Integrand size = 7, antiderivative size = 87 \[ \int \cosh (x) \tanh (6 x) \, dx=-\frac {\text {arctanh}\left (\sqrt {2} \cosh (x)\right )}{3 \sqrt {2}}-\frac {\text {arctanh}\left (2 \sqrt {2-\sqrt {3}} \cosh (x)\right )}{6 \sqrt {2-\sqrt {3}}}-\frac {\text {arctanh}\left (2 \sqrt {2+\sqrt {3}} \cosh (x)\right )}{6 \sqrt {2+\sqrt {3}}}+\cosh (x) \] Output:
-1/6*arctanh(2^(1/2)*cosh(x))*2^(1/2)-1/6*arctanh(2*(1/2*6^(1/2)-1/2*2^(1/ 2))*cosh(x))/(1/2*6^(1/2)-1/2*2^(1/2))-1/6*arctanh(2*(1/2*6^(1/2)+1/2*2^(1 /2))*cosh(x))/(1/2*6^(1/2)+1/2*2^(1/2))+cosh(x)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.23 \[ \int \cosh (x) \tanh (6 x) \, dx=\frac {1}{24} \left (-4 \left (\sqrt {2} \text {arctanh}\left (\sqrt {2}-i \tanh \left (\frac {x}{2}\right )\right )+\sqrt {2} \text {arctanh}\left (\sqrt {2}+i \tanh \left (\frac {x}{2}\right )\right )-6 \cosh (x)\right )+\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 x-4 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )-x \text {$\#$1}^2-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2+x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^4+2 x \text {$\#$1}^6+4 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right ) \] Input:
Integrate[Cosh[x]*Tanh[6*x],x]
Output:
(-4*(Sqrt[2]*ArcTanh[Sqrt[2] - I*Tanh[x/2]] + Sqrt[2]*ArcTanh[Sqrt[2] + I* Tanh[x/2]] - 6*Cosh[x]) + RootSum[1 - #1^4 + #1^8 & , (-2*x - 4*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 + 2*x*#1^6 + 4*Log[-Co sh[x/2] - Sinh[x/2] + Cosh[x/2]*#1 - Sinh[x/2]*#1]*#1^6)/(-#1^3 + 2*#1^7) & ])/24
Time = 0.39 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 26, 4879, 27, 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.
| \(\displaystyle \int \cosh (x) \tanh (6 x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \tan (6 i x)}{\sec (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\tan (6 i x)}{\sec (i x)}dx\) |
| \(\Big \downarrow \) 4879 |
| \(\displaystyle \int -\frac {2 \cosh ^2(x) \left (16 \cosh ^4(x)-16 \cosh ^2(x)+3\right )}{-32 \cosh ^6(x)+48 \cosh ^4(x)-18 \cosh ^2(x)+1}d\cosh (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 \int \frac {\cosh ^2(x) \left (16 \cosh ^4(x)-16 \cosh ^2(x)+3\right )}{-32 \cosh ^6(x)+48 \cosh ^4(x)-18 \cosh ^2(x)+1}d\cosh (x)\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle -2 \int \left (\frac {1-8 \cosh ^2(x)}{3 \left (16 \cosh ^4(x)-16 \cosh ^2(x)+1\right )}-\frac {1}{6 \left (2 \cosh ^2(x)-1\right )}-\frac {1}{2}\right )d\cosh (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\frac {\text {arctanh}\left (\sqrt {2} \cosh (x)\right )}{6 \sqrt {2}}+\frac {1}{12} \sqrt {2-\sqrt {3}} \text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{12} \sqrt {2+\sqrt {3}} \text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {3}}}\right )-\frac {\cosh (x)}{2}\right )\) |
Input:
Int[Cosh[x]*Tanh[6*x],x]
Output:
-2*(ArcTanh[Sqrt[2]*Cosh[x]]/(6*Sqrt[2]) + (Sqrt[2 - Sqrt[3]]*ArcTanh[(2*C osh[x])/Sqrt[2 - Sqrt[3]]])/12 + (Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Cosh[x])/Sq rt[2 + Sqrt[3]]])/12 - Cosh[x]/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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_, 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 = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-\sqrt {2}\, {\mathrm e}^{x}+1\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\sqrt {2}\, {\mathrm e}^{x}+1\right )}{12}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (20736 \textit {\_Z}^{4}-576 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-12 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+1\right )\right )\) | \(79\) |
Input:
int(cosh(x)*tanh(6*x),x,method=_RETURNVERBOSE)
Output:
1/2*exp(x)+1/2*exp(-x)+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)+sum(_R*ln(-12*_R*exp(x)+exp(2*x)+1),_R= RootOf(20736*_Z^4-576*_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (63) = 126\).
