### 3.664 $$\int (-\coth (x)+\text{csch}(x))^5 \, dx$$

Optimal. Leaf size=24 $-\frac{4}{\cosh (x)+1}+\frac{2}{(\cosh (x)+1)^2}-\log (\cosh (x)+1)$

[Out]

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

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Rubi [A]  time = 0.0626796, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {4392, 2667, 43} $-\frac{4}{\cosh (x)+1}+\frac{2}{(\cosh (x)+1)^2}-\log (\cosh (x)+1)$

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 [B]  time = 0.0846704, size = 55, normalized size = 2.29 $\frac{1}{2} \text{sech}^4\left (\frac{x}{2}\right )-2 \text{sech}^2\left (\frac{x}{2}\right )+6 \log \left (\sinh \left (\frac{x}{2}\right )\right )-\log (\sinh (x))-5 \log \left (\tanh \left (\frac{x}{2}\right )\right )-6 \log \left (\cosh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-Coth[x] + Csch[x])^5,x]

[Out]

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

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Maple [B]  time = 0.022, size = 77, normalized size = 3.2 \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*c
sch(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 = 1.23135, size = 321, normalized size = 13.38 \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) + 2
0/(e^(-x) - e^x)^4 - 2*log(e^(-x) + 1)

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Fricas [B]  time = 2.04957, size = 923, normalized size = 38.46 \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)*c
osh(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*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)*si
nh(x) + 4*cosh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int 5 \coth{\left (x \right )} \operatorname{csch}^{4}{\left (x \right )}\, dx - \int - 10 \coth ^{2}{\left (x \right )} \operatorname{csch}^{3}{\left (x \right )}\, dx - \int 10 \coth ^{3}{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )}\, dx - \int - 5 \coth ^{4}{\left (x \right )} \operatorname{csch}{\left (x \right )}\, dx - \int \coth ^{5}{\left (x \right )}\, dx - \int - \operatorname{csch}^{5}{\left (x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(5*coth(x)*csch(x)**4, x) - Integral(-10*coth(x)**2*csch(x)**3, x) - Integral(10*coth(x)**3*csch(x)**
2, x) - Integral(-5*coth(x)**4*csch(x), x) - Integral(coth(x)**5, x) - Integral(-csch(x)**5, x)

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Giac [A]  time = 1.12839, size = 38, normalized size = 1.58 \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 (e^{x} + 1\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(e^x + 1)