Integrand size = 7, antiderivative size = 81 \[ \int \cosh (x) \tanh (5 x) \, dx=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5-\sqrt {5}\right )} \cosh (x)\right )-\sqrt {\frac {2}{5 \left (5+\sqrt {5}\right )}} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cosh (x)\right )+\cosh (x) \] Output:
-1/10*(10+2*5^(1/2))^(1/2)*arctanh(1/5*(50-10*5^(1/2))^(1/2)*cosh(x))-2^(1 /2)/(25+5*5^(1/2))^(1/2)*arctanh(1/5*(50+10*5^(1/2))^(1/2)*cosh(x))+cosh(x )
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 249, normalized size of antiderivative = 3.07 \[ \int \cosh (x) \tanh (5 x) \, dx=\cosh (x)+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4-\text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-x-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )+x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2-x \text {$\#$1}^4-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^4+x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+2 \text {$\#$1}^3-3 \text {$\#$1}^5+4 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[Cosh[x]*Tanh[5*x],x]
Output:
Cosh[x] + RootSum[1 - #1^2 + #1^4 - #1^6 + #1^8 & , (-x - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1] + x*#1^2 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^2 - x*#1^4 - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^4 + x*#1^6 + 2*Log[-Cosh[x/2 ] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(-#1 + 2*#1^3 - 3*#1^5 + 4*#1^7) & ]/4
Time = 0.38 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01, 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, 2205, 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.
\(\displaystyle \int \cosh (x) \tanh (5 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (5 i x)}{\sec (i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (5 i x)}{\sec (i x)}dx\) |
\(\Big \downarrow \) 4879 |
\(\displaystyle \int \frac {16 \cosh ^4(x)-12 \cosh ^2(x)+1}{16 \cosh ^4(x)-20 \cosh ^2(x)+5}d\cosh (x)\) |
\(\Big \downarrow \) 2205 |
\(\displaystyle \int \left (1-\frac {4 \left (1-2 \cosh ^2(x)\right )}{16 \cosh ^4(x)-20 \cosh ^2(x)+5}\right )d\cosh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {arctanh}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cosh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cosh (x)\right )+\cosh (x)\) |
Input:
Int[Cosh[x]*Tanh[5*x],x]
Output:
-1/5*(Sqrt[(5 + Sqrt[5])/2]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Cosh[x]]) - (S qrt[(5 - Sqrt[5])/2]*ArcTanh[Sqrt[(2*(5 + Sqrt[5]))/5]*Cosh[x]])/5 + Cosh[ x]
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[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
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]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2000 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-10 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+1\right )\right )\) | \(42\) |
Input:
int(cosh(x)*tanh(5*x),x,method=_RETURNVERBOSE)
Output:
1/2*exp(x)+1/2*exp(-x)+sum(_R*ln(-10*_R*exp(x)+exp(2*x)+1),_R=RootOf(2000* _Z^4-100*_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (56) = 112\).
Time = 0.10 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.73 \[ \int \cosh (x) \tanh (5 x) \, dx=-\frac {\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) + \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - 5 \, \cosh \left (x\right )^{2} - 10 \, \cosh \left (x\right ) \sinh \left (x\right ) - 5 \, \sinh \left (x\right )^{2} - 5}{10 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \] Input:
integrate(cosh(x)*tanh(5*x),x, algorithm="fricas")
Output:
-1/10*(sqrt(1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh( x)*sinh(x) + sinh(x)^2 + sqrt(1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x)) + 1) - sqrt(1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*si nh(x) + sinh(x)^2 - sqrt(1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x)) + 1) + sqr t(-1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x ) + sinh(x)^2 + sqrt(-1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x)) + 1) - sqrt(- 1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(-1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x)) + 1) - 5*cosh(x) ^2 - 10*cosh(x)*sinh(x) - 5*sinh(x)^2 - 5)/(cosh(x) + sinh(x))
\[ \int \cosh (x) \tanh (5 x) \, dx=\int \cosh {\left (x \right )} \tanh {\left (5 x \right )}\, dx \] Input:
integrate(cosh(x)*tanh(5*x),x)
Output:
Integral(cosh(x)*tanh(5*x), x)
\[ \int \cosh (x) \tanh (5 x) \, dx=\int { \cosh \left (x\right ) \tanh \left (5 \, x\right ) \,d x } \] Input:
integrate(cosh(x)*tanh(5*x),x, algorithm="maxima")
Output:
1/2*(e^(2*x) + 1)*e^(-x) + 1/2*integrate(2*(e^(7*x) - e^(5*x) + e^(3*x) - e^x)/(e^(8*x) - e^(6*x) + e^(4*x) - e^(2*x) + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (56) = 112\).
Time = 0.15 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.57 \[ \int \cosh (x) \tanh (5 x) \, dx=-\frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left (-\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left (-\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \] Input:
integrate(cosh(x)*tanh(5*x),x, algorithm="giac")
Output:
-1/20*sqrt(2*sqrt(5) + 10)*log(sqrt(1/2*sqrt(5) + 5/2) + e^(-x) + e^x) + 1 /20*sqrt(2*sqrt(5) + 10)*log(-sqrt(1/2*sqrt(5) + 5/2) + e^(-x) + e^x) - 1/ 20*sqrt(-2*sqrt(5) + 10)*log(sqrt(-1/2*sqrt(5) + 5/2) + e^(-x) + e^x) + 1/ 20*sqrt(-2*sqrt(5) + 10)*log(-sqrt(-1/2*sqrt(5) + 5/2) + e^(-x) + e^x) + 1 /2*e^(-x) + 1/2*e^x
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.74 \[ \int \cosh (x) \tanh (5 x) \, dx=\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}+\ln \left (4\,{\mathrm {e}}^{2\,x}-40\,{\mathrm {e}}^x\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+4\right )\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}-\ln \left (4\,{\mathrm {e}}^{2\,x}+40\,{\mathrm {e}}^x\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+4\right )\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+\ln \left (4\,{\mathrm {e}}^{2\,x}-40\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+4\right )\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}-\ln \left (4\,{\mathrm {e}}^{2\,x}+40\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+4\right )\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}} \] Input:
int(tanh(5*x)*cosh(x),x)
Output:
exp(-x)/2 + exp(x)/2 + log(4*exp(2*x) - 40*exp(x)*(1/40 - 5^(1/2)/200)^(1/ 2) + 4)*(1/40 - 5^(1/2)/200)^(1/2) - log(4*exp(2*x) + 40*exp(x)*(1/40 - 5^ (1/2)/200)^(1/2) + 4)*(1/40 - 5^(1/2)/200)^(1/2) + log(4*exp(2*x) - 40*exp (x)*(5^(1/2)/200 + 1/40)^(1/2) + 4)*(5^(1/2)/200 + 1/40)^(1/2) - log(4*exp (2*x) + 40*exp(x)*(5^(1/2)/200 + 1/40)^(1/2) + 4)*(5^(1/2)/200 + 1/40)^(1/ 2)
\[ \int \cosh (x) \tanh (5 x) \, dx=\frac {e^{2 x}-2 e^{x} \left (\int \frac {1}{e^{9 x}-e^{7 x}+e^{5 x}-e^{3 x}+e^{x}}d x \right )-1}{2 e^{x}} \] Input:
int(cosh(x)*tanh(5*x),x)
Output:
(e**(2*x) - 2*e**x*int(1/(e**(9*x) - e**(7*x) + e**(5*x) - e**(3*x) + e**x ),x) - 1)/(2*e**x)