Time = 0.10 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.97 \[ \int \cosh (x) \tanh (6 x) \, dx=-\frac {\sqrt {\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt {\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - \sqrt {\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt {\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) + \sqrt {-\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt {-\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - \sqrt {-\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt {-\sqrt {3} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - 6 \, \cosh \left (x\right )^{2} - {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \log \left (\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 2 \, \sqrt {2} \cosh \left (x\right ) + 2}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2}}\right ) - 12 \, \cosh \left (x\right ) \sinh \left (x\right ) - 6 \, \sinh \left (x\right )^{2} - 6}{12 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \] Input:
integrate(cosh(x)*tanh(6*x),x, algorithm="fricas")
Output:
-1/12*(sqrt(sqrt(3) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sin h(x) + sinh(x)^2 + sqrt(sqrt(3) + 2)*(cosh(x) + sinh(x)) + 1) - sqrt(sqrt( 3) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(sqrt(3) + 2)*(cosh(x) + sinh(x)) + 1) + sqrt(-sqrt(3) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + sqrt(-sqrt(3) + 2)*(cosh(x) + sinh(x)) + 1) - sqrt(-sqrt(3) + 2)*(cosh(x) + sinh(x))*log (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(-sqrt(3) + 2)*(cosh(x) + sinh(x)) + 1) - 6*cosh(x)^2 - (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*log((co sh(x)^2 + sinh(x)^2 - 2*sqrt(2)*cosh(x) + 2)/(cosh(x)^2 + sinh(x)^2)) - 12 *cosh(x)*sinh(x) - 6*sinh(x)^2 - 6)/(cosh(x) + sinh(x))
\[ \int \cosh (x) \tanh (6 x) \, dx=\int \cosh {\left (x \right )} \tanh {\left (6 x \right )}\, dx \] Input:
integrate(cosh(x)*tanh(6*x),x)
Output:
Integral(cosh(x)*tanh(6*x), x)
\[ \int \cosh (x) \tanh (6 x) \, dx=\int { \cosh \left (x\right ) \tanh \left (6 \, x\right ) \,d x } \] Input:
integrate(cosh(x)*tanh(6*x),x, algorithm="maxima")
Output:
1/2*(e^(2*x) + 1)*e^(-x) - 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) + 1/2*integrate(2/3*(2*e^(7*x) + e^(5*x) - e^(3*x) - 2*e^x)/(e^(8*x) - e^(4*x) + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (63) = 126\).
Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.80 \[ \int \cosh (x) \tanh (6 x) \, dx=-\frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, \sqrt {6} + \frac {1}{2} \, \sqrt {2} + e^{\left (-x\right )} + e^{x}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, \sqrt {6} - \frac {1}{2} \, \sqrt {2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, \sqrt {6} + \frac {1}{2} \, \sqrt {2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, \sqrt {6} - \frac {1}{2} \, \sqrt {2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - e^{x}}{\sqrt {2} + e^{\left (-x\right )} + e^{x}}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \] Input:
integrate(cosh(x)*tanh(6*x),x, algorithm="giac")
Output:
-1/24*(sqrt(6) + sqrt(2))*log(1/2*sqrt(6) + 1/2*sqrt(2) + e^(-x) + e^x) - 1/24*(sqrt(6) - sqrt(2))*log(1/2*sqrt(6) - 1/2*sqrt(2) + e^(-x) + e^x) + 1 /24*(sqrt(6) - sqrt(2))*log(-1/2*sqrt(6) + 1/2*sqrt(2) + e^(-x) + e^x) + 1 /24*(sqrt(6) + sqrt(2))*log(-1/2*sqrt(6) - 1/2*sqrt(2) + e^(-x) + e^x) + 1 /12*sqrt(2)*log(-(sqrt(2) - e^(-x) - e^x)/(sqrt(2) + e^(-x) + e^x)) + 1/2* e^(-x) + 1/2*e^x
Time = 0.99 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.95 \[ \int \cosh (x) \tanh (6 x) \, dx=\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}-\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}+\sqrt {2}\,{\mathrm {e}}^x+1\right )}{12}+\frac {\sqrt {2}\,\ln \left ({\mathrm {e}}^{2\,x}-\sqrt {2}\,{\mathrm {e}}^x+1\right )}{12}+\ln \left ({\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+1\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-\ln \left ({\mathrm {e}}^{2\,x}+12\,{\mathrm {e}}^x\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+1\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}+\ln \left ({\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+1\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}-\ln \left ({\mathrm {e}}^{2\,x}+12\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}+1\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}} \] Input:
int(tanh(6*x)*cosh(x),x)
Output:
exp(-x)/2 + exp(x)/2 - (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(exp(2*x) - 12*exp(x)* (1/72 - 3^(1/2)/144)^(1/2) + 1)*(1/72 - 3^(1/2)/144)^(1/2) - log(exp(2*x) + 12*exp(x)*(1/72 - 3^(1/2)/144)^(1/2) + 1)*(1/72 - 3^(1/2)/144)^(1/2) + l og(exp(2*x) - 12*exp(x)*(3^(1/2)/144 + 1/72)^(1/2) + 1)*(3^(1/2)/144 + 1/7 2)^(1/2) - log(exp(2*x) + 12*exp(x)*(3^(1/2)/144 + 1/72)^(1/2) + 1)*(3^(1/ 2)/144 + 1/72)^(1/2)
Time = 0.25 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.71 \[ \int \cosh (x) \tanh (6 x) \, dx=\frac {2 e^{x} \sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {3}+2}+e^{2 x}+1\right )-2 e^{x} \sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {3}+2}+e^{2 x}+1\right )+12 e^{2 x}+e^{x} \sqrt {6}\, \mathrm {log}\left (e^{2 x}-\frac {e^{x} \sqrt {6}}{2}-\frac {e^{x} \sqrt {2}}{2}+1\right )-e^{x} \sqrt {6}\, \mathrm {log}\left (e^{2 x}+\frac {e^{x} \sqrt {6}}{2}+\frac {e^{x} \sqrt {2}}{2}+1\right )+2 e^{x} \sqrt {2}\, \mathrm {log}\left (e^{2 x}-e^{x} \sqrt {2}+1\right )-2 e^{x} \sqrt {2}\, \mathrm {log}\left (e^{2 x}+e^{x} \sqrt {2}+1\right )+e^{x} \sqrt {2}\, \mathrm {log}\left (e^{2 x}-\frac {e^{x} \sqrt {6}}{2}-\frac {e^{x} \sqrt {2}}{2}+1\right )-e^{x} \sqrt {2}\, \mathrm {log}\left (e^{2 x}+\frac {e^{x} \sqrt {6}}{2}+\frac {e^{x} \sqrt {2}}{2}+1\right )+12}{24 e^{x}} \] Input:
int(cosh(x)*tanh(6*x),x)
Output:
(2*e**x*sqrt( - sqrt(3) + 2)*log( - e**x*sqrt( - sqrt(3) + 2) + e**(2*x) + 1) - 2*e**x*sqrt( - sqrt(3) + 2)*log(e**x*sqrt( - sqrt(3) + 2) + e**(2*x) + 1) + 12*e**(2*x) + e**x*sqrt(6)*log((2*e**(2*x) - e**x*sqrt(6) - e**x*s qrt(2) + 2)/2) - e**x*sqrt(6)*log((2*e**(2*x) + e**x*sqrt(6) + e**x*sqrt(2 ) + 2)/2) + 2*e**x*sqrt(2)*log(e**(2*x) - e**x*sqrt(2) + 1) - 2*e**x*sqrt( 2)*log(e**(2*x) + e**x*sqrt(2) + 1) + e**x*sqrt(2)*log((2*e**(2*x) - e**x* sqrt(6) - e**x*sqrt(2) + 2)/2) - e**x*sqrt(2)*log((2*e**(2*x) + e**x*sqrt( 6) + e**x*sqrt(2) + 2)/2) + 12)/(24*e**x)