Integrand size = 7, antiderivative size = 85 \[ \int \text {sech}(6 x) \sinh (x) \, dx=\frac {\text {arctanh}\left (\sqrt {2} \cosh (x)\right )}{3 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}-\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \] Output:
1/6*arctanh(2^(1/2)*cosh(x))*2^(1/2)-1/6*arctanh(2*cosh(x)/(1/2*6^(1/2)-1/ 2*2^(1/2)))/(1/2*6^(1/2)-1/2*2^(1/2))-1/6*arctanh(2*cosh(x)/(1/2*6^(1/2)+1 /2*2^(1/2)))/(1/2*6^(1/2)+1/2*2^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.16 \[ \int \text {sech}(6 x) \sinh (x) \, dx=\frac {1}{24} \left (4 \sqrt {2} \left (\text {arctanh}\left (\sqrt {2}-i \tanh \left (\frac {x}{2}\right )\right )+\text {arctanh}\left (\sqrt {2}+i \tanh \left (\frac {x}{2}\right )\right )\right )+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-x-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )+x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2-x \text {$\#$1}^4-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^4+x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right ) \] Input:
Integrate[Sech[6*x]*Sinh[x],x]
Output:
(4*Sqrt[2]*(ArcTanh[Sqrt[2] - I*Tanh[x/2]] + ArcTanh[Sqrt[2] + I*Tanh[x/2] ]) + RootSum[1 - #1^4 + #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] + Cos h[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(-#1^3 + 2*#1^7) & ])/24
Time = 0.32 (sec) , antiderivative size = 85, 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, 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.
| \(\displaystyle \int \sinh (x) \text {sech}(6 x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i x)}{\cos (6 i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\sin (i x)}{\cos (6 i x)}dx\) |
| \(\Big \downarrow \) 4857 |
| \(\displaystyle \int \frac {1}{32 \cosh ^6(x)-48 \cosh ^4(x)+18 \cosh ^2(x)-1}d\cosh (x)\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {4 \left (2 \cosh ^2(x)-1\right )}{3 \left (16 \cosh ^4(x)-16 \cosh ^2(x)+1\right )}-\frac {1}{3 \left (2 \cosh ^2(x)-1\right )}\right )d\cosh (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}\left (\sqrt {2} \cosh (x)\right )}{3 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}-\frac {\text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}}\) |
Input:
Int[Sech[6*x]*Sinh[x],x]
Output:
ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2]) - ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[3 ]]]/(6*Sqrt[2 - Sqrt[3]]) - ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt [2 + Sqrt[3]])
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[(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_)*(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 = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (331776 \textit {\_Z}^{4}-2304 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (13824 \textit {\_R}^{3}-96 \textit {\_R} \right ) {\mathrm e}^{x}+1\right )\right )+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\sqrt {2}\, {\mathrm e}^{x}+1\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-\sqrt {2}\, {\mathrm e}^{x}+1\right )}{12}\) | \(78\) |
Input:
int(sech(6*x)*sinh(x),x,method=_RETURNVERBOSE)
Output:
2*sum(_R*ln(exp(2*x)+(13824*_R^3-96*_R)*exp(x)+1),_R=RootOf(331776*_Z^4-23 04*_Z^2+1))+1/12*2^(1/2)*ln(exp(2*x)+2^(1/2)*exp(x)+1)-1/12*2^(1/2)*ln(exp (2*x)-2^(1/2)*exp(x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (67) = 134\).
