3.5.20 \(\int x^3 \cosh ^2(x) \coth ^2(x) \, dx\) [420]

Optimal. Leaf size=102 \[ \frac {3 x^2}{8}-x^3+\frac {3 x^4}{8}-\frac {3 \cosh ^2(x)}{8}-\frac {3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+3 x \text {PolyLog}\left (2,e^{2 x}\right )-\frac {3}{2} \text {PolyLog}\left (3,e^{2 x}\right )+\frac {3}{4} x \cosh (x) \sinh (x)+\frac {1}{2} x^3 \cosh (x) \sinh (x) \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5558, 3392, 30, 3391, 3801, 3797, 2221, 2611, 2320, 6724} \begin {gather*} 3 x \text {Li}_2\left (e^{2 x}\right )-\frac {3 \text {Li}_3\left (e^{2 x}\right )}{2}+\frac {3 x^4}{8}-x^3-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)+\frac {3 x^2}{8}+3 x^2 \log \left (1-e^{2 x}\right )-\frac {3}{4} x^2 \cosh ^2(x)-\frac {3 \cosh ^2(x)}{8}+\frac {3}{4} x \sinh (x) \cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[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 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 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 3801

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 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 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 x^3 \cosh ^2(x) \coth ^2(x) \, dx &=\int x^3 \cosh ^2(x) \, dx+\int x^3 \coth ^2(x) \, dx\\ &=-\frac {3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+\frac {1}{2} x^3 \cosh (x) \sinh (x)+\frac {\int x^3 \, dx}{2}+\frac {3}{2} \int x \cosh ^2(x) \, dx+3 \int x^2 \coth (x) \, dx+\int x^3 \, dx\\ &=-x^3+\frac {3 x^4}{8}-\frac {3 \cosh ^2(x)}{8}-\frac {3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+\frac {3}{4} x \cosh (x) \sinh (x)+\frac {1}{2} x^3 \cosh (x) \sinh (x)+\frac {3 \int x \, dx}{4}-6 \int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx\\ &=\frac {3 x^2}{8}-x^3+\frac {3 x^4}{8}-\frac {3 \cosh ^2(x)}{8}-\frac {3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+\frac {3}{4} x \cosh (x) \sinh (x)+\frac {1}{2} x^3 \cosh (x) \sinh (x)-6 \int x \log \left (1-e^{2 x}\right ) \, dx\\ &=\frac {3 x^2}{8}-x^3+\frac {3 x^4}{8}-\frac {3 \cosh ^2(x)}{8}-\frac {3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+3 x \text {Li}_2\left (e^{2 x}\right )+\frac {3}{4} x \cosh (x) \sinh (x)+\frac {1}{2} x^3 \cosh (x) \sinh (x)-3 \int \text {Li}_2\left (e^{2 x}\right ) \, dx\\ &=\frac {3 x^2}{8}-x^3+\frac {3 x^4}{8}-\frac {3 \cosh ^2(x)}{8}-\frac {3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+3 x \text {Li}_2\left (e^{2 x}\right )+\frac {3}{4} x \cosh (x) \sinh (x)+\frac {1}{2} x^3 \cosh (x) \sinh (x)-\frac {3}{2} \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 x}\right )\\ &=\frac {3 x^2}{8}-x^3+\frac {3 x^4}{8}-\frac {3 \cosh ^2(x)}{8}-\frac {3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+3 x \text {Li}_2\left (e^{2 x}\right )-\frac {3 \text {Li}_3\left (e^{2 x}\right )}{2}+\frac {3}{4} x \cosh (x) \sinh (x)+\frac {1}{2} x^3 \cosh (x) \sinh (x)\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.11, size = 94, normalized size = 0.92 \begin {gather*} \frac {1}{16} \left (2 i \pi ^3-16 x^3+6 x^4-3 \cosh (2 x)-6 x^2 \cosh (2 x)-16 x^3 \coth (x)+48 x^2 \log \left (1-e^{2 x}\right )+48 x \text {PolyLog}\left (2,e^{2 x}\right )-24 \text {PolyLog}\left (3,e^{2 x}\right )+6 x \sinh (2 x)+4 x^3 \sinh (2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*Pi^3 - 16*x^3 + 6*x^4 - 3*Cosh[2*x] - 6*x^2*Cosh[2*x] - 16*x^3*Coth[x] + 48*x^2*Log[1 - E^(2*x)] + 48*x
*PolyLog[2, E^(2*x)] - 24*PolyLog[3, E^(2*x)] + 6*x*Sinh[2*x] + 4*x^3*Sinh[2*x])/16

