x^3 \sinh (2 c+2 d x)-24 a b d^2 e^2 \sinh (3 c+3 d x)+48 a b d e f \sinh (3 c+3 d x)-48 a b f^2 \sinh (3 c+3 d x)-48 a b d^2 e f x \sinh (3 c+3 d x)+48 a b d f^2 x \sinh (3 c+3 d x)-24 a b d^2 f^2 x^2 \sinh (3 c+3 d x)+6 b^2 d^2 e^2 \sinh (4 c+4 d x)-6 b^2 d e f \sinh (4 c+4 d x)+3 b^2 f^2 \sinh (4 c+4 d x)+12 b^2 d^2 e f x \sinh (4 c+4 d x)-6 b^2 d f^2 x \sinh (4 c+4 d x)+6 b^2 d^2 f^2 x^2 \sinh (4 c+4 d x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(2*d^2*e^2*Sqrt[(a^2 + b^2)*E^(2*c)]*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*Sqrt[a^2 + b^2]*d^2
*e*E^c*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - Sqrt[a^2 + b^2]*d^2*E^c*f^2*x^2*Lo
g[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*Sqrt[a^2 + b^2]*d^2*e*E^c*f*x*Log[1 + (b*E^(2
*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + Sqrt[a^2 + b^2]*d^2*E^c*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a
*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*Sqrt[a^2 + b^2]*d*E^c*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c
- Sqrt[(a^2 + b^2)*E^(2*c)]))] + 2*Sqrt[a^2 + b^2]*d*E^c*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c +
Sqrt[(a^2 + b^2)*E^(2*c)]))] + 2*Sqrt[a^2 + b^2]*E^c*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b
^2)*E^(2*c)]))] - 2*Sqrt[a^2 + b^2]*E^c*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])
)]))/(b^3*Sqrt[a^2 + b^2]*d^3*Sqrt[(a^2 + b^2)*E^(2*c)]) + (Cosh[2*c + 2*d*x]/(48*b^3*d^3) - Sinh[2*c + 2*d*x]
/(48*b^3*d^3))*(-6*b^2*d^2*e^2 - 6*b^2*d*e*f - 3*b^2*f^2 - 12*b^2*d^2*e*f*x - 6*b^2*d*f^2*x - 6*b^2*d^2*f^2*x^
2 - 24*a*b*d^2*e^2*Cosh[c + d*x] - 48*a*b*d*e*f*Cosh[c + d*x] - 48*a*b*f^2*Cosh[c + d*x] - 48*a*b*d^2*e*f*x*Co
sh[c + d*x] - 48*a*b*d*f^2*x*Cosh[c + d*x] - 24*a*b*d^2*f^2*x^2*Cosh[c + d*x] + 48*a^2*d^3*e^2*x*Cosh[2*c + 2*
d*x] - 24*b^2*d^3*e^2*x*Cosh[2*c + 2*d*x] + 48*a^2*d^3*e*f*x^2*Cosh[2*c + 2*d*x] - 24*b^2*d^3*e*f*x^2*Cosh[2*c
+ 2*d*x] + 16*a^2*d^3*f^2*x^3*Cosh[2*c + 2*d*x] - 8*b^2*d^3*f^2*x^3*Cosh[2*c + 2*d*x] - 24*a*b*d^2*e^2*Cosh[3
*c + 3*d*x] + 48*a*b*d*e*f*Cosh[3*c + 3*d*x] - 48*a*b*f^2*Cosh[3*c + 3*d*x] - 48*a*b*d^2*e*f*x*Cosh[3*c + 3*d*
x] + 48*a*b*d*f^2*x*Cosh[3*c + 3*d*x] - 24*a*b*d^2*f^2*x^2*Cosh[3*c + 3*d*x] + 6*b^2*d^2*e^2*Cosh[4*c + 4*d*x]
- 6*b^2*d*e*f*Cosh[4*c + 4*d*x] + 3*b^2*f^2*Cosh[4*c + 4*d*x] + 12*b^2*d^2*e*f*x*Cosh[4*c + 4*d*x] - 6*b^2*d*
f^2*x*Cosh[4*c + 4*d*x] + 6*b^2*d^2*f^2*x^2*Cosh[4*c + 4*d*x] - 24*a*b*d^2*e^2*Sinh[c + d*x] - 48*a*b*d*e*f*Si
nh[c + d*x] - 48*a*b*f^2*Sinh[c + d*x] - 48*a*b*d^2*e*f*x*Sinh[c + d*x] - 48*a*b*d*f^2*x*Sinh[c + d*x] - 24*a*
b*d^2*f^2*x^2*Sinh[c + d*x] + 48*a^2*d^3*e^2*x*Sinh[2*c + 2*d*x] - 24*b^2*d^3*e^2*x*Sinh[2*c + 2*d*x] + 48*a^2
*d^3*e*f*x^2*Sinh[2*c + 2*d*x] - 24*b^2*d^3*e*f*x^2*Sinh[2*c + 2*d*x] + 16*a^2*d^3*f^2*x^3*Sinh[2*c + 2*d*x] -
8*b^2*d^3*f^2*x^3*Sinh[2*c + 2*d*x] - 24*a*b*d^2*e^2*Sinh[3*c + 3*d*x] + 48*a*b*d*e*f*Sinh[3*c + 3*d*x] - 48*
a*b*f^2*Sinh[3*c + 3*d*x] - 48*a*b*d^2*e*f*x*Sinh[3*c + 3*d*x] + 48*a*b*d*f^2*x*Sinh[3*c + 3*d*x] - 24*a*b*d^2
*f^2*x^2*Sinh[3*c + 3*d*x] + 6*b^2*d^2*e^2*Sinh[4*c + 4*d*x] - 6*b^2*d*e*f*Sinh[4*c + 4*d*x] + 3*b^2*f^2*Sinh[
4*c + 4*d*x] + 12*b^2*d^2*e*f*x*Sinh[4*c + 4*d*x] - 6*b^2*d*f^2*x*Sinh[4*c + 4*d*x] + 6*b^2*d^2*f^2*x^2*Sinh[4
*c + 4*d*x])