Time = 0.10 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.94 \[ \int \text {sech}(6 x) \sinh (x) \, dx=\frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + {\left ({\left (\sqrt {3} - 2\right )} \cosh \left (x\right ) + {\left (\sqrt {3} - 2\right )} \sinh \left (x\right )\right )} \sqrt {\sqrt {3} + 2} + 1\right ) - \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - {\left ({\left (\sqrt {3} - 2\right )} \cosh \left (x\right ) + {\left (\sqrt {3} - 2\right )} \sinh \left (x\right )\right )} \sqrt {\sqrt {3} + 2} + 1\right ) - \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + {\left ({\left (\sqrt {3} + 2\right )} \cosh \left (x\right ) + {\left (\sqrt {3} + 2\right )} \sinh \left (x\right )\right )} \sqrt {-\sqrt {3} + 2} + 1\right ) + \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - {\left ({\left (\sqrt {3} + 2\right )} \cosh \left (x\right ) + {\left (\sqrt {3} + 2\right )} \sinh \left (x\right )\right )} \sqrt {-\sqrt {3} + 2} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\right ) \] Input:
integrate(sech(6*x)*sinh(x),x, algorithm="fricas")
Output:
1/12*sqrt(sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + ((s qrt(3) - 2)*cosh(x) + (sqrt(3) - 2)*sinh(x))*sqrt(sqrt(3) + 2) + 1) - 1/12 *sqrt(sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqrt( 3) - 2)*cosh(x) + (sqrt(3) - 2)*sinh(x))*sqrt(sqrt(3) + 2) + 1) - 1/12*sqr t(-sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + ((sqrt(3) + 2)*cosh(x) + (sqrt(3) + 2)*sinh(x))*sqrt(-sqrt(3) + 2) + 1) + 1/12*sqrt( -sqrt(3) + 2)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - ((sqrt(3) + 2)*cosh(x) + (sqrt(3) + 2)*sinh(x))*sqrt(-sqrt(3) + 2) + 1) + 1/12*sqrt(2) *log((cosh(x)^2 + sinh(x)^2 + 2*sqrt(2)*cosh(x) + 2)/(cosh(x)^2 + sinh(x)^ 2))
\[ \int \text {sech}(6 x) \sinh (x) \, dx=\int \sinh {\left (x \right )} \operatorname {sech}{\left (6 x \right )}\, dx \] Input:
integrate(sech(6*x)*sinh(x),x)
Output:
Integral(sinh(x)*sech(6*x), x)
\[ \int \text {sech}(6 x) \sinh (x) \, dx=\int { \operatorname {sech}\left (6 \, x\right ) \sinh \left (x\right ) \,d x } \] Input:
integrate(sech(6*x)*sinh(x),x, algorithm="maxima")
Output:
1/12*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) - 1/12*sqrt(2)*log(-sqrt(2)*e^ x + e^(2*x) + 1) + integrate(1/3*(e^(7*x) - e^(5*x) + e^(3*x) - e^x)/(e^(8 *x) - e^(4*x) + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (67) = 134\).
Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.81 \[ \int \text {sech}(6 x) \sinh (x) \, dx=-\frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{12} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \] Input:
integrate(sech(6*x)*sinh(x),x, algorithm="giac")
Output:
-1/24*(sqrt(6) - sqrt(2))*log(1/2*(sqrt(6) + sqrt(2))*e^x + e^(2*x) + 1) + 1/24*(sqrt(6) - sqrt(2))*log(-1/2*(sqrt(6) + sqrt(2))*e^x + e^(2*x) + 1) - 1/24*(sqrt(6) + sqrt(2))*log(1/2*(sqrt(6) - sqrt(2))*e^x + e^(2*x) + 1) + 1/24*(sqrt(6) + sqrt(2))*log(-1/2*(sqrt(6) - sqrt(2))*e^x + e^(2*x) + 1) + 1/12*sqrt(2)*log(sqrt(2)*e^x + e^(2*x) + 1) - 1/12*sqrt(2)*log(-sqrt(2) *e^x + e^(2*x) + 1)
