### 3.237 $$\int \cosh (x) \coth (5 x) \, dx$$

Optimal. Leaf size=110 $\cosh (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-4 \cosh (x)-\sqrt{5}+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-4 \cosh (x)+\sqrt{5}+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (4 \cosh (x)-\sqrt{5}+1\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (4 \cosh (x)+\sqrt{5}+1\right )-\frac{1}{5} \tanh ^{-1}(\cosh (x))$

[Out]

-ArcTanh[Cosh[x]]/5 + Cosh[x] + ((1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Cosh[x]])/20 + ((1 + Sqrt[5])*Log[1 + Sqrt[
5] - 4*Cosh[x]])/20 - ((1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Cosh[x]])/20 - ((1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos
h[x]])/20

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Rubi [A]  time = 0.176159, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.571, Rules used = {2075, 207, 632, 31} $\cosh (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-4 \cosh (x)-\sqrt{5}+1\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-4 \cosh (x)+\sqrt{5}+1\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (4 \cosh (x)-\sqrt{5}+1\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (4 \cosh (x)+\sqrt{5}+1\right )-\frac{1}{5} \tanh ^{-1}(\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[x]*Coth[5*x],x]

[Out]

-ArcTanh[Cosh[x]]/5 + Cosh[x] + ((1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Cosh[x]])/20 + ((1 + Sqrt[5])*Log[1 + Sqrt[
5] - 4*Cosh[x]])/20 - ((1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Cosh[x]])/20 - ((1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos
h[x]])/20

Rule 2075

Int[(P_)^(p_)*(Qm_), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Qm, x], x] /; QuadraticProdu
ctQ[PP, x]] /; PolyQ[Qm, x] && PolyQ[P, x] && ILtQ[p, 0]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \cosh (x) \coth (5 x) \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (5-20 x^2+16 x^4\right )}{1-13 x^2+28 x^4-16 x^6} \, dx,x,\cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-1-\frac{1}{5 \left (-1+x^2\right )}-\frac{2 (1+x)}{5 \left (-1-2 x+4 x^2\right )}+\frac{2 (-1+x)}{5 \left (-1+2 x+4 x^2\right )}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cosh (x)\right )+\frac{2}{5} \operatorname{Subst}\left (\int \frac{1+x}{-1-2 x+4 x^2} \, dx,x,\cosh (x)\right )-\frac{2}{5} \operatorname{Subst}\left (\int \frac{-1+x}{-1+2 x+4 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{5} \tanh ^{-1}(\cosh (x))+\cosh (x)-\frac{1}{5} \left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{5}+4 x} \, dx,x,\cosh (x)\right )+\frac{1}{5} \left (1-\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\sqrt{5}+4 x} \, dx,x,\cosh (x)\right )+\frac{1}{5} \left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\sqrt{5}+4 x} \, dx,x,\cosh (x)\right )-\frac{1}{5} \left (1+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{5}+4 x} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{5} \tanh ^{-1}(\cosh (x))+\cosh (x)+\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (1-\sqrt{5}-4 \cosh (x)\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (1+\sqrt{5}-4 \cosh (x)\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (1-\sqrt{5}+4 \cosh (x)\right )-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (1+\sqrt{5}+4 \cosh (x)\right )\\ \end{align*}

Mathematica [A]  time = 0.114537, size = 133, normalized size = 1.21 $\frac{1}{100} \left (100 \cosh (x)+20 \log \left (\sinh \left (\frac{x}{2}\right )\right )-20 \log \left (\cosh \left (\frac{x}{2}\right )\right )+\sqrt{5} \left (\sqrt{5}-5\right ) \log \left (-4 \cosh (x)-\sqrt{5}+1\right )+\sqrt{5} \left (5+\sqrt{5}\right ) \log \left (-4 \cosh (x)+\sqrt{5}+1\right )-\sqrt{5} \left (\sqrt{5}-5\right ) \log \left (4 \cosh (x)-\sqrt{5}+1\right )-\sqrt{5} \left (5+\sqrt{5}\right ) \log \left (4 \cosh (x)+\sqrt{5}+1\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[x]*Coth[5*x],x]

[Out]