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Maple [A]
time = 0.97, size = 117, normalized size = 1.15

method	result	size
risch	\(\frac {3 x^{4}}{8}+\left (-\frac {3}{32}+\frac {3}{16} x -\frac {3}{16} x^{2}+\frac {1}{8} x^{3}\right ) {\mathrm e}^{2 x}+\left (-\frac {3}{32}-\frac {3}{16} x -\frac {3}{16} x^{2}-\frac {1}{8} x^{3}\right ) {\mathrm e}^{-2 x}-\frac {2 x^{3}}{{\mathrm e}^{2 x}-1}-2 x^{3}+3 x^{2} \ln \left (1-{\mathrm e}^{x}\right )+6 x \polylog \left (2, {\mathrm e}^{x}\right )-6 \polylog \left (3, {\mathrm e}^{x}\right )+3 x^{2} \ln \left ({\mathrm e}^{x}+1\right )+6 x \polylog \left (2, -{\mathrm e}^{x}\right )-6 \polylog \left (3, -{\mathrm e}^{x}\right )\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(x)^2*coth(x)^2,x,method=_RETURNVERBOSE)

[Out]

3/8*x^4+(-3/32+3/16*x-3/16*x^2+1/8*x^3)*exp(2*x)+(-3/32-3/16*x-3/16*x^2-1/8*x^3)*exp(-2*x)-2*x^3/(exp(2*x)-1)-
2*x^3+3*x^2*ln(1-exp(x))+6*x*polylog(2,exp(x))-6*polylog(3,exp(x))+3*x^2*ln(exp(x)+1)+6*x*polylog(2,-exp(x))-6
*polylog(3,-exp(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (84) = 168\).
time = 0.38, size = 875, normalized size = 8.58 \begin {gather*} \frac {{\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} \cosh \left (x\right )^{6} + 6 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} \sinh \left (x\right )^{6} + {\left (12 \, x^{4} - 68 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} \cosh \left (x\right )^{4} + {\left (12 \, x^{4} - 68 \, x^{3} + 15 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} \cosh \left (x\right )^{2} + 6 \, x^{2} - 6 \, x + 3\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} \cosh \left (x\right )^{3} + {\left (12 \, x^{4} - 68 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, x^{3} - {\left (12 \, x^{4} + 4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} \cosh \left (x\right )^{2} + {\left (15 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} \cosh \left (x\right )^{4} - 12 \, x^{4} - 4 \, x^{3} + 6 \, {\left (12 \, x^{4} - 68 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} \cosh \left (x\right )^{2} - 6 \, x^{2} - 6 \, x - 3\right )} \sinh \left (x\right )^{2} + 6 \, x^{2} + 192 \, {\left (x \cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + x \sinh \left (x\right )^{4} - x \cosh \left (x\right )^{2} + {\left (6 \, x \cosh \left (x\right )^{2} - x\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, x \cosh \left (x\right )^{3} - x \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 192 \, {\left (x \cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + x \sinh \left (x\right )^{4} - x \cosh \left (x\right )^{2} + {\left (6 \, x \cosh \left (x\right )^{2} - x\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, x \cosh \left (x\right )^{3} - x \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) + 96 \, {\left (x^{2} \cosh \left (x\right )^{4} + 4 \, x^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + x^{2} \sinh \left (x\right )^{4} - x^{2} \cosh \left (x\right )^{2} + {\left (6 \, x^{2} \cosh \left (x\right )^{2} - x^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, x^{2} \cosh \left (x\right )^{3} - x^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 