Time = 1.44 (sec) , antiderivative size = 288, normalized size of antiderivative = 3.39 \[ \int \text {sech}(6 x) \sinh (x) \, dx=\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}+\sqrt {2}\,{\mathrm {e}}^x+1\right )}{12}-\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}-\sqrt {2}\,{\mathrm {e}}^x+1\right )}{12}+\ln \left (7\,{\mathrm {e}}^{2\,x}-4\,\sqrt {3}-24\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+12\,\sqrt {3}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+7\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-\ln \left (7\,{\mathrm {e}}^{2\,x}-4\,\sqrt {3}+24\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-12\,\sqrt {3}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+7\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+\ln \left (7\,{\mathrm {e}}^{2\,x}+4\,\sqrt {3}-24\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}-12\,\sqrt {3}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+7\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}-\ln \left (7\,{\mathrm {e}}^{2\,x}+4\,\sqrt {3}+24\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+4\,\sqrt {3}\,{\mathrm {e}}^{2\,x}+12\,\sqrt {3}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+7\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}} \] Input:
int(sinh(x)/cosh(6*x),x)
Output:
(2^(1/2)*log(exp(2*x) + 2^(1/2)*exp(x) + 1))/12 - (2^(1/2)*log(exp(2*x) - 2^(1/2)*exp(x) + 1))/12 + log(7*exp(2*x) - 4*3^(1/2) - 24*exp(x)*(1/72 - 3 ^(1/2)/144)^(1/2) - 4*3^(1/2)*exp(2*x) + 12*3^(1/2)*exp(x)*(1/72 - 3^(1/2) /144)^(1/2) + 7)*(1/72 - 3^(1/2)/144)^(1/2) - log(7*exp(2*x) - 4*3^(1/2) + 24*exp(x)*(1/72 - 3^(1/2)/144)^(1/2) - 4*3^(1/2)*exp(2*x) - 12*3^(1/2)*ex p(x)*(1/72 - 3^(1/2)/144)^(1/2) + 7)*(1/72 - 3^(1/2)/144)^(1/2) + log(7*ex p(2*x) + 4*3^(1/2) - 24*exp(x)*(3^(1/2)/144 + 1/72)^(1/2) + 4*3^(1/2)*exp( 2*x) - 12*3^(1/2)*exp(x)*(3^(1/2)/144 + 1/72)^(1/2) + 7)*(3^(1/2)/144 + 1/ 72)^(1/2) - log(7*exp(2*x) + 4*3^(1/2) + 24*exp(x)*(3^(1/2)/144 + 1/72)^(1 /2) + 4*3^(1/2)*exp(2*x) + 12*3^(1/2)*exp(x)*(3^(1/2)/144 + 1/72)^(1/2) + 7)*(3^(1/2)/144 + 1/72)^(1/2)
Time = 0.26 (sec) , antiderivative size = 260, normalized size of antiderivative = 3.06 \[ \int \text {sech}(6 x) \sinh (x) \, dx=\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {3}+2}+e^{2 x}+1\right )}{12}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {3}+2}+e^{2 x}+1\right )}{12}+\frac {\sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {3}+2}+e^{2 x}+1\right )}{6}-\frac {\sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {3}+2}+e^{2 x}+1\right )}{6}+\frac {\sqrt {6}\, \mathrm {log}\left (e^{2 x}-\frac {e^{x} \sqrt {6}}{2}-\frac {e^{x} \sqrt {2}}{2}+1\right )}{24}-\frac {\sqrt {6}\, \mathrm {log}\left (e^{2 x}+\frac {e^{x} \sqrt {6}}{2}+\frac {e^{x} \sqrt {2}}{2}+1\right )}{24}-\frac {\sqrt {2}\, \mathrm {log}\left (e^{2 x}-e^{x} \sqrt {2}+1\right )}{12}+\frac {\sqrt {2}\, \mathrm {log}\left (e^{2 x}+e^{x} \sqrt {2}+1\right )}{12}-\frac {\sqrt {2}\, \mathrm {log}\left (e^{2 x}-\frac {e^{x} \sqrt {6}}{2}-\frac {e^{x} \sqrt {2}}{2}+1\right )}{24}+\frac {\sqrt {2}\, \mathrm {log}\left (e^{2 x}+\frac {e^{x} \sqrt {6}}{2}+\frac {e^{x} \sqrt {2}}{2}+1\right )}{24} \] Input:
int(sech(6*x)*sinh(x),x)
Output:
(2*sqrt( - sqrt(3) + 2)*sqrt(3)*log( - e**x*sqrt( - sqrt(3) + 2) + e**(2*x ) + 1) - 2*sqrt( - sqrt(3) + 2)*sqrt(3)*log(e**x*sqrt( - sqrt(3) + 2) + e* *(2*x) + 1) + 4*sqrt( - sqrt(3) + 2)*log( - e**x*sqrt( - sqrt(3) + 2) + e* *(2*x) + 1) - 4*sqrt( - sqrt(3) + 2)*log(e**x*sqrt( - sqrt(3) + 2) + e**(2 *x) + 1) + sqrt(6)*log((2*e**(2*x) - e**x*sqrt(6) - e**x*sqrt(2) + 2)/2) - sqrt(6)*log((2*e**(2*x) + e**x*sqrt(6) + e**x*sqrt(2) + 2)/2) - 2*sqrt(2) *log(e**(2*x) - e**x*sqrt(2) + 1) + 2*sqrt(2)*log(e**(2*x) + e**x*sqrt(2) + 1) - sqrt(2)*log((2*e**(2*x) - e**x*sqrt(6) - e**x*sqrt(2) + 2)/2) + sqr t(2)*log((2*e**(2*x) + e**x*sqrt(6) + e**x*sqrt(2) + 2)/2))/24