3.654 \(\int (\coth (x)+\text {csch}(x))^5 \, dx\)

Optimal. Leaf size=28 \[ \frac {4}{1-\cosh (x)}-\frac {2}{(1-\cosh (x))^2}+\log (1-\cosh (x)) \]

[Out]

-2/(1-cosh(x))^2+4/(1-cosh(x))+ln(1-cosh(x))

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Rubi [A]  time = 0.07, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4392, 2667, 43} \[ \frac {4}{1-\cosh (x)}-\frac {2}{(1-\cosh (x))^2}+\log (1-\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Coth[x] + Csch[x])^5,x]

[Out]

-2/(1 - Cosh[x])^2 + 4/(1 - Cosh[x]) + Log[1 - Cosh[x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

\begin {align*} \int (\coth (x)+\text {csch}(x))^5 \, dx &=-\left (i \int (i+i \cosh (x))^5 \text {csch}^5(x) \, dx\right )\\ &=-\operatorname {Subst}\left (\int \frac {(i+x)^2}{(i-x)^3} \, dx,x,i \cosh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{i-x}+\frac {4}{(-i+x)^3}-\frac {4 i}{(-i+x)^2}\right ) \, dx,x,i \cosh (x)\right )\\ &=\frac {2}{(i-i \cosh (x))^2}+\frac {4 i}{i-i \cosh (x)}+\log (1-\cosh (x))\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 53, normalized size = 1.89 \[ -\frac {1}{2} \text {csch}^4\left (\frac {x}{2}\right )-2 \text {csch}^2\left (\frac {x}{2}\right )+6 \log \left (\sinh \left (\frac {x}{2}\right )\right )+\log (\sinh (x))-5 \log \left (\tanh \left (\frac {x}{2}\right )\right )-6 \log \left (\cosh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Coth[x] + Csch[x])^5,x]

[Out]

-2*Csch[x/2]^2 - Csch[x/2]^4/2 - 6*Log[Cosh[x/2]] + 6*Log[Sinh[x/2]] + Log[Sinh[x]] - 5*Log[Tanh[x/2]]

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fricas [B]  time = 0.43, size = 270, normalized size = 9.64 \[ -\frac {x \cosh \relax (x)^{4} + x \sinh \relax (x)^{4} - 4 \, {\left (x - 2\right )} \cosh \relax (x)^{3} + 4 \, {\left (x \cosh \relax (x) - x + 2\right )} \sinh \relax (x)^{3} + 2 \, {\left (3 \, x - 4\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, x \cosh \relax (x)^{2} - 6 \, {\left (x - 2\right )} \cosh \relax (x) + 3 \, x - 4\right )} \sinh \relax (x)^{2} - 4 \, {\left (x - 2\right )} \cosh \relax (x) - 2 \, {\left (\cosh \relax (x)^{4} + 4 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 4 \, \cosh \relax (x)^{3} + 6 \, {\left (\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{2} + 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 3 \, \cosh \relax (x)^{2} + 3 \, \cosh \relax (x) - 1\right )} \sinh \relax (x) - 4 \, \cosh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 4 \, {\left (x \cosh \relax (x)^{3} - 3 \, {\left (x - 2\right )} \cosh \relax (x)^{2} + {\left (3 \, x - 4\right )} \cosh \relax (x) - x + 2\right )} \sinh \relax (x) + x}{\cosh \relax (x)^{4} + 4 \, {\left (\cosh \relax (x) - 1\right )} \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 4 \, \cosh \relax (x)^{3} + 6 \, {\left (\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{2} + 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 3 \, \cosh \relax (x)^{2} + 3 \, \cosh \relax (x) - 1\right )} \sinh \relax (x) - 4 \, \cosh \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^5,x, algorithm="fricas")

[Out]

-(x*cosh(x)^4 + x*sinh(x)^4 - 4*(x - 2)*cosh(x)^3 + 4*(x*cosh(x) - x + 2)*sinh(x)^3 + 2*(3*x - 4)*cosh(x)^2 +
2*(3*x*cosh(x)^2 - 6*(x - 2)*cosh(x) + 3*x - 4)*sinh(x)^2 - 4*(x - 2)*cosh(x) - 2*(cosh(x)^4 + 4*(cosh(x) - 1)
*sinh(x)^3 + sinh(x)^4 - 4*cosh(x)^3 + 6*(cosh(x)^2 - 2*cosh(x) + 1)*sinh(x)^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 -
3*cosh(x)^2 + 3*cosh(x) - 1)*sinh(x) - 4*cosh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 4*(x*cosh(x)^3 - 3*(x - 2)*
cosh(x)^2 + (3*x - 4)*cosh(x) - x + 2)*sinh(x) + x)/(cosh(x)^4 + 4*(cosh(x) - 1)*sinh(x)^3 + sinh(x)^4 - 4*cos
h(x)^3 + 6*(cosh(x)^2 - 2*cosh(x) + 1)*sinh(x)^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x)^2 + 3*cosh(x) - 1)*s
inh(x) - 4*cosh(x) + 1)

