Integrand size = 7, antiderivative size = 84 \[ \int \cosh (x) \coth (5 x) \, dx=-\frac {1}{5} \text {arctanh}(\cosh (x))-\frac {1}{5} \sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {2 \left (3-\sqrt {5}\right )} \cosh (x)\right )-\frac {1}{5} \sqrt {\frac {2}{3+\sqrt {5}}} \text {arctanh}\left (\sqrt {2 \left (3+\sqrt {5}\right )} \cosh (x)\right )+\cosh (x) \] Output:
-1/5*arctanh(cosh(x))-1/5*(1/2+1/2*5^(1/2))*arctanh((5^(1/2)-1)*cosh(x))-1 /5*2^(1/2)/(3+5^(1/2))^(1/2)*arctanh((5^(1/2)+1)*cosh(x))+cosh(x)
Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.58 \[ \int \cosh (x) \coth (5 x) \, dx=\frac {1}{100} \left (100 \cosh (x)-20 \log \left (\cosh \left (\frac {x}{2}\right )\right )+\sqrt {5} \left (-5+\sqrt {5}\right ) \log \left (1-\sqrt {5}-4 \cosh (x)\right )+\sqrt {5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}-4 \cosh (x)\right )-\sqrt {5} \left (-5+\sqrt {5}\right ) \log \left (1-\sqrt {5}+4 \cosh (x)\right )-\sqrt {5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+4 \cosh (x)\right )+20 \log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \] Input:
Integrate[Cosh[x]*Coth[5*x],x]
Output:
(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])*Log[ 1 + Sqrt[5] + 4*Cosh[x]] + 20*Log[Sinh[x/2]])/100
Time = 0.43 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 26, 4879, 2460, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cosh (x) \coth (5 x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i \cos (i x) \cot (5 i x)dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \cos (i x) \cot (5 i x)dx\) |
| \(\Big \downarrow \) 4879 |
| \(\displaystyle -\int \frac {\cosh ^2(x) \left (16 \cosh ^4(x)-20 \cosh ^2(x)+5\right )}{-16 \cosh ^6(x)+28 \cosh ^4(x)-13 \cosh ^2(x)+1}d\cosh (x)\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle -\int \left (\frac {2 (\cosh (x)-1)}{5 \left (4 \cosh ^2(x)+2 \cosh (x)-1\right )}-\frac {1}{5 \left (\cosh ^2(x)-1\right )}-\frac {2 (\cosh (x)+1)}{5 \left (4 \cosh ^2(x)-2 \cosh (x)-1\right )}-1\right )d\cosh (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{5} \text {arctanh}(\cosh (x))+\cosh (x)+\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (-4 \cosh (x)-\sqrt {5}+1\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (-4 \cosh (x)+\sqrt {5}+1\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (4 \cosh (x)-\sqrt {5}+1\right )-\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (4 \cosh (x)+\sqrt {5}+1\right )\) |
Input:
Int[Cosh[x]*Coth[5*x],x]
Output:
-1/5*ArcTanh[Cosh[x]] + 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
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[Q x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Cos[v], x]}, -d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Cos[v]/d , u/Sin[v], x], x], x, Cos[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[Nonfree Factors[Cos[v], x], u/Sin[v], x]]
Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(47)=94\).
Time = 0.12 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.26
| method | result | size |
| risch | \(\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{5}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{5}-\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right )}{20}+\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right ) \sqrt {5}}{20}-\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right )}{20}-\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right ) \sqrt {5}}{20}+\frac {\ln \left ({\mathrm e}^{2 x}+\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right )}{20}+\frac {\ln \left ({\mathrm e}^{2 x}+\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{x}+1\right ) \sqrt {5}}{20}+\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) {\mathrm e}^{x}+1\right )}{20}-\frac {\ln \left ({\mathrm e}^{2 x}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) {\mathrm e}^{x}+1\right ) \sqrt {5}}{20}\) | \(190\) |
Input:
int(cosh(x)*coth(5*x),x,method=_RETURNVERBOSE)
Output:
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+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)+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)
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (45) = 90\).