96 \, {\left (x^{2} \cosh \left (x\right )^{4} + 4 \, x^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + x^{2} \sinh \left (x\right )^{4} - x^{2} \cosh \left (x\right )^{2} + {\left (6 \, x^{2} \cosh \left (x\right )^{2} - x^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, x^{2} \cosh \left (x\right )^{3} - x^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) - 192 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 192 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) + 2 \, {\left (3 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} \cosh \left (x\right )^{5} + 2 \, {\left (12 \, x^{4} - 68 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} \cosh \left (x\right )^{3} - {\left (12 \, x^{4} + 4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 6 \, x + 3}{32 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*((4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^6 + 6*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)*sinh(x)^5 + (4*x^3 - 6*x^2 + 6
*x - 3)*sinh(x)^6 + (12*x^4 - 68*x^3 + 6*x^2 - 6*x + 3)*cosh(x)^4 + (12*x^4 - 68*x^3 + 15*(4*x^3 - 6*x^2 + 6*x
- 3)*cosh(x)^2 + 6*x^2 - 6*x + 3)*sinh(x)^4 + 4*(5*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^3 + (12*x^4 - 68*x^3 + 6
*x^2 - 6*x + 3)*cosh(x))*sinh(x)^3 + 4*x^3 - (12*x^4 + 4*x^3 + 6*x^2 + 6*x + 3)*cosh(x)^2 + (15*(4*x^3 - 6*x^2
+ 6*x - 3)*cosh(x)^4 - 12*x^4 - 4*x^3 + 6*(12*x^4 - 68*x^3 + 6*x^2 - 6*x + 3)*cosh(x)^2 - 6*x^2 - 6*x - 3)*si
nh(x)^2 + 6*x^2 + 192*(x*cosh(x)^4 + 4*x*cosh(x)*sinh(x)^3 + x*sinh(x)^4 - x*cosh(x)^2 + (6*x*cosh(x)^2 - x)*s
inh(x)^2 + 2*(2*x*cosh(x)^3 - x*cosh(x))*sinh(x))*dilog(cosh(x) + sinh(x)) + 192*(x*cosh(x)^4 + 4*x*cosh(x)*si
nh(x)^3 + x*sinh(x)^4 - x*cosh(x)^2 + (6*x*cosh(x)^2 - x)*sinh(x)^2 + 2*(2*x*cosh(x)^3 - x*cosh(x))*sinh(x))*d
ilog(-cosh(x) - sinh(x)) + 96*(x^2*cosh(x)^4 + 4*x^2*cosh(x)*sinh(x)^3 + x^2*sinh(x)^4 - x^2*cosh(x)^2 + (6*x^
2*cosh(x)^2 - x^2)*sinh(x)^2 + 2*(2*x^2*cosh(x)^3 - x^2*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 96*(x^2
*cosh(x)^4 + 4*x^2*cosh(x)*sinh(x)^3 + x^2*sinh(x)^4 - x^2*cosh(x)^2 + (6*x^2*cosh(x)^2 - x^2)*sinh(x)^2 + 2*(
2*x^2*cosh(x)^3 - x^2*cosh(x))*sinh(x))*log(-cosh(x) - sinh(x) + 1) - 192*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + s
inh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cosh(x)^3 - cosh(x))*sinh(x))*polylog(3, cosh(x) + s
inh(x)) - 192*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*co
sh(x)^3 - cosh(x))*sinh(x))*polylog(3, -cosh(x) - sinh(x)) + 2*(3*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^5 + 2*(12*
x^4 - 68*x^3 + 6*x^2 - 6*x + 3)*cosh(x)^3 - (12*x^4 + 4*x^3 + 6*x^2 + 6*x + 3)*cosh(x))*sinh(x) + 6*x + 3)/(co
sh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cosh(x)^3 - cosh(x)
)*sinh(x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \cosh ^{2}{\left (x \right )} \coth ^{2}{\left (x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(x^3*cosh(x)^2*coth(x)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {coth}\left (x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*cosh(x)^2*coth(x)^2, x)

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