3.5.30 \(\int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) [430]

Optimal. Leaf size=656 \[ -\frac {(e+f x)^4}{4 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^3}-\frac {3 f^2 (e+f x) \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {6 \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^4}+\frac {3 f^3 \text {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.91, antiderivative size = 656, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 17, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5704, 5558, 5554, 3392, 32, 2715, 8, 3797, 2221, 2611, 6744, 2320, 6724, 5684, 3377, 2718, 5680} \begin {gather*} -\frac {6 f^3 \left (a^2+b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^4}-\frac {6 f^3 \left (a^2+b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^4}+\frac {6 f^2 \left (a^2+b^2\right ) (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^3}+\frac {6 f^2 \left (a^2+b^2\right ) (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^3}-\frac {3 f \left (a^2+b^2\right ) (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {3 f \left (a^2+b^2\right ) (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a b^2 d}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {3 f^3 \text {Li}_4\left (e^{2 (c+d x)}\right )}{4 a d^4}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {(e+f x)^4}{4 a f}-\frac {6 f^3 \cosh (c+d x)}{b d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}+\frac {(e+f x)^3 \sinh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

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 5558

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, 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 5704

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Cosh[c + d*x]^(p + 1)*(Coth[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]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cosh ^2(c+d x) \coth (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x)^3 \coth (c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \, dx}{b}-\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a b}\\ &=-\frac {(e+f x)^4}{4 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}-\frac {2 \int \frac {e^{2 (c+d x)} (e+f x)^3}{1-e^{2 (c+d x)}} \, dx}{a}-\frac {\left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a b}-\frac {\left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a b}-\frac {(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{b d}\\ &=-\frac {(e+f x)^4}{4 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d}+\frac {\left (3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a b^2 d}+\frac {\left (3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a b^2 d}+\frac {\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b d^2}\\ &=-\frac {(e+f x)^4}{4 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}-\frac {\left (3 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a b^2 d^2}+\frac {\left (6 \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a b^2 d^2}-\frac {\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{b d^3}\\ &=-\frac {(e+f x)^4}{4 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}+\frac {\left (3 f^3\right ) \int \text {Li}_3\left (e^{2 (c+d x)}\right ) \, dx}{2 a d^3}-\frac {\left (6 \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a b^2 d^3}-\frac {\left (6 \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a b^2 d^3}\\ &=-\frac {(e+f x)^4}{4 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 a d^4}-\frac {\left (6 \left (a^2+b^2\right ) f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a b^2 d^4}-\frac {\left (6 \left (a^2+b^2\right ) f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a b^2 d^4}\\ &=-\frac {(e+f x)^4}{4 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a b^2 f}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a d^3}-\frac {6 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^4}+\frac {3 f^3 \text {Li}_4\left (e^{2 (c+d x)}\right )}{4 a d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 27.74, size = 14209, normalized size = 21.66 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 5.77, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\cosh ^{2}\left (d x +c \right )\right ) \coth \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/2*(2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(-d*x - c)/(b*d) - 2*log(e^(-d*x - c) + 1)/(a*d) - 2*log(e
^(-d*x - c) - 1)/(a*d) + 2*(a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a*b^2*d))*e^3 + 3*(d*x
*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*f*e^2/(a*d^2) + 3*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c))
)*f*e^2/(a*d^2) + 3*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*f^
2*e/(a*d^3) + 3*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*f^2*e/(
a*d^3) + (d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*po
lylog(4, -e^(d*x + c)))*f^3/(a*d^4) + (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*po
lylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*f^3/(a*d^4) - 1/2*(d^4*f^3*x^4 + 4*d^4*f^2*x^3*e + 6*d^4*f*
x^2*e^2)/(a*d^4) - 1/4*(a*d^4*f^3*x^4*e^c + 4*a*d^4*f^2*x^3*e^(c + 1) + 6*a*d^4*f*x^2*e^(c + 2) - 2*(b*d^3*f^3
*x^3*e^(2*c) - 6*b*f^3*e^(2*c) - 3*b*d^2*f*e^(2*c + 2) + 6*b*d*f^2*e^(2*c + 1) - 3*(b*d^2*f^3*e^(2*c) - b*d^3*
f^2*e^(2*c + 1))*x^2 + 3*(2*b*d*f^3*e^(2*c) + b*d^3*f*e^(2*c + 2) - 2*b*d^2*f^2*e^(2*c + 1))*x)*e^(d*x) + 2*(b
*d^3*f^3*x^3 + 3*b*d^2*f*e^2 + 6*b*d*f^2*e + 6*b*f^3 + 3*(b*d^3*f^2*e + b*d^2*f^3)*x^2 + 3*(b*d^3*f*e^2 + 2*b*
d^2*f^2*e + 2*b*d*f^3)*x)*e^(-d*x))*e^(-c)/(b^2*d^4) + integrate(-2*((a^2*b*f^3 + b^3*f^3)*x^3 + 3*(a^2*b*f^2
+ b^3*f^2)*x^2*e + 3*(a^2*b*f + b^3*f)*x*e^2 - ((a^3*f^3*e^c + a*b^2*f^3*e^c)*x^3 + 3*(a^3*f^2*e^c + a*b^2*f^2
*e^c)*x^2*e + 3*(a^3*f*e^c + a*b^2*f*e^c)*x*e^2)*e^(d*x))/(a*b^3*e^(2*d*x + 2*c) + 2*a^2*b^2*e^(d*x + c) - a*b
^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5885 vs. \(2 (633) = 1266\).
