Integrand size = 7, antiderivative size = 71 \[ \int \text {sech}(4 x) \sinh (x) \, dx=\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \] Output:
1/2*arctanh(2*cosh(x)/(2-2^(1/2))^(1/2))/(4-2*2^(1/2))^(1/2)-1/2*arctanh(2 *cosh(x)/(2+2^(1/2))^(1/2))/(4+2*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.55 \[ \int \text {sech}(4 x) \sinh (x) \, dx=\frac {1}{16} \text {RootSum}\left [1+\text {$\#$1}^8\&,\frac {-x-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )+x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2}{\text {$\#$1}^5}\&\right ] \] Input:
Integrate[Sech[4*x]*Sinh[x],x]
Output:
RootSum[1 + #1^8 & , (-x - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - S inh[x/2]*#1] + x*#1^2 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh [x/2]*#1]*#1^2)/#1^5 & ]/16
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 26, 4857, 1406, 220}
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.
\(\displaystyle \int \sinh (x) \text {sech}(4 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i x)}{\cos (4 i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i x)}{\cos (4 i x)}dx\) |
\(\Big \downarrow \) 4857 |
\(\displaystyle \int \frac {1}{8 \cosh ^4(x)-8 \cosh ^2(x)+1}d\cosh (x)\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \sqrt {2} \int \frac {1}{8 \cosh ^2(x)-2 \left (2+\sqrt {2}\right )}d\cosh (x)-\sqrt {2} \int \frac {1}{8 \cosh ^2(x)-2 \left (2-\sqrt {2}\right )}d\cosh (x)\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}\) |
Input:
Int[Sech[4*x]*Sinh[x],x]
Output:
ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTanh [(2*Cosh[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
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])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.56
method | result | size |
risch | \(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (32768 \textit {\_Z}^{4}-512 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (4096 \textit {\_R}^{3}-48 \textit {\_R} \right ) {\mathrm e}^{x}+1\right )\right )\) | \(40\) |
Input:
int(sech(4*x)*sinh(x),x,method=_RETURNVERBOSE)
Output:
2*sum(_R*ln(exp(2*x)+(4096*_R^3-48*_R)*exp(x)+1),_R=RootOf(32768*_Z^4-512* _Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (49) = 98\).
Time = 0.10 (sec) , antiderivative size = 215, normalized size of antiderivative = 3.03 \[ \int \text {sech}(4 x) \sinh (x) \, dx=\frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + {\left ({\left (\sqrt {2} - 1\right )} \cosh \left (x\right ) + {\left (\sqrt {2} - 1\right )} \sinh \left (x\right )\right )} \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - {\left ({\left (\sqrt {2} - 1\right )} \cosh \left (x\right ) + {\left (\sqrt {2} - 1\right )} \sinh \left (x\right )\right )} \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + {\left ({\left (\sqrt {2} + 1\right )} \cosh \left (x\right ) + {\left (\sqrt {2} + 1\right )} \sinh \left (x\right )\right )} \sqrt {-\sqrt {2} + 2} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - {\left ({\left (\sqrt {2} + 1\right )} \cosh \left (x\right ) + {\left (\sqrt {2} + 1\right )} \sinh \left (x\right )\right )} \sqrt {-\sqrt {2} + 2} + 1\right ) \] Input:
integrate(sech(4*x)*sinh(x),x, algorithm="fricas")
Output:
1/8*sqrt(sqrt(2) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + ((sq rt(2) - 1)*cosh(x) + (sqrt(2) - 1)*sinh(x))*sqrt(sqrt(2) + 2) + 1) - 1/8*s qrt(sqrt(2) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqrt(2) - 1)*cosh(x) + (sqrt(2) - 1)*sinh(x))*sqrt(sqrt(2) + 2) + 1) - 1/8*sqrt(- sqrt(2) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + ((sqrt(2) + 1 )*cosh(x) + (sqrt(2) + 1)*sinh(x))*sqrt(-sqrt(2) + 2) + 1) + 1/8*sqrt(-sqr t(2) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqrt(2) + 1)*c osh(x) + (sqrt(2) + 1)*sinh(x))*sqrt(-sqrt(2) + 2) + 1)
\[ \int \text {sech}(4 x) \sinh (x) \, dx=\int \sinh {\left (x \right )} \operatorname {sech}{\left (4 x \right )}\, dx \] Input:
integrate(sech(4*x)*sinh(x),x)
Output:
Integral(sinh(x)*sech(4*x), x)
\[ \int \text {sech}(4 x) \sinh (x) \, dx=\int { \operatorname {sech}\left (4 \, x\right ) \sinh \left (x\right ) \,d x } \] Input:
integrate(sech(4*x)*sinh(x),x, algorithm="maxima")
Output:
integrate(sech(4*x)*sinh(x), x)
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (49) = 98\).
Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.62 \[ \int \text {sech}(4 x) \sinh (x) \, dx=-\frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \] Input:
integrate(sech(4*x)*sinh(x),x, algorithm="giac")
Output:
-1/8*sqrt(-sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)*e^x + e^(2*x) + 1) + 1/8*sqr t(-sqrt(2) + 2)*log(-sqrt(sqrt(2) + 2)*e^x + e^(2*x) + 1) + 1/8*sqrt(sqrt( 2) + 2)*log(sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) - 1/8*sqrt(sqrt(2) + 2)* log(-sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1)
Time = 1.25 (sec) , antiderivative size = 251, normalized size of antiderivative = 3.54 \[ \int \text {sech}(4 x) \sinh (x) \, dx=\ln \left (3\,{\mathrm {e}}^{2\,x}-2\,\sqrt {2}+8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+3\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-\ln \left (3\,{\mathrm {e}}^{2\,x}-2\,\sqrt {2}-8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}+8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+3\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-\ln \left (3\,{\mathrm {e}}^{2\,x}+2\,\sqrt {2}-8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+3\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+\ln \left (3\,{\mathrm {e}}^{2\,x}+2\,\sqrt {2}+8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}+8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+3\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}} \] Input:
int(sinh(x)/cosh(4*x),x)
Output:
log(3*exp(2*x) - 2*2^(1/2) + 8*exp(x)*(1/32 - 2^(1/2)/64)^(1/2) - 2*2^(1/2 )*exp(2*x) - 8*2^(1/2)*exp(x)*(1/32 - 2^(1/2)/64)^(1/2) + 3)*(1/32 - 2^(1/ 2)/64)^(1/2) - log(3*exp(2*x) - 2*2^(1/2) - 8*exp(x)*(1/32 - 2^(1/2)/64)^( 1/2) - 2*2^(1/2)*exp(2*x) + 8*2^(1/2)*exp(x)*(1/32 - 2^(1/2)/64)^(1/2) + 3 )*(1/32 - 2^(1/2)/64)^(1/2) - log(3*exp(2*x) + 2*2^(1/2) - 8*exp(x)*(2^(1/ 2)/64 + 1/32)^(1/2) + 2*2^(1/2)*exp(2*x) - 8*2^(1/2)*exp(x)*(2^(1/2)/64 + 1/32)^(1/2) + 3)*(2^(1/2)/64 + 1/32)^(1/2) + log(3*exp(2*x) + 2*2^(1/2) + 8*exp(x)*(2^(1/2)/64 + 1/32)^(1/2) + 2*2^(1/2)*exp(2*x) + 8*2^(1/2)*exp(x) *(2^(1/2)/64 + 1/32)^(1/2) + 3)*(2^(1/2)/64 + 1/32)^(1/2)
Time = 0.24 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.11 \[ \int \text {sech}(4 x) \sinh (x) \, dx=-\frac {\sqrt {-\sqrt {2}+2}\, \sqrt {2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )}{8}+\frac {\sqrt {-\sqrt {2}+2}\, \sqrt {2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )}{8}-\frac {\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )}{8}+\frac {\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )}{8}+\frac {\sqrt {\sqrt {2}+2}\, \sqrt {2}\, \mathrm {log}\left (-e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )}{8}-\frac {\sqrt {\sqrt {2}+2}\, \sqrt {2}\, \mathrm {log}\left (e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )}{8}-\frac {\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )}{8}+\frac {\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )}{8} \] Input:
int(sech(4*x)*sinh(x),x)
Output:
( - sqrt( - sqrt(2) + 2)*sqrt(2)*log( - e**x*sqrt( - sqrt(2) + 2) + e**(2* x) + 1) + sqrt( - sqrt(2) + 2)*sqrt(2)*log(e**x*sqrt( - sqrt(2) + 2) + e** (2*x) + 1) - sqrt( - sqrt(2) + 2)*log( - e**x*sqrt( - sqrt(2) + 2) + e**(2 *x) + 1) + sqrt( - sqrt(2) + 2)*log(e**x*sqrt( - sqrt(2) + 2) + e**(2*x) + 1) + sqrt(sqrt(2) + 2)*sqrt(2)*log( - e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) - sqrt(sqrt(2) + 2)*sqrt(2)*log(e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) - sqrt(sqrt(2) + 2)*log( - e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) + sqrt( sqrt(2) + 2)*log(e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1))/8