Integrand size = 7, antiderivative size = 71 \[ \int \text {sech}(5 x) \sinh (x) \, dx=\frac {1}{5} \log (\cosh (x))-\frac {\log \left (5-\sqrt {5}-8 \cosh ^2(x)\right )}{\sqrt {5} \left (5-\sqrt {5}\right )}+\frac {\log \left (5+\sqrt {5}-8 \cosh ^2(x)\right )}{\sqrt {5} \left (5+\sqrt {5}\right )} \] Output:
1/5*ln(cosh(x))-1/5*ln(5-5^(1/2)-8*cosh(x)^2)*5^(1/2)/(5-5^(1/2))+1/5*ln(5 +5^(1/2)-8*cosh(x)^2)*5^(1/2)/(5+5^(1/2))
Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.80 \[ \int \text {sech}(5 x) \sinh (x) \, dx=\frac {1}{20} \left (4 \log (\cosh (x))+\left (-1+\sqrt {5}\right ) \log \left (3-\sqrt {5}+8 \sinh ^2(x)\right )-\left (1+\sqrt {5}\right ) \log \left (3+\sqrt {5}+8 \sinh ^2(x)\right )\right ) \] Input:
Integrate[Sech[5*x]*Sinh[x],x]
Output:
(4*Log[Cosh[x]] + (-1 + Sqrt[5])*Log[3 - Sqrt[5] + 8*Sinh[x]^2] - (1 + Sqr t[5])*Log[3 + Sqrt[5] + 8*Sinh[x]^2])/20
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 26, 4857, 1434, 1141, 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.
\(\displaystyle \int \sinh (x) \text {sech}(5 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i x)}{\cos (5 i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i x)}{\cos (5 i x)}dx\) |
\(\Big \downarrow \) 4857 |
\(\displaystyle \int \frac {\text {sech}(x)}{16 \cosh ^4(x)-20 \cosh ^2(x)+5}d\cosh (x)\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {\text {sech}(x)}{16 \cosh ^4(x)-20 \cosh ^2(x)+5}d\cosh ^2(x)\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle 8 \int \left (\frac {\text {sech}(x)}{80}+\frac {1}{\sqrt {5} \left (5-\sqrt {5}\right ) \left (-8 \cosh ^2(x)-\sqrt {5}+5\right )}-\frac {1}{\sqrt {5} \left (5+\sqrt {5}\right ) \left (-8 \cosh ^2(x)+\sqrt {5}+5\right )}\right )d\cosh ^2(x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \left (\frac {1}{80} \log \left (\cosh ^2(x)\right )-\frac {\log \left (-8 \cosh ^2(x)-\sqrt {5}+5\right )}{8 \sqrt {5} \left (5-\sqrt {5}\right )}+\frac {\log \left (-8 \cosh ^2(x)+\sqrt {5}+5\right )}{8 \sqrt {5} \left (5+\sqrt {5}\right )}\right )\) |
Input:
Int[Sech[5*x]*Sinh[x],x]
Output:
8*(Log[Cosh[x]^2]/80 - Log[5 - Sqrt[5] - 8*Cosh[x]^2]/(8*Sqrt[5]*(5 - Sqrt [5])) + Log[5 + Sqrt[5] - 8*Cosh[x]^2]/(8*Sqrt[5]*(5 + Sqrt[5])))
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[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFacto rs[Cos[c*(a + b*x)], x]}, Simp[-d/(b*c) Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b* x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.42
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{2 x}+1\right )}{5}-\frac {\ln \left ({\mathrm e}^{4 x}+\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{2 x}+1\right )}{20}+\frac {\ln \left ({\mathrm e}^{4 x}+\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{2 x}+1\right ) \sqrt {5}}{20}-\frac {\ln \left ({\mathrm e}^{4 x}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) {\mathrm e}^{2 x}+1\right )}{20}-\frac {\ln \left ({\mathrm e}^{4 x}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) {\mathrm e}^{2 x}+1\right ) \sqrt {5}}{20}\) | \(101\) |
Input:
int(sech(5*x)*sinh(x),x,method=_RETURNVERBOSE)
Output:
1/5*ln(exp(2*x)+1)-1/20*ln(exp(4*x)+(-1/2-1/2*5^(1/2))*exp(2*x)+1)+1/20*ln (exp(4*x)+(-1/2-1/2*5^(1/2))*exp(2*x)+1)*5^(1/2)-1/20*ln(exp(4*x)+(1/2*5^( 1/2)-1/2)*exp(2*x)+1)-1/20*ln(exp(4*x)+(1/2*5^(1/2)-1/2)*exp(2*x)+1)*5^(1/ 2)
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (56) = 112\).
