### 3.422 $$\int x^2 \cosh ^2(x) \coth ^3(x) \, dx$$

Optimal. Leaf size=96 $2 x \text{PolyLog}\left (2,e^{2 x}\right )-\text{PolyLog}\left (3,e^{2 x}\right )-\frac{2 x^3}{3}+\frac{3 x^2}{4}+2 x^2 \log \left (1-e^{2 x}\right )+\frac{1}{2} x^2 \sinh ^2(x)-\frac{1}{2} x^2 \coth ^2(x)+\frac{\sinh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))-\frac{1}{2} x \sinh (x) \cosh (x)$

[Out]

(3*x^2)/4 - (2*x^3)/3 - x*Coth[x] - (x^2*Coth[x]^2)/2 + 2*x^2*Log[1 - E^(2*x)] + Log[Sinh[x]] + 2*x*PolyLog[2,
E^(2*x)] - PolyLog[3, E^(2*x)] - (x*Cosh[x]*Sinh[x])/2 + Sinh[x]^2/4 + (x^2*Sinh[x]^2)/2

________________________________________________________________________________________

Rubi [A]  time = 0.280165, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.917, Rules used = {5450, 5372, 3310, 30, 3716, 2190, 2531, 2282, 6589, 3720, 3475} $2 x \text{PolyLog}\left (2,e^{2 x}\right )-\text{PolyLog}\left (3,e^{2 x}\right )-\frac{2 x^3}{3}+\frac{3 x^2}{4}+2 x^2 \log \left (1-e^{2 x}\right )+\frac{1}{2} x^2 \sinh ^2(x)-\frac{1}{2} x^2 \coth ^2(x)+\frac{\sinh ^2(x)}{4}-x \coth (x)+\log (\sinh (x))-\frac{1}{2} x \sinh (x) \cosh (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2*Cosh[x]^2*Coth[x]^3,x]

[Out]

(3*x^2)/4 - (2*x^3)/3 - x*Coth[x] - (x^2*Coth[x]^2)/2 + 2*x^2*Log[1 - E^(2*x)] + Log[Sinh[x]] + 2*x*PolyLog[2,
E^(2*x)] - PolyLog[3, E^(2*x)] - (x*Cosh[x]*Sinh[x])/2 + Sinh[x]^2/4 + (x^2*Sinh[x]^2)/2

Rule 5450

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 5372

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m -
n + 1)*Sinh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 3310

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3716

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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2531

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 2282

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 6589

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 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^2 \cosh ^2(x) \coth ^3(x) \, dx &=\int x^2 \cosh ^2(x) \coth (x) \, dx+\int x^2 \coth ^3(x) \, dx\\ &=-\frac{1}{2} x^2 \coth ^2(x)+2 \int x^2 \coth (x) \, dx+\int x \coth ^2(x) \, dx+\int x^2 \cosh (x) \sinh (x) \, dx\\ &=-x \coth (x)-\frac{1}{2} x^2 \coth ^2(x)+\frac{1}{2} x^2 \sinh ^2(x)+2 \left (-\frac{x^3}{3}-2 \int \frac{e^{2 x} x^2}{1-e^{2 x}} \, dx\right )+\int x \, dx+\int \coth (x) \, dx-\int x \sinh ^2(x) \, dx\\ &=\frac{x^2}{2}-x \coth (x)-\frac{1}{2} x^2 \coth ^2(x)+\log (\sinh (x))-\frac{1}{2} x \cosh (x) \sinh (x)+\frac{\sinh ^2(x)}{4}+\frac{1}{2} x^2 \sinh ^2(x)+\frac{\int x \, dx}{2}+2 \left (-\frac{x^3}{3}+x^2 \log \left (1-e^{2 x}\right )-2 \int x \log \left (1-e^{2 x}\right ) \, dx\right )\\ &=\frac{3 x^2}{4}-x \coth (x)-\frac{1}{2} x^2 \coth ^2(x)+\log (\sinh (x))-\frac{1}{2} x \cosh (x) \sinh (x)+\frac{\sinh ^2(x)}{4}+\frac{1}{2} x^2 \sinh ^2(x)+2 \left (-\frac{x^3}{3}+x^2 \log \left (1-e^{2 x}\right )+x \text{Li}_2\left (e^{2 x}\right )-\int \text{Li}_2\left (e^{2 x}\right ) \, dx\right )\\ &=\frac{3 x^2}{4}-x \coth (x)-\frac{1}{2} x^2 \coth ^2(x)+\log (\sinh (x))-\frac{1}{2} x \cosh (x) \sinh (x)+\frac{\sinh ^2(x)}{4}+\frac{1}{2} x^2 \sinh ^2(x)+2 \left (-\frac{x^3}{3}+x^2 \log \left (1-e^{2 x}\right )+x \text{Li}_2\left (e^{2 x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 x}\right )\right )\\ &=\frac{3 x^2}{4}-x \coth (x)-\frac{1}{2} x^2 \coth ^2(x)+\log (\sinh (x))+2 \left (-\frac{x^3}{3}+x^2 \log \left (1-e^{2 x}\right )+x \text{Li}_2\left (e^{2 x}\right )-\frac{\text{Li}_3\left (e^{2 x}\right )}{2}\right )-\frac{1}{2} x \cosh (x) \sinh (x)+\frac{\sinh ^2(x)}{4}+\frac{1}{2} x^2 \sinh ^2(x)\\ \end{align*}