time = 0.52, size = 5885, normalized size = 8.97 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*b*d^3*f^3*x^3 + 6*a*b*d^2*f^3*x^2 + 2*a*b*d^3*cosh(1)^3 + 2*a*b*d^3*sinh(1)^3 + 12*a*b*d*f^3*x + 12*
a*b*f^3 + 6*(a*b*d^3*f*x + a*b*d^2*f)*cosh(1)^2 - 2*(a*b*d^3*f^3*x^3 - 3*a*b*d^2*f^3*x^2 + a*b*d^3*cosh(1)^3 +
a*b*d^3*sinh(1)^3 + 6*a*b*d*f^3*x - 6*a*b*f^3 + 3*(a*b*d^3*f*x - a*b*d^2*f)*cosh(1)^2 + 3*(a*b*d^3*f*x + a*b*
d^3*cosh(1) - a*b*d^2*f)*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - 2*a*b*d^2*f^2*x + 2*a*b*d*f^2)*cosh(1) + 3*(a*b*d^3*
f^2*x^2 - 2*a*b*d^2*f^2*x + a*b*d^3*cosh(1)^2 + 2*a*b*d*f^2 + 2*(a*b*d^3*f*x - a*b*d^2*f)*cosh(1))*sinh(1))*co
sh(d*x + c)^2 + 6*(a*b*d^3*f*x + a*b*d^3*cosh(1) + a*b*d^2*f)*sinh(1)^2 - 2*(a*b*d^3*f^3*x^3 - 3*a*b*d^2*f^3*x
^2 + a*b*d^3*cosh(1)^3 + a*b*d^3*sinh(1)^3 + 6*a*b*d*f^3*x - 6*a*b*f^3 + 3*(a*b*d^3*f*x - a*b*d^2*f)*cosh(1)^2
+ 3*(a*b*d^3*f*x + a*b*d^3*cosh(1) - a*b*d^2*f)*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - 2*a*b*d^2*f^2*x + 2*a*b*d*f^
2)*cosh(1) + 3*(a*b*d^3*f^2*x^2 - 2*a*b*d^2*f^2*x + a*b*d^3*cosh(1)^2 + 2*a*b*d*f^2 + 2*(a*b*d^3*f*x - a*b*d^2
*f)*cosh(1))*sinh(1))*sinh(d*x + c)^2 + 6*(a*b*d^3*f^2*x^2 + 2*a*b*d^2*f^2*x + 2*a*b*d*f^2)*cosh(1) - (a^2*d^4
*f^3*x^4 - 2*a^2*c^4*f^3 + 4*(a^2*d^4*x + 2*a^2*c*d^3)*cosh(1)^3 + 4*(a^2*d^4*x + 2*a^2*c*d^3)*sinh(1)^3 + 6*(
a^2*d^4*f*x^2 - 2*a^2*c^2*d^2*f)*cosh(1)^2 + 6*(a^2*d^4*f*x^2 - 2*a^2*c^2*d^2*f + 2*(a^2*d^4*x + 2*a^2*c*d^3)*
cosh(1))*sinh(1)^2 + 4*(a^2*d^4*f^2*x^3 + 2*a^2*c^3*d*f^2)*cosh(1) + 4*(a^2*d^4*f^2*x^3 + 2*a^2*c^3*d*f^2 + 3*
(a^2*d^4*x + 2*a^2*c*d^3)*cosh(1)^2 + 3*(a^2*d^4*f*x^2 - 2*a^2*c^2*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 12
*(((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f
*sinh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c) + ((a^2 + b^2)*d^2*f
^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*sinh(1)^2 + 2*((a^2
+ b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*sinh(d*x + c))*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) + 12*(((a^2 + b^2)*d^2*f^3*x^2 + 2*(
a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*sinh(1)^2 + 2*((a^2 + b^2)*d^2*
f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c) + ((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x
*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*sinh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*
