Integrand size = 7, antiderivative size = 38 \[ \int \cosh (x) \coth (6 x) \, dx=-\frac {1}{6} \text {arctanh}(\cosh (x))-\frac {1}{6} \text {arctanh}(2 \cosh (x))-\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt {3}}+\cosh (x) \] Output:
-1/6*arctanh(cosh(x))-1/6*arctanh(2*cosh(x))-1/6*arctanh(2/3*cosh(x)*3^(1/ 2))*3^(1/2)+cosh(x)
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.50 \[ \int \cosh (x) \coth (6 x) \, dx=\frac {1}{12} \left (-2 \sqrt {3} \text {arctanh}\left (\frac {2-i \tanh \left (\frac {x}{2}\right )}{\sqrt {3}}\right )-2 \sqrt {3} \text {arctanh}\left (\frac {2+i \tanh \left (\frac {x}{2}\right )}{\sqrt {3}}\right )+12 \cosh (x)-2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+\log (1-2 \cosh (x))-\log (1+2 \cosh (x))+2 \log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \] Input:
Integrate[Cosh[x]*Coth[6*x],x]
Output:
(-2*Sqrt[3]*ArcTanh[(2 - I*Tanh[x/2])/Sqrt[3]] - 2*Sqrt[3]*ArcTanh[(2 + I* Tanh[x/2])/Sqrt[3]] + 12*Cosh[x] - 2*Log[Cosh[x/2]] + Log[1 - 2*Cosh[x]] - Log[1 + 2*Cosh[x]] + 2*Log[Sinh[x/2]])/12
Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 26, 4879, 27, 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.
\(\displaystyle \int \cosh (x) \coth (6 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \cos (i x) \cot (6 i x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \cos (i x) \cot (6 i x)dx\) |
\(\Big \downarrow \) 4879 |
\(\displaystyle -\int -\frac {-32 \cosh ^6(x)+48 \cosh ^4(x)-18 \cosh ^2(x)+1}{2 \left (-16 \cosh ^6(x)+32 \cosh ^4(x)-19 \cosh ^2(x)+3\right )}d\cosh (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {-32 \cosh ^6(x)+48 \cosh ^4(x)-18 \cosh ^2(x)+1}{-16 \cosh ^6(x)+32 \cosh ^4(x)-19 \cosh ^2(x)+3}d\cosh (x)\) |
\(\Big \downarrow \) 2460 |
\(\displaystyle \frac {1}{2} \int \left (\frac {2}{4 \cosh ^2(x)-3}+\frac {2}{3 \left (4 \cosh ^2(x)-1\right )}+2+\frac {1}{3 \left (\cosh ^2(x)-1\right )}\right )d\cosh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{3} \text {arctanh}(\cosh (x))-\frac {1}{3} \text {arctanh}(2 \cosh (x))-\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {3}}\right )}{\sqrt {3}}+2 \cosh (x)\right )\) |
Input:
Int[Cosh[x]*Coth[6*x],x]
Output:
(-1/3*ArcTanh[Cosh[x]] - ArcTanh[2*Cosh[x]]/3 - ArcTanh[(2*Cosh[x])/Sqrt[3 ]]/Sqrt[3] + 2*Cosh[x])/2
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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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. \(86\) vs. \(2(28)=56\).
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.29
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{6}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{6}+\frac {\ln \left ({\mathrm e}^{2 x}-{\mathrm e}^{x}+1\right )}{12}-\frac {\ln \left ({\mathrm e}^{2 x}+{\mathrm e}^{x}+1\right )}{12}+\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 x}-\sqrt {3}\, {\mathrm e}^{x}+1\right )}{12}-\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 x}+\sqrt {3}\, {\mathrm e}^{x}+1\right )}{12}\) | \(87\) |
Input:
int(cosh(x)*coth(6*x),x,method=_RETURNVERBOSE)
Output:
1/2*exp(x)+1/2*exp(-x)+1/6*ln(exp(x)-1)-1/6*ln(exp(x)+1)+1/12*ln(exp(2*x)- exp(x)+1)-1/12*ln(exp(2*x)+exp(x)+1)+1/12*3^(1/2)*ln(exp(2*x)-3^(1/2)*exp( x)+1)-1/12*3^(1/2)*ln(exp(2*x)+3^(1/2)*exp(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (28) = 56\).
Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 4.13 \[ \int \cosh (x) \coth (6 x) \, dx=\frac {6 \, \cosh \left (x\right )^{2} + {\left (\sqrt {3} \cosh \left (x\right ) + \sqrt {3} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 4 \, \sqrt {3} \cosh \left (x\right ) + 5}{2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 1}\right ) - {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right ) + 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right ) - 1}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 12 \, \cosh \left (x\right ) \sinh \left (x\right ) + 6 \, \sinh \left (x\right )^{2} + 6}{12 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \] Input:
integrate(cosh(x)*coth(6*x),x, algorithm="fricas")
Output:
1/12*(6*cosh(x)^2 + (sqrt(3)*cosh(x) + sqrt(3)*sinh(x))*log((2*cosh(x)^2 + 2*sinh(x)^2 - 4*sqrt(3)*cosh(x) + 5)/(2*cosh(x)^2 + 2*sinh(x)^2 - 1)) - ( cosh(x) + sinh(x))*log((2*cosh(x) + 1)/(cosh(x) - sinh(x))) + (cosh(x) + s inh(x))*log((2*cosh(x) - 1)/(cosh(x) - sinh(x))) - 2*(cosh(x) + sinh(x))*l og(cosh(x) + sinh(x) + 1) + 2*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x) - 1) + 12*cosh(x)*sinh(x) + 6*sinh(x)^2 + 6)/(cosh(x) + sinh(x))
\[ \int \cosh (x) \coth (6 x) \, dx=\int \cosh {\left (x \right )} \coth {\left (6 x \right )}\, dx \] Input:
integrate(cosh(x)*coth(6*x),x)
Output:
Integral(cosh(x)*coth(6*x), x)
\[ \int \cosh (x) \coth (6 x) \, dx=\int { \cosh \left (x\right ) \coth \left (6 \, x\right ) \,d x } \] Input:
integrate(cosh(x)*coth(6*x),x, algorithm="maxima")
Output:
1/2*(e^(2*x) + 1)*e^(-x) + 1/2*integrate((e^(3*x) - e^x)/(e^(4*x) - e^(2*x ) + 1), x) - 1/12*log(e^(2*x) + e^x + 1) + 1/12*log(e^(2*x) - e^x + 1) - 1 /6*log(e^x + 1) + 1/6*log(e^x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (28) = 56\).
Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.34 \[ \int \cosh (x) \coth (6 x) \, dx=\frac {1}{12} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - e^{\left (-x\right )} - e^{x}}{\sqrt {3} + e^{\left (-x\right )} + e^{x}}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} + 1\right ) + \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} - 1\right ) + \frac {1}{12} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \] Input:
integrate(cosh(x)*coth(6*x),x, algorithm="giac")
Output:
1/12*sqrt(3)*log(-(sqrt(3) - e^(-x) - e^x)/(sqrt(3) + e^(-x) + e^x)) + 1/2 *e^(-x) + 1/2*e^x - 1/12*log(e^(-x) + e^x + 2) - 1/12*log(e^(-x) + e^x + 1 ) + 1/12*log(e^(-x) + e^x - 1) + 1/12*log(e^(-x) + e^x - 2)
Time = 0.96 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.66 \[ \int \cosh (x) \coth (6 x) \, dx=\frac {\ln \left (\frac {1}{3}-\frac {{\mathrm {e}}^x}{3}\right )}{6}-\frac {\ln \left (-\frac {{\mathrm {e}}^x}{3}-\frac {1}{3}\right )}{6}+\frac {{\mathrm {e}}^{-x}}{2}-\frac {\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{36}-\frac {{\mathrm {e}}^x}{36}-\frac {1}{36}\right )}{12}+\frac {\ln \left (\frac {{\mathrm {e}}^x}{36}-\frac {{\mathrm {e}}^{2\,x}}{36}-\frac {1}{36}\right )}{12}+\frac {{\mathrm {e}}^x}{2}-\frac {\sqrt {3}\,\ln \left (-\frac {{\mathrm {e}}^{2\,x}}{12}-\frac {\sqrt {3}\,{\mathrm {e}}^x}{12}-\frac {1}{12}\right )}{12}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}\,{\mathrm {e}}^x}{12}-\frac {{\mathrm {e}}^{2\,x}}{12}-\frac {1}{12}\right )}{12} \] Input:
int(coth(6*x)*cosh(x),x)
Output:
log(1/3 - exp(x)/3)/6 - log(- exp(x)/3 - 1/3)/6 + exp(-x)/2 - log(- exp(2* x)/36 - exp(x)/36 - 1/36)/12 + log(exp(x)/36 - exp(2*x)/36 - 1/36)/12 + ex p(x)/2 - (3^(1/2)*log(- exp(2*x)/12 - (3^(1/2)*exp(x))/12 - 1/12))/12 + (3 ^(1/2)*log((3^(1/2)*exp(x))/12 - exp(2*x)/12 - 1/12))/12
Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.97 \[ \int \cosh (x) \coth (6 x) \, dx=\frac {6 e^{2 x}+e^{x} \sqrt {3}\, \mathrm {log}\left (e^{2 x}-e^{x} \sqrt {3}+1\right )-e^{x} \sqrt {3}\, \mathrm {log}\left (e^{2 x}+e^{x} \sqrt {3}+1\right )-e^{x} \mathrm {log}\left (e^{2 x}+e^{x}+1\right )+e^{x} \mathrm {log}\left (e^{2 x}-e^{x}+1\right )+2 e^{x} \mathrm {log}\left (e^{x}-1\right )-2 e^{x} \mathrm {log}\left (e^{x}+1\right )+6}{12 e^{x}} \] Input:
int(cosh(x)*coth(6*x),x)
Output:
(6*e**(2*x) + e**x*sqrt(3)*log(e**(2*x) - e**x*sqrt(3) + 1) - e**x*sqrt(3) *log(e**(2*x) + e**x*sqrt(3) + 1) - e**x*log(e**(2*x) + e**x + 1) + e**x*l og(e**(2*x) - e**x + 1) + 2*e**x*log(e**x - 1) - 2*e**x*log(e**x + 1) + 6) /(12*e**x)