Integrand size = 7, antiderivative size = 81 \[ \int \coth (5 x) \sinh (x) \, dx=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{5} \left (5-\sqrt {5}\right )} \sinh (x)\right )-\sqrt {\frac {2}{5 \left (5+\sqrt {5}\right )}} \arctan \left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sinh (x)\right )+\sinh (x) \] Output:
-1/10*(10+2*5^(1/2))^(1/2)*arctan(1/5*(50-10*5^(1/2))^(1/2)*sinh(x))-2^(1/ 2)/(25+5*5^(1/2))^(1/2)*arctan(1/5*(50+10*5^(1/2))^(1/2)*sinh(x))+sinh(x)
Time = 0.16 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \coth (5 x) \sinh (x) \, dx=\frac {1}{10} \left (-\sqrt {10-2 \sqrt {5}} \arctan \left (\sqrt {2+\frac {2}{\sqrt {5}}} \sinh (x)\right )-\sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sinh (x)\right )+10 \sinh (x)\right ) \] Input:
Integrate[Coth[5*x]*Sinh[x],x]
Output:
(-(Sqrt[10 - 2*Sqrt[5]]*ArcTan[Sqrt[2 + 2/Sqrt[5]]*Sinh[x]]) - Sqrt[2*(5 + Sqrt[5])]*ArcTan[2*Sqrt[2/(5 + Sqrt[5])]*Sinh[x]] + 10*Sinh[x])/10
Time = 0.39 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4878, 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.
\(\displaystyle \int \sinh (x) \coth (5 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cot (5 i x)}{\csc (i x)}dx\) |
\(\Big \downarrow \) 4878 |
\(\displaystyle \int \frac {16 \sinh ^4(x)+12 \sinh ^2(x)+1}{16 \sinh ^4(x)+20 \sinh ^2(x)+5}d\sinh (x)\) |
\(\Big \downarrow \) 2205 |
\(\displaystyle \int \left (1-\frac {4 \left (2 \sinh ^2(x)+1\right )}{16 \sinh ^4(x)+20 \sinh ^2(x)+5}\right )d\sinh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sinh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sinh (x)\right )+\sinh (x)\) |
Input:
Int[Coth[5*x]*Sinh[x],x]
Output:
-1/5*(Sqrt[(5 + Sqrt[5])/2]*ArcTan[2*Sqrt[2/(5 + Sqrt[5])]*Sinh[x]]) - (Sq rt[(5 - Sqrt[5])/2]*ArcTan[Sqrt[(2*(5 + Sqrt[5]))/5]*Sinh[x]])/5 + Sinh[x]
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[Sin[v], x]}, d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeF actors[Sin[v], x], u/Cos[v], x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (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(coth(5*x)*sinh(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. 303 vs. \(2 (56) = 112\).
Time = 0.10 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.74 \[ \int \coth (5 x) \sinh (x) \, dx=\frac {2 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\frac {1}{10} \, {\left ({\left (\sqrt {5} - 5\right )} \cosh \left (x\right )^{3} + 3 \, {\left (\sqrt {5} - 5\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (\sqrt {5} - 5\right )} \sinh \left (x\right )^{3} - {\left (\sqrt {5} + 5\right )} \cosh \left (x\right ) + {\left (3 \, {\left (\sqrt {5} - 5\right )} \cosh \left (x\right )^{2} - \sqrt {5} - 5\right )} \sinh \left (x\right )\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) + 2 \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\frac {1}{10} \, {\left ({\left (\sqrt {5} - 5\right )} \cosh \left (x\right ) + {\left (\sqrt {5} - 5\right )} \sinh \left (x\right )\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) - 2 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\frac {1}{10} \, {\left ({\left (\sqrt {5} + 5\right )} \cosh \left (x\right )^{3} + 3 \, {\left (\sqrt {5} + 5\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (\sqrt {5} + 5\right )} \sinh \left (x\right )^{3} - {\left (\sqrt {5} - 5\right )} \cosh \left (x\right ) + {\left (3 \, {\left (\sqrt {5} + 5\right )} \cosh \left (x\right )^{2} - \sqrt {5} + 5\right )} \sinh \left (x\right )\right )} \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) - 2 \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\frac {1}{10} \, {\left ({\left (\sqrt {5} + 5\right )} \cosh \left (x\right ) + {\left (\sqrt {5} + 5\right )} \sinh \left (x\right )\right )} \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\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(coth(5*x)*sinh(x),x, algorithm="fricas")