Time = 0.09 (sec) , antiderivative size = 272, normalized size of antiderivative = 3.24 \[ \int \cosh (x) \coth (5 x) \, dx=\frac {10 \, \cosh \left (x\right )^{2} + {\left (\sqrt {5} \cosh \left (x\right ) + \sqrt {5} \sinh \left (x\right )\right )} \log \left (-\frac {4 \, {\left (\sqrt {5} - 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} - 4 \, \sinh \left (x\right )^{2} + \sqrt {5} - 7}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1}\right ) + {\left (\sqrt {5} \cosh \left (x\right ) + \sqrt {5} \sinh \left (x\right )\right )} \log \left (-\frac {4 \, {\left (\sqrt {5} + 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} - 4 \, \sinh \left (x\right )^{2} - \sqrt {5} - 7}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1}\right ) - {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) - 4 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 4 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 20 \, \cosh \left (x\right ) \sinh \left (x\right ) + 10 \, \sinh \left (x\right )^{2} + 10}{20 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \] Input:
integrate(cosh(x)*coth(5*x),x, algorithm="fricas")
Output:
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*s inh(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)/(c osh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 4*(cosh(x) + sinh(x))*log(cos h(x) + sinh(x) + 1) + 4*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x) - 1) + 2 0*cosh(x)*sinh(x) + 10*sinh(x)^2 + 10)/(cosh(x) + sinh(x))
\[ \int \cosh (x) \coth (5 x) \, dx=\int \cosh {\left (x \right )} \coth {\left (5 x \right )}\, dx \] Input:
integrate(cosh(x)*coth(5*x),x)
Output:
Integral(cosh(x)*coth(5*x), x)
\[ \int \cosh (x) \coth (5 x) \, dx=\int { \cosh \left (x\right ) \coth \left (5 \, x\right ) \,d x } \] Input:
integrate(cosh(x)*coth(5*x),x, algorithm="maxima")
Output:
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*in tegrate(e^(3*x)/(e^(4*x) + e^(3*x) + e^(2*x) + e^x + 1), x) + 3/10*integra te(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)
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (45) = 90\).
Time = 0.14 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.87 \[ \int \cosh (x) \coth (5 x) \, dx=\frac {1}{20} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, e^{\left (-x\right )} - 2 \, e^{x} + 1}{\sqrt {5} + 2 \, e^{\left (-x\right )} + 2 \, e^{x} - 1}\right ) + \frac {1}{20} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, e^{\left (-x\right )} - 2 \, e^{x} - 1}{\sqrt {5} + 2 \, e^{\left (-x\right )} + 2 \, e^{x} + 1}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{20} \, \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + e^{\left (-x\right )} + e^{x} - 1\right ) + \frac {1}{20} \, \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - e^{\left (-x\right )} - e^{x} - 1\right ) - \frac {1}{10} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \frac {1}{10} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \] Input:
integrate(cosh(x)*coth(5*x),x, algorithm="giac")
Output:
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) - 1/1 0*log(e^(-x) + e^x + 2) + 1/10*log(e^(-x) + e^x - 2)
Time = 0.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.70 \[ \int \cosh (x) \coth (5 x) \, dx=\frac {\ln \left (10-10\,{\mathrm {e}}^x\right )}{5}-\frac {\ln \left (-10\,{\mathrm {e}}^x-10\right )}{5}+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}-\ln \left (-{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^x\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-1\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )+\ln \left (10\,{\mathrm {e}}^x\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-{\mathrm {e}}^{2\,x}-1\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )-\ln \left (-{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^x\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-1\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )+\ln \left (10\,{\mathrm {e}}^x\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )-{\mathrm {e}}^{2\,x}-1\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right ) \] Input:
int(coth(5*x)*cosh(x),x)
Output:
log(10 - 10*exp(x))/5 - log(- 10*exp(x) - 10)/5 + exp(-x)/2 + exp(x)/2 - l og(- exp(2*x) - 10*exp(x)*(5^(1/2)/20 - 1/20) - 1)*(5^(1/2)/20 - 1/20) + l og(10*exp(x)*(5^(1/2)/20 - 1/20) - exp(2*x) - 1)*(5^(1/2)/20 - 1/20) - log (- exp(2*x) - 10*exp(x)*(5^(1/2)/20 + 1/20) - 1)*(5^(1/2)/20 + 1/20) + log (10*exp(x)*(5^(1/2)/20 + 1/20) - exp(2*x) - 1)*(5^(1/2)/20 + 1/20)
\[ \int \cosh (x) \coth (5 x) \, dx=\frac {e^{2 x}+2 e^{x} \left (\int \frac {e^{x}}{e^{10 x}-1}d x \right )+2 e^{x} \left (\int \frac {1}{e^{11 x}-e^{x}}d x \right )-1}{2 e^{x}} \] Input:
int(cosh(x)*coth(5*x),x)
Output:
(e**(2*x) + 2*e**x*int(e**x/(e**(10*x) - 1),x) + 2*e**x*int(1/(e**(11*x) - e**x),x) - 1)/(2*e**x)