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Maple [F]
time = 1.61, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\sinh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/8*(8*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)))/(sqrt(a^2 + b^2
)*b^3*d) + (4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) - 4*(2*a^2 - b^2)*(d*x + c)/(b^3*d) + (4*a*e^(-d*x -
c) + b*e^(-2*d*x - 2*c))/(b^2*d))*e^2 + 1/48*(8*(2*a^2*d^3*f^2*e^(2*c) - b^2*d^3*f^2*e^(2*c))*x^3 + 24*(2*a^2
*d^3*f*e^(2*c) - b^2*d^3*f*e^(2*c))*x^2*e + 3*(2*b^2*d^2*f^2*x^2*e^(4*c) + b^2*f^2*e^(4*c) - 2*b^2*d*f*e^(4*c
+ 1) - 2*(b^2*d*f^2*e^(4*c) - 2*b^2*d^2*f*e^(4*c + 1))*x)*e^(2*d*x) - 24*(a*b*d^2*f^2*x^2*e^(3*c) + 2*a*b*f^2*
e^(3*c) - 2*a*b*d*f*e^(3*c + 1) - 2*(a*b*d*f^2*e^(3*c) - a*b*d^2*f*e^(3*c + 1))*x)*e^(d*x) - 24*(a*b*d^2*f^2*x
^2*e^c + 2*a*b*d*f*e^(c + 1) + 2*a*b*f^2*e^c + 2*(a*b*d^2*f*e^(c + 1) + a*b*d*f^2*e^c)*x)*e^(-d*x) - 3*(2*b^2*
d^2*f^2*x^2 + 2*b^2*d*f*e + b^2*f^2 + 2*(2*b^2*d^2*f*e + b^2*d*f^2)*x)*e^(-2*d*x))*e^(-2*c)/(b^3*d^3) - integr
ate(2*(a^3*f^2*x^2*e^c + 2*a^3*f*x*e^(c + 1))*e^(d*x)/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4857 vs. \(2 (488) = 976\).
time = 0.43, size = 4857, normalized size = 9.30 \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*sinh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/48*(6*(a^2*b^2 + b^4)*d^2*f^2*x^2 + 6*(a^2*b^2 + b^4)*d*f^2*x + 6*(a^2*b^2 + b^4)*d^2*cosh(1)^2 - 3*(2*(a^2
*b^2 + b^4)*d^2*f^2*x^2 - 2*(a^2*b^2 + b^4)*d*f^2*x + 2*(a^2*b^2 + b^4)*d^2*cosh(1)^2 + 2*(a^2*b^2 + b^4)*d^2*
sinh(1)^2 + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4)*d^2*f*x - (a^2*b^2 + b^4)*d*f)*cosh(1) + 2*(2*(a^2*b^2
+ b^4)*d^2*f*x + 2*(a^2*b^2 + b^4)*d^2*cosh(1) - (a^2*b^2 + b^4)*d*f)*sinh(1))*cosh(d*x + c)^4 + 6*(a^2*b^2 +
b^4)*d^2*sinh(1)^2 - 3*(2*(a^2*b^2 + b^4)*d^2*f^2*x^2 - 2*(a^2*b^2 + b^4)*d*f^2*x + 2*(a^2*b^2 + b^4)*d^2*cosh
(1)^2 + 2*(a^2*b^2 + b^4)*d^2*sinh(1)^2 + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4)*d^2*f*x - (a^2*b^2 + b^4)
*d*f)*cosh(1) + 2*(2*(a^2*b^2 + b^4)*d^2*f*x + 2*(a^2*b^2 + b^4)*d^2*cosh(1) - (a^2*b^2 + b^4)*d*f)*sinh(1))*s