Mathematica [C]  time = 0.277203, size = 98, normalized size = 1.02 $2 x \text{PolyLog}\left (2,e^{2 x}\right )-\text{PolyLog}\left (3,e^{2 x}\right )-\frac{2 x^3}{3}+2 x^2 \log \left (1-e^{2 x}\right )+\frac{1}{4} x^2 \cosh (2 x)-\frac{1}{2} x^2 \text{csch}^2(x)-\frac{1}{4} x \sinh (2 x)+\frac{1}{8} \cosh (2 x)-x \coth (x)+\log (\sinh (x))+\frac{i \pi ^3}{12}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2*Cosh[x]^2*Coth[x]^3,x]

[Out]

(I/12)*Pi^3 - (2*x^3)/3 + Cosh[2*x]/8 + (x^2*Cosh[2*x])/4 - x*Coth[x] - (x^2*Csch[x]^2)/2 + 2*x^2*Log[1 - E^(2
*x)] + Log[Sinh[x]] + 2*x*PolyLog[2, E^(2*x)] - PolyLog[3, E^(2*x)] - (x*Sinh[2*x])/4

________________________________________________________________________________________

Maple [A]  time = 0.061, size = 127, normalized size = 1.3 \begin{align*} -{\frac{2\,{x}^{3}}{3}}+ \left ({\frac{1}{16}}-{\frac{x}{8}}+{\frac{{x}^{2}}{8}} \right ){{\rm e}^{2\,x}}+ \left ({\frac{1}{16}}+{\frac{x}{8}}+{\frac{{x}^{2}}{8}} \right ){{\rm e}^{-2\,x}}-2\,{\frac{x \left ( x{{\rm e}^{2\,x}}+{{\rm e}^{2\,x}}-1 \right ) }{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}+\ln \left ({{\rm e}^{x}}+1 \right ) -2\,\ln \left ({{\rm e}^{x}} \right ) +\ln \left ({{\rm e}^{x}}-1 \right ) +2\,{x}^{2}\ln \left ({{\rm e}^{x}}+1 \right ) +4\,x{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) -4\,{\it polylog} \left ( 3,-{{\rm e}^{x}} \right ) +2\,{x}^{2}\ln \left ( 1-{{\rm e}^{x}} \right ) +4\,x{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) -4\,{\it polylog} \left ( 3,{{\rm e}^{x}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(x)^2*coth(x)^3,x)

[Out]

-2/3*x^3+(1/16-1/8*x+1/8*x^2)*exp(2*x)+(1/16+1/8*x+1/8*x^2)*exp(-2*x)-2*x*(x*exp(2*x)+exp(2*x)-1)/(exp(2*x)-1)
^2+ln(exp(x)+1)-2*ln(exp(x))+ln(exp(x)-1)+2*x^2*ln(exp(x)+1)+4*x*polylog(2,-exp(x))-4*polylog(3,-exp(x))+2*x^2
*ln(1-exp(x))+4*x*polylog(2,exp(x))-4*polylog(3,exp(x))