d^2*f*cosh(1))*sinh(1))*sinh(d*x + c))*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) - 12*((b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)
^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c) + (b^2*d^2*f^3*x^2 + 2
*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sin
h(1))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) - 12*((b^2*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b
^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c) + (b^2
*d^2*f^3*x^2 + 2*b^2*d^2*f^2*x*cosh(1) + b^2*d^2*f*cosh(1)^2 + b^2*d^2*f*sinh(1)^2 + 2*(b^2*d^2*f^2*x + b^2*d^
2*f*cosh(1))*sinh(1))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) - 4*(((a^2 + b^2)*c^3*f^3 - 3*(a^2
+ b^2)*c^2*d*f^2*cosh(1) + 3*(a^2 + b^2)*c*d^2*f*cosh(1)^2 - (a^2 + b^2)*d^3*cosh(1)^3 - (a^2 + b^2)*d^3*sinh(
1)^3 + 3*((a^2 + b^2)*c*d^2*f - (a^2 + b^2)*d^3*cosh(1))*sinh(1)^2 - 3*((a^2 + b^2)*c^2*d*f^2 - 2*(a^2 + b^2)*
c*d^2*f*cosh(1) + (a^2 + b^2)*d^3*cosh(1)^2)*sinh(1))*cosh(d*x + c) + ((a^2 + b^2)*c^3*f^3 - 3*(a^2 + b^2)*c^2
*d*f^2*cosh(1) + 3*(a^2 + b^2)*c*d^2*f*cosh(1)^2 - (a^2 + b^2)*d^3*cosh(1)^3 - (a^2 + b^2)*d^3*sinh(1)^3 + 3*(
(a^2 + b^2)*c*d^2*f - (a^2 + b^2)*d^3*cosh(1))*sinh(1)^2 - 3*((a^2 + b^2)*c^2*d*f^2 - 2*(a^2 + b^2)*c*d^2*f*co
sh(1) + (a^2 + b^2)*d^3*cosh(1)^2)*sinh(1))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqr
t((a^2 + b^2)/b^2) + 2*a) - 4*(((a^2 + b^2)*c^3*f^3 - 3*(a^2 + b^2)*c^2*d*f^2*cosh(1) + 3*(a^2 + b^2)*c*d^2*f*
cosh(1)^2 - (a^2 + b^2)*d^3*cosh(1)^3 - (a^2 + b^2)*d^3*sinh(1)^3 + 3*((a^2 + b^2)*c*d^2*f - (a^2 + b^2)*d^3*c
osh(1))*sinh(1)^2 - 3*((a^2 + b^2)*c^2*d*f^2 - 2*(a^2 + b^2)*c*d^2*f*cosh(1) + (a^2 + b^2)*d^3*cosh(1)^2)*sinh
(1))*cosh(d*x + c) + ((a^2 + b^2)*c^3*f^3 - 3*(a^2 + b^2)*c^2*d*f^2*cosh(1) + 3*(a^2 + b^2)*c*d^2*f*cosh(1)^2
- (a^2 + b^2)*d^3*cosh(1)^3 - (a^2 + b^2)*d^3*sinh(1)^3 + 3*((a^2 + b^2)*c*d^2*f - (a^2 + b^2)*d^3*cosh(1))*si
nh(1)^2 - 3*((a^2 + b^2)*c^2*d*f^2 - 2*(a^2 + b^2)*c*d^2*f*cosh(1) + (a^2 + b^2)*d^3*cosh(1)^2)*sinh(1))*sinh(
d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 4*(((a^2 + b^2)*d^3*f
^3*x^3 + (a^2 + b^2)*c^3*f^3 + 3*((a^2 + b^2)*d^3*f*x + (a^2 + b^2)*c*d^2*f)*cosh(1)^2 + 3*((a^2 + b^2)*d^3*f*
x + (a^2 + b^2)*c*d^2*f)*sinh(1)^2 + 3*((a^2 + b^2)*d^3*f^2*x^2 - (a^2 + b^2)*c^2*d*f^2)*cosh(1) + 3*((a^2 + b
^2)*d^3*f^2*x^2 - (a^2 + b^2)*c^2*d*f^2 + 2*((a...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 624.79Not invertible Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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