inh(d*x + c)^4 + 24*((a^3*b + a*b^3)*d^2*f^2*x^2 - 2*(a^3*b + a*b^3)*d*f^2*x + (a^3*b + a*b^3)*d^2*cosh(1)^2 +
(a^3*b + a*b^3)*d^2*sinh(1)^2 + 2*(a^3*b + a*b^3)*f^2 + 2*((a^3*b + a*b^3)*d^2*f*x - (a^3*b + a*b^3)*d*f)*cos
h(1) + 2*((a^3*b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d^2*cosh(1) - (a^3*b + a*b^3)*d*f)*sinh(1))*cosh(d*x + c)^
3 + 12*(2*(a^3*b + a*b^3)*d^2*f^2*x^2 - 4*(a^3*b + a*b^3)*d*f^2*x + 2*(a^3*b + a*b^3)*d^2*cosh(1)^2 + 2*(a^3*b
+ a*b^3)*d^2*sinh(1)^2 + 4*(a^3*b + a*b^3)*f^2 + 4*((a^3*b + a*b^3)*d^2*f*x - (a^3*b + a*b^3)*d*f)*cosh(1) -
(2*(a^2*b^2 + b^4)*d^2*f^2*x^2 - 2*(a^2*b^2 + b^4)*d*f^2*x + 2*(a^2*b^2 + b^4)*d^2*cosh(1)^2 + 2*(a^2*b^2 + b^
4)*d^2*sinh(1)^2 + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4)*d^2*f*x - (a^2*b^2 + b^4)*d*f)*cosh(1) + 2*(2*(a
^2*b^2 + b^4)*d^2*f*x + 2*(a^2*b^2 + b^4)*d^2*cosh(1) - (a^2*b^2 + b^4)*d*f)*sinh(1))*cosh(d*x + c) + 4*((a^3*
b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d^2*cosh(1) - (a^3*b + a*b^3)*d*f)*sinh(1))*sinh(d*x + c)^3 + 3*(a^2*b^2
+ b^4)*f^2 - 8*((2*a^4 + a^2*b^2 - b^4)*d^3*f^2*x^3 + 3*(2*a^4 + a^2*b^2 - b^4)*d^3*f*x^2*cosh(1) + 3*(2*a^4 +
a^2*b^2 - b^4)*d^3*x*cosh(1)^2 + 3*(2*a^4 + a^2*b^2 - b^4)*d^3*x*sinh(1)^2 + 3*((2*a^4 + a^2*b^2 - b^4)*d^3*f
*x^2 + 2*(2*a^4 + a^2*b^2 - b^4)*d^3*x*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*(4*(2*a^4 + a^2*b^2 - b^4)*d^3*f^
2*x^3 + 12*(2*a^4 + a^2*b^2 - b^4)*d^3*f*x^2*cosh(1) + 12*(2*a^4 + a^2*b^2 - b^4)*d^3*x*cosh(1)^2 + 12*(2*a^4
+ a^2*b^2 - b^4)*d^3*x*sinh(1)^2 + 9*(2*(a^2*b^2 + b^4)*d^2*f^2*x^2 - 2*(a^2*b^2 + b^4)*d*f^2*x + 2*(a^2*b^2 +
b^4)*d^2*cosh(1)^2 + 2*(a^2*b^2 + b^4)*d^2*sinh(1)^2 + (a^2*b^2 + b^4)*f^2 + 2*(2*(a^2*b^2 + b^4)*d^2*f*x - (
a^2*b^2 + b^4)*d*f)*cosh(1) + 2*(2*(a^2*b^2 + b^4)*d^2*f*x + 2*(a^2*b^2 + b^4)*d^2*cosh(1) - (a^2*b^2 + b^4)*d
*f)*sinh(1))*cosh(d*x + c)^2 - 36*((a^3*b + a*b^3)*d^2*f^2*x^2 - 2*(a^3*b + a*b^3)*d*f^2*x + (a^3*b + a*b^3)*d
^2*cosh(1)^2 + (a^3*b + a*b^3)*d^2*sinh(1)^2 + 2*(a^3*b + a*b^3)*f^2 + 2*((a^3*b + a*b^3)*d^2*f*x - (a^3*b + a
*b^3)*d*f)*cosh(1) + 2*((a^3*b + a*b^3)*d^2*f*x + (a^3*b + a*b^3)*d^2*cosh(1) - (a^3*b + a*b^3)*d*f)*sinh(1))*