________________________________________________________________________________________

Maxima [B]  time = 1.33457, size = 235, normalized size = 2.45 \begin{align*} -\frac{4}{3} \, x^{3} + 2 \, x^{2} \log \left (e^{x} + 1\right ) + 2 \, x^{2} \log \left (-e^{x} + 1\right ) + 4 \, x{\rm Li}_2\left (-e^{x}\right ) + 4 \, x{\rm Li}_2\left (e^{x}\right ) - 2 \, x + \frac{32 \, x^{3} - 12 \, x^{2} + 3 \,{\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (6 \, x\right )} + 2 \,{\left (16 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (4 \, x\right )} - 2 \,{\left (32 \, x^{3} + 42 \, x^{2} + 48 \, x - 3\right )} e^{\left (2 \, x\right )} + 3 \,{\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} + 84 \, x - 6}{48 \,{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} + \log \left (e^{x} + 1\right ) + \log \left (e^{x} - 1\right ) - 4 \,{\rm Li}_{3}(-e^{x}) - 4 \,{\rm Li}_{3}(e^{x}) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(x)^2*coth(x)^3,x, algorithm="maxima")

[Out]

-4/3*x^3 + 2*x^2*log(e^x + 1) + 2*x^2*log(-e^x + 1) + 4*x*dilog(-e^x) + 4*x*dilog(e^x) - 2*x + 1/48*(32*x^3 -
12*x^2 + 3*(2*x^2 - 2*x + 1)*e^(6*x) + 2*(16*x^3 - 6*x^2 + 6*x - 3)*e^(4*x) - 2*(32*x^3 + 42*x^2 + 48*x - 3)*e
^(2*x) + 3*(2*x^2 + 2*x + 1)*e^(-2*x) + 84*x - 6)/(e^(4*x) - 2*e^(2*x) + 1) + log(e^x + 1) + log(e^x - 1) - 4*
polylog(3, -e^x) - 4*polylog(3, e^x)

________________________________________________________________________________________

Fricas [C]  time = 2.3186, size = 4618, normalized size = 48.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(x)^2*coth(x)^3,x, algorithm="fricas")

[Out]