(100*Cosh[x] - 20*Log[Cosh[x/2]] + Sqrt[5]*(-5 + Sqrt[5])*Log[1 - Sqrt[5] - 4*Cosh[x]] + Sqrt[5]*(5 + Sqrt[5])
*Log[1 + Sqrt[5] - 4*Cosh[x]] - Sqrt[5]*(-5 + Sqrt[5])*Log[1 - Sqrt[5] + 4*Cosh[x]] - Sqrt[5]*(5 + Sqrt[5])*Lo
g[1 + Sqrt[5] + 4*Cosh[x]] + 20*Log[Sinh[x/2]])/100

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Maple [B]  time = 0.083, size = 190, normalized size = 1.7 \begin{align*}{\frac{{{\rm e}^{x}}}{2}}+{\frac{{{\rm e}^{-x}}}{2}}-{\frac{\ln \left ({{\rm e}^{x}}+1 \right ) }{5}}+{\frac{\ln \left ({{\rm e}^{x}}-1 \right ) }{5}}+{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ( -{\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) }{20}}+{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ( -{\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) \sqrt{5}}{20}}+{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{\sqrt{5}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{x}}+1 \right ) }{20}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{\sqrt{5}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{x}}+1 \right ) \sqrt{5}}{20}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) }{20}}+{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) \sqrt{5}}{20}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{1}{2}}+{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) }{20}}-{\frac{\ln \left ({{\rm e}^{2\,x}}+ \left ({\frac{1}{2}}+{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{x}}+1 \right ) \sqrt{5}}{20}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*coth(5*x),x)

[Out]

1/2*exp(x)+1/2*exp(-x)-1/5*ln(exp(x)+1)+1/5*ln(exp(x)-1)+1/20*ln(exp(2*x)+(-1/2-1/2*5^(1/2))*exp(x)+1)+1/20*ln
(exp(2*x)+(-1/2-1/2*5^(1/2))*exp(x)+1)*5^(1/2)+1/20*ln(exp(2*x)+(1/2*5^(1/2)-1/2)*exp(x)+1)-1/20*ln(exp(2*x)+(
1/2*5^(1/2)-1/2)*exp(x)+1)*5^(1/2)-1/20*ln(exp(2*x)+(1/2-1/2*5^(1/2))*exp(x)+1)+1/20*ln(exp(2*x)+(1/2-1/2*5^(1
/2))*exp(x)+1)*5^(1/2)-1/20*ln(exp(2*x)+(1/2+1/2*5^(1/2))*exp(x)+1)-1/20*ln(exp(2*x)+(1/2+1/2*5^(1/2))*exp(x)+
1)*5^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} - \frac{1}{5} \, \int \frac{{\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1\right )} e^{x}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} + \frac{1}{5} \, \int \frac{{\left (e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + e^{x} - 1\right )} e^{x}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} + \frac{3}{10} \, \int \frac{e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} + \frac{3}{10} \, \int \frac{e^{\left (3 \, x\right )}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} + \frac{1}{10} \, \int \frac{e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} - \frac{1}{10} \, \int \frac{e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} - \frac{1}{10} \, \int \frac{e^{x}}{e^{\left (4 \, x\right )} + e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x} + 1}\,{d x} - \frac{1}{10} \, \int \frac{e^{x}}{e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - e^{x} + 1}\,{d x} - \frac{1}{5} \, \log \left (e^{x} + 1\right ) + \frac{1}{5} \, \log \left (e^{x} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(5*x),x, algorithm="maxima")

[Out]

1/2*(e^(2*x) + 1)*e^(-x) - 1/5*integrate((e^(3*x) + e^(2*x) + e^x + 1)*e^x/(e^(4*x) + e^(3*x) + e^(2*x) + e^x
+ 1), x) + 1/5*integrate((e^(3*x) - e^(2*x) + e^x - 1)*e^x/(e^(4*x) - e^(3*x) + e^(2*x) - e^x + 1), x) + 3/10*
integrate(e^(3*x)/(e^(4*x) + e^(3*x) + e^(2*x) + e^x + 1), x) + 3/10*integrate(e^(3*x)/(e^(4*x) - e^(3*x) + e^
(2*x) - e^x + 1), x) + 1/10*integrate(e^(2*x)/(e^(4*x) + e^(3*x) + e^(2*x) + e^x + 1), x) - 1/10*integrate(e^(
2*x)/(e^(4*x) - e^(3*x) + e^(2*x) - e^x + 1), x) - 1/10*integrate(e^x/(e^(4*x) + e^(3*x) + e^(2*x) + e^x + 1),
x) - 1/10*integrate(e^x/(e^(4*x) - e^(3*x) + e^(2*x) - e^x + 1), x) - 1/5*log(e^x + 1) + 1/5*log(e^x - 1)

