3.5.60 \(\int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [460]

Optimal. Leaf size=518 \[ \frac {b (e+f x)^3}{3 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}-\frac {b f (e+f x) \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {2 \left (a^2+b^2\right ) f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^3}-\frac {2 \left (a^2+b^2\right ) f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^3}+\frac {b f^2 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.89, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 17, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5704, 5558, 3377, 2717, 5560, 4267, 2317, 2438, 5554, 3391, 3797, 2221, 2611, 2320, 6724, 5684, 5680} \begin {gather*} -\frac {2 f^2 \left (a^2+b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^3}-\frac {2 f^2 \left (a^2+b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^3}+\frac {2 f \left (a^2+b^2\right ) (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {2 f \left (a^2+b^2\right ) (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}+\frac {b f^2 \text {Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b (e+f x)^3}{3 a^2 f}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && 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 5560

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 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]

Rubi steps

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

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Mathematica [A]
time = 10.13, size = 978, normalized size = 1.89 \begin {gather*} \frac {-12 b e^2 x+\frac {12 b e^2 e^{2 c} x}{-1+e^{2 c}}+\frac {12 b e f x^2}{-1+e^{2 c}}+\frac {4 b f^2 x^3}{-1+e^{2 c}}-\frac {24 a e f \tanh ^{-1}\left (e^{c+d x}\right )}{d^2}+\frac {6 b e^2 \left (2 d x-\log \left (1-e^{2 (c+d x)}\right )\right )}{d}+\frac {12 a f^2 \left (d x \left (\log \left (1-e^{c+d x}\right )-\log \left (1+e^{c+d x}\right )\right )-\text {PolyLog}\left (2,-e^{c+d x}\right )+\text {PolyLog}\left (2,e^{c+d x}\right )\right )}{d^3}+\frac {6 b e f \left (2 d x \left (d x-\log \left (1-e^{2 (c+d x)}\right )\right )-\text {PolyLog}\left (2,e^{2 (c+d x)}\right )\right )}{d^2}+\frac {b f^2 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1-e^{2 (c+d x)}\right )\right )-6 d x \text {PolyLog}\left (2,e^{2 (c+d x)}\right )+3 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )\right )}{d^3}}{6 a^2}+\frac {\left (a^2+b^2\right ) \left (-\frac {2 e^{2 c} x \left (3 e^2+3 e f x+f^2 x^2\right )}{-1+e^{2 c}}+\frac {3 \left (d^2 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 d^2 e 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 )+d^2 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 d^2 e 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 )+d^2 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 d 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 d 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 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 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 )}{d^3}\right )}{3 a^2 b}+\frac {\left (-3 b e^2-6 b e f x-3 b f^2 x^2+3 a d e^2 x \cosh (c)+3 a d e f x^2 \cosh (c)+a d f^2 x^3 \cosh (c)\right ) \text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right )}{6 a b d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-12*b*e^2*x + (12*b*e^2*E^(2*c)*x)/(-1 + E^(2*c)) + (12*b*e*f*x^2)/(-1 + E^(2*c)) + (4*b*f^2*x^3)/(-1 + E^(2*
c)) - (24*a*e*f*ArcTanh[E^(c + d*x)])/d^2 + (6*b*e^2*(2*d*x - Log[1 - E^(2*(c + d*x))]))/d + (12*a*f^2*(d*x*(L
og[1 - E^(c + d*x)] - Log[1 + E^(c + d*x)]) - PolyLog[2, -E^(c + d*x)] + PolyLog[2, E^(c + d*x)]))/d^3 + (6*b*
e*f*(2*d*x*(d*x - Log[1 - E^(2*(c + d*x))]) - PolyLog[2, E^(2*(c + d*x))]))/d^2 + (b*f^2*(2*d^2*x^2*(2*d*x - 3
*Log[1 - E^(2*(c + d*x))]) - 6*d*x*PolyLog[2, E^(2*(c + d*x))] + 3*PolyLog[3, E^(2*(c + d*x))]))/d^3)/(6*a^2)
+ ((a^2 + b^2)*((-2*E^(2*c)*x*(3*e^2 + 3*e*f*x + f^2*x^2))/(-1 + E^(2*c)) + (3*(d^2*e^2*Log[2*a*E^(c + d*x) +
b*(-1 + E^(2*(c + d*x)))] + 2*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + d^2*f
^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/
(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)
])] + 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 2*d*f*(e + f*x)*P
olyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*
E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))
]))/d^3))/(3*a^2*b) + ((-3*b*e^2 - 6*b*e*f*x - 3*b*f^2*x^2 + 3*a*d*e^2*x*Cosh[c] + 3*a*d*e*f*x^2*Cosh[c] + a*d
*f^2*x^3*Cosh[c])*Csch[c/2]*Sech[c/2])/(6*a*b*d) + (Csch[c/2]*Csch[c/2 + (d*x)/2]*(e^2*Sinh[(d*x)/2] + 2*e*f*x
*Sinh[(d*x)/2] + f^2*x^2*Sinh[(d*x)/2]))/(2*a*d) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(e^2*Sinh[(d*x)/2] + 2*e*f*x
*Sinh[(d*x)/2] + f^2*x^2*Sinh[(d*x)/2]))/(2*a*d)

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Maple [F]
time = 1.58, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \left (\coth ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5069 vs. \(2 (496) = 992\).
time = 0.45, size = 5069, normalized size = 9.79 \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)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [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)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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

[Out]

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

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