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giac [A]  time = 0.12, size = 33, normalized size = 1.18 \[ -x - \frac {8 \, {\left (e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} - 1\right )}^{4}} + 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^5,x, algorithm="giac")

[Out]

-x - 8*(e^(3*x) - e^(2*x) + e^x)/(e^x - 1)^4 + 2*log(abs(e^x - 1))

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maple [B]  time = 0.32, size = 71, normalized size = 2.54 \[ \ln \left (\sinh \relax (x )\right )-\frac {\left (\coth ^{2}\relax (x )\right )}{2}-\frac {\left (\coth ^{4}\relax (x )\right )}{4}-\frac {5 \left (\cosh ^{3}\relax (x )\right )}{\sinh \relax (x )^{4}}+\frac {5 \cosh \relax (x )}{3 \sinh \relax (x )^{4}}+\frac {8 \left (-\frac {\mathrm {csch}\relax (x )^{3}}{4}+\frac {3 \,\mathrm {csch}\relax (x )}{8}\right ) \coth \relax (x )}{3}-2 \arctanh \left ({\mathrm e}^{x}\right )-\frac {5 \left (\cosh ^{2}\relax (x )\right )}{\sinh \relax (x )^{4}}+\frac {5}{4 \sinh \relax (x )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(x)+csch(x))^5,x)

[Out]

ln(sinh(x))-1/2*coth(x)^2-1/4*coth(x)^4-5/sinh(x)^4*cosh(x)^3+5/3/sinh(x)^4*cosh(x)+8/3*(-1/4*csch(x)^3+3/8*cs
ch(x))*coth(x)-2*arctanh(exp(x))-5/sinh(x)^4*cosh(x)^2+5/4/sinh(x)^4

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maxima [B]  time = 0.58, size = 236, normalized size = 8.43 \[ -\frac {5}{2} \, \coth \relax (x)^{4} + x + \frac {5 \, {\left (5 \, e^{\left (-x\right )} + 3 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-7 \, x\right )}\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} - \frac {3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {5 \, {\left (e^{\left (-x\right )} + 7 \, e^{\left (-3 \, x\right )} + 7 \, e^{\left (-5 \, x\right )} + e^{\left (-7 \, x\right )}\right )}}{2 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {4 \, {\left (e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac {20}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{4}} + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))^5,x, algorithm="maxima")

[Out]

-5/2*coth(x)^4 + x + 5/4*(5*e^(-x) + 3*e^(-3*x) + 3*e^(-5*x) + 5*e^(-7*x))/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*
x) - e^(-8*x) - 1) - 1/4*(3*e^(-x) - 11*e^(-3*x) - 11*e^(-5*x) + 3*e^(-7*x))/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-
6*x) - e^(-8*x) - 1) + 5/2*(e^(-x) + 7*e^(-3*x) + 7*e^(-5*x) + e^(-7*x))/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x)
- e^(-8*x) - 1) + 4*(e^(-2*x) - e^(-4*x) + e^(-6*x))/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) -
20/(e^(-x) - e^x)^4 + 2*log(e^(-x) - 1)

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mupad [B]  time = 1.52, size = 81, normalized size = 2.89 \[ 2\,\ln \left ({\mathrm {e}}^x-1\right )-x+\frac {16}{3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1}-\frac {16}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1}-\frac {8}{6\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x+1}-\frac {8}{{\mathrm {e}}^x-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(x) + 1/sinh(x))^5,x)

[Out]

2*log(exp(x) - 1) - x + 16/(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1) - 16/(exp(2*x) - 2*exp(x) + 1) - 8/(6*exp(2*
x) - 4*exp(3*x) + exp(4*x) - 4*exp(x) + 1) - 8/(exp(x) - 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )^{5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((coth(x)+csch(x))**5,x)

[Out]

Integral((coth(x) + csch(x))**5, x)

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