cosh(d*x + c) + 12*((2*a^4 + a^2*b^2 - b^4)*d^3*f*x^2 + 2*(2*a^4 + a^2*b^2 - b^4)*d^3*x*cosh(1))*sinh(1))*sinh
(d*x + c)^2 + 96*((a^3*b*d*f^2*x + a^3*b*d*f*cosh(1) + a^3*b*d*f*sinh(1))*cosh(d*x + c)^2 + 2*(a^3*b*d*f^2*x +
a^3*b*d*f*cosh(1) + a^3*b*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f^2*x + a^3*b*d*f*cosh(1) + a^3
*b*d*f*sinh(1))*sinh(d*x + c)^2)*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) - 96*((a^3*b*d*f^2*x + a^3*b*d*f*cosh(1) + a^3*b*d*f
*sinh(1))*cosh(d*x + c)^2 + 2*(a^3*b*d*f^2*x + a^3*b*d*f*cosh(1) + a^3*b*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x +
c) + (a^3*b*d*f^2*x + a^3*b*d*f*cosh(1) + a^3*b*d*f*sinh(1))*sinh(d*x + c)^2)*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) - 48*(
(a^3*b*c^2*f^2 - 2*a^3*b*c*d*f*cosh(1) + a^3*b*d^2*cosh(1)^2 + a^3*b*d^2*sinh(1)^2 - 2*(a^3*b*c*d*f - a^3*b*d^
2*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*(a^3*b*c^2*f^2 - 2*a^3*b*c*d*f*cosh(1) + a^3*b*d^2*cosh(1)^2 + a^3*b*d
^2*sinh(1)^2 - 2*(a^3*b*c*d*f - a^3*b*d^2*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*c^2*f^2 - 2*a
^3*b*c*d*f*cosh(1) + a^3*b*d^2*cosh(1)^2 + a^3*b*d^2*sinh(1)^2 - 2*(a^3*b*c*d*f - a^3*b*d^2*cosh(1))*sinh(1))*
sinh(d*x + c)^2)*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) + 48*((a^3*b*c^2*f^2 - 2*a^3*b*c*d*f*cosh(1) + a^3*b*d^2*cosh(1)^2 + a^3*b*d^2*sinh(1)^2 - 2*(a^3*b*c*d*
f - a^3*b*d^2*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*(a^3*b*c^2*f^2 - 2*a^3*b*c*d*f*cosh(1) + a^3*b*d^2*cosh(1)
^2 + a^3*b*d^2*sinh(1)^2 - 2*(a^3*b*c*d*f - a^3*b*d^2*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*c
^2*f^2 - 2*a^3*b*c*d*f*cosh(1) + a^3*b*d^2*cosh(1)^2 + a^3*b*d^2*sinh(1)^2 - 2*(a^3*b*c*d*f - a^3*b*d^2*cosh(1
))*sinh(1))*sinh(d*x + c)^2)*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) + 48*((a^3*b*d^2*f^2*x^2 - a^3*b*c^2*f^2 + 2*(a^3*b*d^2*f*x + a^3*b*c*d*f)*cosh(1) + 2*(a^3*
b*d^2*f*x + a^3*b*c*d*f)*sinh(1))*cosh(d*x + c)...

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

[Out]

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

[Out]

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

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