### 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]) + Log[1 - Cosh[x]]

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Rubi [A]  time = 0.0670859, antiderivative size = 28, normalized size of antiderivative = 1., 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 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]

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 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])

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*}

Mathematica [A]  time = 0.0870054, 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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Maple [B]  time = 0.021, size = 75, normalized size = 2.7 \begin{align*} \ln \left ( \sinh \left ( x \right ) \right ) -{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{2}}{2}}-{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{4}}{4}}-5\,{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{3}}{ \left ( \sinh \left ( x \right ) \right ) ^{4}}}+{\frac{5\,\cosh \left ( x \right ) }{3\, \left ( \sinh \left ( x \right ) \right ) ^{4}}}+{\frac{8\,{\rm coth} \left (x\right )}{3} \left ( -{\frac{ \left ({\rm csch} \left (x\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm csch} \left (x\right )}{8}} \right ) }-2\,{\it Artanh} \left ({{\rm e}^{x}} \right ) -{\frac{15\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{4\, \left ( \sinh \left ( x \right ) \right ) ^{4}}}-{\frac{5\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{4\, \left ( \sinh \left ( x \right ) \right ) ^{2}}} \end{align*}

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))-15/4/sinh(x)^4*cosh(x)^2-5/4*cosh(x)^2/sinh(x)^2

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Maxima [B]  time = 1.27624, size = 319, normalized size = 11.39 \begin{align*} -\frac{5}{2} \, \coth \left (x\right )^{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 ) \end{align*}

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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Fricas [B]  time = 2.08206, size = 925, normalized size = 33.04 \begin{align*} -\frac{x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} - 4 \,{\left (x - 2\right )} \cosh \left (x\right )^{3} + 4 \,{\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right )^{3} + 2 \,{\left (3 \, x - 4\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, x \cosh \left (x\right )^{2} - 6 \,{\left (x - 2\right )} \cosh \left (x\right ) + 3 \, x - 4\right )} \sinh \left (x\right )^{2} - 4 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 4 \,{\left (x \cosh \left (x\right )^{3} - 3 \,{\left (x - 2\right )} \cosh \left (x\right )^{2} +{\left (3 \, x - 4\right )} \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \,{\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \,{\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1} \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 1.12696, size = 45, normalized size = 1.61 \begin{align*} -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 ) \end{align*}

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))