Output:
1/10*(2*sqrt(1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x))*arctan(1/10*((sqrt(5) - 5)*cosh(x)^3 + 3*(sqrt(5) - 5)*cosh(x)*sinh(x)^2 + (sqrt(5) - 5)*sinh(x) ^3 - (sqrt(5) + 5)*cosh(x) + (3*(sqrt(5) - 5)*cosh(x)^2 - sqrt(5) - 5)*sin h(x))*sqrt(1/2*sqrt(5) + 5/2)) + 2*sqrt(1/2*sqrt(5) + 5/2)*(cosh(x) + sinh (x))*arctan(1/10*((sqrt(5) - 5)*cosh(x) + (sqrt(5) - 5)*sinh(x))*sqrt(1/2* sqrt(5) + 5/2)) - 2*sqrt(-1/2*sqrt(5) + 5/2)*(cosh(x) + sinh(x))*arctan(1/ 10*((sqrt(5) + 5)*cosh(x)^3 + 3*(sqrt(5) + 5)*cosh(x)*sinh(x)^2 + (sqrt(5) + 5)*sinh(x)^3 - (sqrt(5) - 5)*cosh(x) + (3*(sqrt(5) + 5)*cosh(x)^2 - sqr t(5) + 5)*sinh(x))*sqrt(-1/2*sqrt(5) + 5/2)) - 2*sqrt(-1/2*sqrt(5) + 5/2)* (cosh(x) + sinh(x))*arctan(1/10*((sqrt(5) + 5)*cosh(x) + (sqrt(5) + 5)*sin h(x))*sqrt(-1/2*sqrt(5) + 5/2)) + 5*cosh(x)^2 + 10*cosh(x)*sinh(x) + 5*sin h(x)^2 - 5)/(cosh(x) + sinh(x))
\[ \int \coth (5 x) \sinh (x) \, dx=\int \sinh {\left (x \right )} \coth {\left (5 x \right )}\, dx \] Input:
integrate(coth(5*x)*sinh(x),x)
Output:
Integral(sinh(x)*coth(5*x), x)
\[ \int \coth (5 x) \sinh (x) \, dx=\int { \coth \left (5 \, x\right ) \sinh \left (x\right ) \,d x } \] Input:
integrate(coth(5*x)*sinh(x),x, algorithm="maxima")
Output:
1/2*(e^(2*x) - 1)*e^(-x) - 1/2*integrate((e^(3*x) + e^(2*x) + e^x)/(e^(4*x ) + e^(3*x) + e^(2*x) + e^x + 1), x) - 1/2*integrate((e^(3*x) - e^(2*x) + e^x)/(e^(4*x) - e^(3*x) + e^(2*x) - e^x + 1), x)
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \coth (5 x) \sinh (x) \, dx=-\frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {e^{\left (-x\right )} - e^{x}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}}\right ) - \frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (-\frac {e^{\left (-x\right )} - e^{x}}{\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \] Input:
integrate(coth(5*x)*sinh(x),x, algorithm="giac")
Output:
-1/10*sqrt(2*sqrt(5) + 10)*arctan(-(e^(-x) - e^x)/sqrt(1/2*sqrt(5) + 5/2)) - 1/10*sqrt(-2*sqrt(5) + 10)*arctan(-(e^(-x) - e^x)/sqrt(-1/2*sqrt(5) + 5 /2)) - 1/2*e^(-x) + 1/2*e^x
Time = 2.93 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.74 \[ \int \coth (5 x) \sinh (x) \, dx=\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}+\ln \left (40\,{\mathrm {e}}^x\,\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}-4\,{\mathrm {e}}^{2\,x}+4\right )\,\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}+\ln \left (40\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}-4\,{\mathrm {e}}^{2\,x}+4\right )\,\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}-\ln \left (4\,{\mathrm {e}}^{2\,x}+40\,{\mathrm {e}}^x\,\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}-4\right )\,\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}-\ln \left (4\,{\mathrm {e}}^{2\,x}+40\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}-4\right )\,\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}} \] Input:
int(coth(5*x)*sinh(x),x)
Output:
exp(x)/2 - exp(-x)/2 + log(40*exp(x)*(- 5^(1/2)/200 - 1/40)^(1/2) - 4*exp( 2*x) + 4)*(- 5^(1/2)/200 - 1/40)^(1/2) + log(40*exp(x)*(5^(1/2)/200 - 1/40 )^(1/2) - 4*exp(2*x) + 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) - l og(4*exp(2*x) + 40*exp(x)*(5^(1/2)/200 - 1/40)^(1/2) - 4)*(5^(1/2)/200 - 1 /40)^(1/2)
\[ \int \coth (5 x) \sinh (x) \, dx=\frac {e^{2 x}+2 e^{x} \left (\int \frac {1}{e^{9 x}+e^{7 x}+e^{5 x}+e^{3 x}+e^{x}}d x \right )+1}{2 e^{x}} \] Input:
int(coth(5*x)*sinh(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)