1/48*(3*(2*x^2 - 2*x + 1)*cosh(x)^8 + 24*(2*x^2 - 2*x + 1)*cosh(x)*sinh(x)^7 + 3*(2*x^2 - 2*x + 1)*sinh(x)^8 -
2*(16*x^3 + 6*x^2 + 42*x + 3)*cosh(x)^6 - 2*(16*x^3 - 42*(2*x^2 - 2*x + 1)*cosh(x)^2 + 6*x^2 + 42*x + 3)*sinh
(x)^6 + 12*(14*(2*x^2 - 2*x + 1)*cosh(x)^3 - (16*x^3 + 6*x^2 + 42*x + 3)*cosh(x))*sinh(x)^5 + 2*(32*x^3 - 42*x
^2 + 48*x + 3)*cosh(x)^4 + 2*(105*(2*x^2 - 2*x + 1)*cosh(x)^4 + 32*x^3 - 15*(16*x^3 + 6*x^2 + 42*x + 3)*cosh(x
)^2 - 42*x^2 + 48*x + 3)*sinh(x)^4 + 8*(21*(2*x^2 - 2*x + 1)*cosh(x)^5 - 5*(16*x^3 + 6*x^2 + 42*x + 3)*cosh(x)
^3 + (32*x^3 - 42*x^2 + 48*x + 3)*cosh(x))*sinh(x)^3 - 2*(16*x^3 + 6*x^2 + 6*x + 3)*cosh(x)^2 + 2*(42*(2*x^2 -
2*x + 1)*cosh(x)^6 - 15*(16*x^3 + 6*x^2 + 42*x + 3)*cosh(x)^4 - 16*x^3 + 6*(32*x^3 - 42*x^2 + 48*x + 3)*cosh(
x)^2 - 6*x^2 - 6*x - 3)*sinh(x)^2 + 6*x^2 + 192*(x*cosh(x)^6 + 6*x*cosh(x)*sinh(x)^5 + x*sinh(x)^6 - 2*x*cosh(
x)^4 + (15*x*cosh(x)^2 - 2*x)*sinh(x)^4 + 4*(5*x*cosh(x)^3 - 2*x*cosh(x))*sinh(x)^3 + x*cosh(x)^2 + (15*x*cosh
(x)^4 - 12*x*cosh(x)^2 + x)*sinh(x)^2 + 2*(3*x*cosh(x)^5 - 4*x*cosh(x)^3 + x*cosh(x))*sinh(x))*dilog(cosh(x) +
sinh(x)) + 192*(x*cosh(x)^6 + 6*x*cosh(x)*sinh(x)^5 + x*sinh(x)^6 - 2*x*cosh(x)^4 + (15*x*cosh(x)^2 - 2*x)*si
nh(x)^4 + 4*(5*x*cosh(x)^3 - 2*x*cosh(x))*sinh(x)^3 + x*cosh(x)^2 + (15*x*cosh(x)^4 - 12*x*cosh(x)^2 + x)*sinh
(x)^2 + 2*(3*x*cosh(x)^5 - 4*x*cosh(x)^3 + x*cosh(x))*sinh(x))*dilog(-cosh(x) - sinh(x)) + 48*((2*x^2 + 1)*cos
h(x)^6 + 6*(2*x^2 + 1)*cosh(x)*sinh(x)^5 + (2*x^2 + 1)*sinh(x)^6 - 2*(2*x^2 + 1)*cosh(x)^4 + (15*(2*x^2 + 1)*c
osh(x)^2 - 4*x^2 - 2)*sinh(x)^4 + 4*(5*(2*x^2 + 1)*cosh(x)^3 - 2*(2*x^2 + 1)*cosh(x))*sinh(x)^3 + (2*x^2 + 1)*
cosh(x)^2 + (15*(2*x^2 + 1)*cosh(x)^4 - 12*(2*x^2 + 1)*cosh(x)^2 + 2*x^2 + 1)*sinh(x)^2 + 2*(3*(2*x^2 + 1)*cos
h(x)^5 - 4*(2*x^2 + 1)*cosh(x)^3 + (2*x^2 + 1)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 48*(cosh(x)^6 +
6*cosh(x)*sinh(x)^5 + sinh(x)^6 + (15*cosh(x)^2 - 2)*sinh(x)^4 - 2*cosh(x)^4 + 4*(5*cosh(x)^3 - 2*cosh(x))*sin
h(x)^3 + (15*cosh(x)^4 - 12*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(3*cosh(x)^5 - 4*cosh(x)^3 + cosh(x))*sin
h(x))*log(cosh(x) + sinh(x) - 1) + 96*(x^2*cosh(x)^6 + 6*x^2*cosh(x)*sinh(x)^5 + x^2*sinh(x)^6 - 2*x^2*cosh(x)
^4 + (15*x^2*cosh(x)^2 - 2*x^2)*sinh(x)^4 + x^2*cosh(x)^2 + 4*(5*x^2*cosh(x)^3 - 2*x^2*cosh(x))*sinh(x)^3 + (1
5*x^2*cosh(x)^4 - 12*x^2*cosh(x)^2 + x^2)*sinh(x)^2 + 2*(3*x^2*cosh(x)^5 - 4*x^2*cosh(x)^3 + x^2*cosh(x))*sinh
(x))*log(-cosh(x) - sinh(x) + 1) - 192*(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + (15*cosh(x)^2 - 2)*sinh(
x)^4 - 2*cosh(x)^4 + 4*(5*cosh(x)^3 - 2*cosh(x))*sinh(x)^3 + (15*cosh(x)^4 - 12*cosh(x)^2 + 1)*sinh(x)^2 + cos
h(x)^2 + 2*(3*cosh(x)^5 - 4*cosh(x)^3 + cosh(x))*sinh(x))*polylog(3, cosh(x) + sinh(x)) - 192*(cosh(x)^6 + 6*c
osh(x)*sinh(x)^5 + sinh(x)^6 + (15*cosh(x)^2 - 2)*sinh(x)^4 - 2*cosh(x)^4 + 4*(5*cosh(x)^3 - 2*cosh(x))*sinh(x
)^3 + (15*cosh(x)^4 - 12*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(3*cosh(x)^5 - 4*cosh(x)^3 + cosh(x))*sinh(x
))*polylog(3, -cosh(x) - sinh(x)) + 4*(6*(2*x^2 - 2*x + 1)*cosh(x)^7 - 3*(16*x^3 + 6*x^2 + 42*x + 3)*cosh(x)^5
+ 2*(32*x^3 - 42*x^2 + 48*x + 3)*cosh(x)^3 - (16*x^3 + 6*x^2 + 6*x + 3)*cosh(x))*sinh(x) + 6*x + 3)/(cosh(x)^
6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + (15*cosh(x)^2 - 2)*sinh(x)^4 - 2*cosh(x)^4 + 4*(5*cosh(x)^3 - 2*cosh(x))
*sinh(x)^3 + (15*cosh(x)^4 - 12*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(3*cosh(x)^5 - 4*cosh(x)^3 + cosh(x))
*sinh(x))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cosh(x)**2*coth(x)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cosh \left (x\right )^{2} \coth \left (x\right )^{3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(x)^2*coth(x)^3,x, algorithm="giac")

[Out]

integrate(x^2*cosh(x)^2*coth(x)^3, x)