Time = 0.09 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.56 \[ \int \text {sech}(5 x) \sinh (x) \, dx=\frac {1}{20} \, \sqrt {5} \log \left (\frac {4 \, \cosh \left (x\right )^{4} + 4 \, \sinh \left (x\right )^{4} - 4 \, {\left (\sqrt {5} + 1\right )} \cosh \left (x\right )^{2} + 4 \, {\left (6 \, \cosh \left (x\right )^{2} - \sqrt {5} - 1\right )} \sinh \left (x\right )^{2} + \sqrt {5} + 7}{2 \, \cosh \left (x\right )^{4} + 2 \, \sinh \left (x\right )^{4} + 2 \, {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 1}\right ) - \frac {1}{20} \, \log \left (\frac {2 \, \cosh \left (x\right )^{4} + 2 \, \sinh \left (x\right )^{4} + 2 \, {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}}\right ) + \frac {1}{5} \, \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \] Input:
integrate(sech(5*x)*sinh(x),x, algorithm="fricas")
Output:
1/20*sqrt(5)*log((4*cosh(x)^4 + 4*sinh(x)^4 - 4*(sqrt(5) + 1)*cosh(x)^2 + 4*(6*cosh(x)^2 - sqrt(5) - 1)*sinh(x)^2 + sqrt(5) + 7)/(2*cosh(x)^4 + 2*si nh(x)^4 + 2*(6*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 1)) - 1/20*log((2* cosh(x)^4 + 2*sinh(x)^4 + 2*(6*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 1) /(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh (x)^3 + sinh(x)^4)) + 1/5*log(2*cosh(x)/(cosh(x) - sinh(x)))
\[ \int \text {sech}(5 x) \sinh (x) \, dx=\int \sinh {\left (x \right )} \operatorname {sech}{\left (5 x \right )}\, dx \] Input:
integrate(sech(5*x)*sinh(x),x)
Output:
Integral(sinh(x)*sech(5*x), x)
\[ \int \text {sech}(5 x) \sinh (x) \, dx=\int { \operatorname {sech}\left (5 \, x\right ) \sinh \left (x\right ) \,d x } \] Input:
integrate(sech(5*x)*sinh(x),x, algorithm="maxima")
Output:
-2/5*integrate((e^(6*x) - e^(4*x) + e^(2*x) - 1)*e^(2*x)/(e^(8*x) - e^(6*x ) + e^(4*x) - e^(2*x) + 1), x) + 2/5*integrate(e^(6*x)/(e^(8*x) - e^(6*x) + e^(4*x) - e^(2*x) + 1), x) + 1/5*integrate(e^(4*x)/(e^(8*x) - e^(6*x) + e^(4*x) - e^(2*x) + 1), x) - 4/5*integrate(e^(2*x)/(e^(8*x) - e^(6*x) + e^ (4*x) - e^(2*x) + 1), x) + 1/5*log(e^(2*x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (56) = 112\).
Time = 0.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.66 \[ \int \text {sech}(5 x) \sinh (x) \, dx=\frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (\frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (\frac {1}{2} \, \sqrt {-2 \, \sqrt {5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (-\frac {1}{2} \, \sqrt {-2 \, \sqrt {5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{5} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \] Input:
integrate(sech(5*x)*sinh(x),x, algorithm="giac")
Output:
1/20*(sqrt(5) - 1)*log(1/2*sqrt(2*sqrt(5) + 10)*e^x + e^(2*x) + 1) + 1/20* (sqrt(5) - 1)*log(-1/2*sqrt(2*sqrt(5) + 10)*e^x + e^(2*x) + 1) - 1/20*(sqr t(5) + 1)*log(1/2*sqrt(-2*sqrt(5) + 10)*e^x + e^(2*x) + 1) - 1/20*(sqrt(5) + 1)*log(-1/2*sqrt(-2*sqrt(5) + 10)*e^x + e^(2*x) + 1) + 1/5*log(e^(2*x) + 1)
Time = 1.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int \text {sech}(5 x) \sinh (x) \, dx=\frac {\ln \left (5\,{\mathrm {e}}^{2\,x}+5\right )}{5}-\ln \left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )\,\left (20\,{\mathrm {e}}^{2\,x}+30\,{\mathrm {e}}^{4\,x}+30\right )+2\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )+\ln \left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}-\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )\,\left (20\,{\mathrm {e}}^{2\,x}+30\,{\mathrm {e}}^{4\,x}+30\right )+2\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right ) \] Input:
int(sinh(x)/cosh(5*x),x)
Output:
log(5*exp(2*x) + 5)/5 - log(exp(2*x) + 2*exp(4*x) + (5^(1/2)/20 + 1/20)*(2 0*exp(2*x) + 30*exp(4*x) + 30) + 2)*(5^(1/2)/20 + 1/20) + log(exp(2*x) + 2 *exp(4*x) - (5^(1/2)/20 - 1/20)*(20*exp(2*x) + 30*exp(4*x) + 30) + 2)*(5^( 1/2)/20 - 1/20)
\[ \int \text {sech}(5 x) \sinh (x) \, dx=\int \mathrm {sech}\left (5 x \right ) \sinh \left (x \right )d x \] Input:
int(sech(5*x)*sinh(x),x)
Output:
int(sech(5*x)*sinh(x),x)