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Fricas [B]  time = 2.23793, size = 980, normalized size = 8.91 \begin{align*} \frac{10 \, \cosh \left (x\right )^{2} +{\left (\sqrt{5} \cosh \left (x\right ) + \sqrt{5} \sinh \left (x\right )\right )} \log \left (-\frac{4 \,{\left (\sqrt{5} - 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} - 4 \, \sinh \left (x\right )^{2} + \sqrt{5} - 7}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1}\right ) +{\left (\sqrt{5} \cosh \left (x\right ) + \sqrt{5} \sinh \left (x\right )\right )} \log \left (-\frac{4 \,{\left (\sqrt{5} + 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} - 4 \, \sinh \left (x\right )^{2} - \sqrt{5} - 7}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1}\right ) -{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) +{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - 4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 20 \, \cosh \left (x\right ) \sinh \left (x\right ) + 10 \, \sinh \left (x\right )^{2} + 10}{20 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(5*x),x, algorithm="fricas")

[Out]

1/20*(10*cosh(x)^2 + (sqrt(5)*cosh(x) + sqrt(5)*sinh(x))*log(-(4*(sqrt(5) - 1)*cosh(x) - 4*cosh(x)^2 - 4*sinh(
x)^2 + sqrt(5) - 7)/(2*cosh(x)^2 + 2*sinh(x)^2 + 2*cosh(x) + 1)) + (sqrt(5)*cosh(x) + sqrt(5)*sinh(x))*log(-(4
*(sqrt(5) + 1)*cosh(x) - 4*cosh(x)^2 - 4*sinh(x)^2 - sqrt(5) - 7)/(2*cosh(x)^2 + 2*sinh(x)^2 - 2*cosh(x) + 1))
- (cosh(x) + sinh(x))*log((2*cosh(x)^2 + 2*sinh(x)^2 + 2*cosh(x) + 1)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x
)^2)) + (cosh(x) + sinh(x))*log((2*cosh(x)^2 + 2*sinh(x)^2 - 2*cosh(x) + 1)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + s
inh(x)^2)) - 4*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x) + 1) + 4*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x) -
1) + 20*cosh(x)*sinh(x) + 10*sinh(x)^2 + 10)/(cosh(x) + sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \coth{\left (5 x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(5*x),x)

[Out]

Integral(cosh(x)*coth(5*x), x)

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Giac [A]  time = 1.18942, size = 212, normalized size = 1.93 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, e^{\left (-x\right )} - 2 \, e^{x} + 1}{\sqrt{5} + 2 \, e^{\left (-x\right )} + 2 \, e^{x} - 1}\right ) + \frac{1}{20} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, e^{\left (-x\right )} - 2 \, e^{x} - 1}{\sqrt{5} + 2 \, e^{\left (-x\right )} + 2 \, e^{x} + 1}\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} - \frac{1}{20} \, \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + e^{\left (-x\right )} + e^{x} - 1\right ) + \frac{1}{20} \, \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - e^{\left (-x\right )} - e^{x} - 1\right ) - \frac{1}{10} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac{1}{10} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*coth(5*x),x, algorithm="giac")

[Out]

1/20*sqrt(5)*log(-(sqrt(5) - 2*e^(-x) - 2*e^x + 1)/(sqrt(5) + 2*e^(-x) + 2*e^x - 1)) + 1/20*sqrt(5)*log(-(sqrt
(5) - 2*e^(-x) - 2*e^x - 1)/(sqrt(5) + 2*e^(-x) + 2*e^x + 1)) + 1/2*e^(-x) + 1/2*e^x - 1/20*log((e^(-x) + e^x)
^2 + e^(-x) + e^x - 1) + 1/20*log((e^(-x) + e^x)^2 - e^(-x) - e^x - 1) - 1/10*log(e^(-x) + e^x + 2) + 1/10*log
(e^(-x) + e^x - 2)