Integrand size = 7, antiderivative size = 69 \[ \int \cosh (x) \tanh (4 x) \, dx=-\frac {1}{4} \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right )+\cosh (x) \]
cosh(x)-1/4*arctanh(2*cosh(x)/(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)-1/4*arc tanh(2*cosh(x)/(2+2^(1/2))^(1/2))*(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 = 113, normalized size of antiderivative = 1.64 \[ \int \cosh (x) \tanh (4 x) \, dx=\cosh (x)+\frac {1}{16} \text {RootSum}\left [1+\text {$\#$1}^8\&,\frac {-x-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )+x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{\text {$\#$1}^7}\&\right ] \]
Cosh[x] + RootSum[1 + #1^8 & , (-x - 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x /2]*#1 - Sinh[x/2]*#1] + x*#1^6 + 2*Log[-Cosh[x/2] - Sinh[x/2] + Cosh[x/2] *#1 - Sinh[x/2]*#1]*#1^6)/#1^7 & ]/16
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3042, 26, 4879, 27, 1602, 27, 1480, 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.
\(\displaystyle \int \cosh (x) \tanh (4 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (4 i x)}{\sec (i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (4 i x)}{\sec (i x)}dx\) |
\(\Big \downarrow \) 4879 |
\(\displaystyle \int -\frac {4 \cosh ^2(x) \left (1-2 \cosh ^2(x)\right )}{8 \cosh ^4(x)-8 \cosh ^2(x)+1}d\cosh (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -4 \int \frac {\cosh ^2(x) \left (1-2 \cosh ^2(x)\right )}{8 \cosh ^4(x)-8 \cosh ^2(x)+1}d\cosh (x)\) |
\(\Big \downarrow \) 1602 |
\(\displaystyle -4 \left (-\frac {1}{8} \int -\frac {2 \left (1-4 \cosh ^2(x)\right )}{8 \cosh ^4(x)-8 \cosh ^2(x)+1}d\cosh (x)-\frac {\cosh (x)}{4}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -4 \left (\frac {1}{4} \int \frac {1-4 \cosh ^2(x)}{8 \cosh ^4(x)-8 \cosh ^2(x)+1}d\cosh (x)-\frac {\cosh (x)}{4}\right )\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle -4 \left (\frac {1}{4} \left (-\left (\left (2-\sqrt {2}\right ) \int \frac {1}{8 \cosh ^2(x)-2 \left (2-\sqrt {2}\right )}d\cosh (x)\right )-\left (2+\sqrt {2}\right ) \int \frac {1}{8 \cosh ^2(x)-2 \left (2+\sqrt {2}\right )}d\cosh (x)\right )-\frac {\cosh (x)}{4}\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -4 \left (\frac {1}{4} \left (\frac {1}{4} \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right )\right )-\frac {\cosh (x)}{4}\right )\) |
-4*(((Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[2]]])/4 + (Sqrt[ 2 + Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[2]]])/4)/4 - Cosh[x]/4)
3.3.31.3.1 Defintions of rubi rules used
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[((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[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)* (a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c , 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | | IntegerQ[m])
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 = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.61
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2048 \textit {\_Z}^{4}-128 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-8 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+1\right )\right )\) | \(42\) |
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (49) = 98\).
Time = 0.26 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.09 \[ \int \cosh (x) \tanh (4 x) \, dx=-\frac {\sqrt {\sqrt {2} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt {\sqrt {2} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - \sqrt {\sqrt {2} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt {\sqrt {2} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) + \sqrt {-\sqrt {2} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \sqrt {-\sqrt {2} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - \sqrt {-\sqrt {2} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \sqrt {-\sqrt {2} + 2} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 1\right ) - 4 \, \cosh \left (x\right )^{2} - 8 \, \cosh \left (x\right ) \sinh \left (x\right ) - 4 \, \sinh \left (x\right )^{2} - 4}{8 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]
-1/8*(sqrt(sqrt(2) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh (x) + sinh(x)^2 + sqrt(sqrt(2) + 2)*(cosh(x) + sinh(x)) + 1) - sqrt(sqrt(2 ) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(sqrt(2) + 2)*(cosh(x) + sinh(x)) + 1) + sqrt(-sqrt(2) + 2)*(cosh(x) + sinh(x))*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + sqrt(-sqrt(2) + 2)*(cosh(x) + sinh(x)) + 1) - sqrt(-sqrt(2) + 2)*(cosh(x) + sinh(x))*log( cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(-sqrt(2) + 2)*(cosh(x) + sinh(x)) + 1) - 4*cosh(x)^2 - 8*cosh(x)*sinh(x) - 4*sinh(x)^2 - 4)/(cosh(x ) + sinh(x))
\[ \int \cosh (x) \tanh (4 x) \, dx=\int \cosh {\left (x \right )} \tanh {\left (4 x \right )}\, dx \]
\[ \int \cosh (x) \tanh (4 x) \, dx=\int { \cosh \left (x\right ) \tanh \left (4 \, x\right ) \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (49) = 98\).
Time = 0.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.72 \[ \int \cosh (x) \tanh (4 x) \, dx=-\frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} + e^{\left (-x\right )} + e^{x}\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
-1/8*sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2) + e^(-x) + e^x) + 1/8*sqrt(sq rt(2) + 2)*log(-sqrt(sqrt(2) + 2) + e^(-x) + e^x) - 1/8*sqrt(-sqrt(2) + 2) *log(sqrt(-sqrt(2) + 2) + e^(-x) + e^x) + 1/8*sqrt(-sqrt(2) + 2)*log(-sqrt (-sqrt(2) + 2) + e^(-x) + e^x) + 1/2*e^(-x) + 1/2*e^x
Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.93 \[ \int \cosh (x) \tanh (4 x) \, dx=\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}+\ln \left ({\mathrm {e}}^{2\,x}-8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+1\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-\ln \left ({\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+1\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+\ln \left ({\mathrm {e}}^{2\,x}-8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+1\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}-\ln \left ({\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+1\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}} \]
exp(-x)/2 + exp(x)/2 + log(exp(2*x) - 8*exp(x)*(1/32 - 2^(1/2)/64)^(1/2) + 1)*(1/32 - 2^(1/2)/64)^(1/2) - log(exp(2*x) + 8*exp(x)*(1/32 - 2^(1/2)/64 )^(1/2) + 1)*(1/32 - 2^(1/2)/64)^(1/2) + log(exp(2*x) - 8*exp(x)*(2^(1/2)/ 64 + 1/32)^(1/2) + 1)*(2^(1/2)/64 + 1/32)^(1/2) - log(exp(2*x) + 8*exp(x)* (2^(1/2)/64 + 1/32)^(1/2) + 1)*(2^(1/2)/64 + 1/32)^(1/2)