Integrand size = 31, antiderivative size = 69 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\sqrt {2} \arctan \left (\text {sech}(x) \sqrt {\cosh (x) \sinh (x)}\right )+\frac {1}{6} \arctan \left (\frac {\sinh (x)}{\sqrt {\sinh (2 x)}}\right )-\frac {1}{3} \sqrt {2} \text {arctanh}\left (\text {sech}(x) \sqrt {\cosh (x) \sinh (x)}\right )+\frac {\cosh (x)}{\sqrt {\sinh (2 x)}} \]
1/6*arctan(sinh(x)/sinh(2*x)^(1/2))+arctan(sech(x)*(cosh(x)*sinh(x))^(1/2) )*2^(1/2)-1/3*arctanh(sech(x)*(cosh(x)*sinh(x))^(1/2))*2^(1/2)+cosh(x)/sin h(2*x)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(69)=138\).
Time = 18.50 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.32 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\frac {\sqrt {\sinh (2 x)} \left (6 \sqrt {2} \arctan \left (\frac {\sqrt {\tanh \left (\frac {x}{2}\right )}}{\sqrt {\frac {\cosh (x)}{1+\cosh (x)}}}\right )+\arctan \left (\frac {\sqrt {\tanh \left (\frac {x}{2}\right )}}{\sqrt {1+\tanh ^2\left (\frac {x}{2}\right )}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\tanh \left (\frac {x}{2}\right )}}{\sqrt {\frac {\cosh (x)}{1+\cosh (x)}}}\right )+\frac {3 \sqrt {\cosh (x) \text {sech}^2\left (\frac {x}{2}\right )}}{\sqrt {\tanh \left (\frac {x}{2}\right )}}\right )}{6 (1+\cosh (x)) \sqrt {\tanh \left (\frac {x}{2}\right )} \sqrt {1+\tanh ^2\left (\frac {x}{2}\right )}} \]
(Sqrt[Sinh[2*x]]*(6*Sqrt[2]*ArcTan[Sqrt[Tanh[x/2]]/Sqrt[Cosh[x]/(1 + Cosh[ x])]] + ArcTan[Sqrt[Tanh[x/2]]/Sqrt[1 + Tanh[x/2]^2]] - 2*Sqrt[2]*ArcTanh[ Sqrt[Tanh[x/2]]/Sqrt[Cosh[x]/(1 + Cosh[x])]] + (3*Sqrt[Cosh[x]*Sech[x/2]^2 ])/Sqrt[Tanh[x/2]]))/(6*(1 + Cosh[x])*Sqrt[Tanh[x/2]]*Sqrt[1 + Tanh[x/2]^2 ])
Time = 0.81 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {3042, 4890, 26, 4889, 25, 2035, 2247, 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.
| \(\displaystyle \int \frac {\cosh (x) (\tanh (x)-\cosh (2 x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i x) (-\cos (2 i x)-i \tan (i x))}{\left (-\sin (i x)^2-i \sin (2 i x)\right ) \sqrt {-i \sin (2 i x)}}dx\) |
| \(\Big \downarrow \) 4890 |
| \(\displaystyle \frac {i \sinh (x) \int -\frac {i \cos (i x) \text {csch}(x) (\cos (2 i x)+i \tan (i x)) \sqrt {\tanh (x)}}{\sin (i x)^2+i \sin (2 i x)}dx}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\sinh (x) \int \frac {\cos (i x) \text {csch}(x) (\cos (2 i x)+i \tan (i x)) \sqrt {\tanh (x)}}{\sin (i x)^2+i \sin (2 i x)}dx}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\) |
| \(\Big \downarrow \) 4889 |
| \(\displaystyle \frac {\sinh (x) \int -\frac {\tanh ^3(x)+\tanh ^2(x)-\tanh (x)+1}{\tanh ^{\frac {3}{2}}(x) (\tanh (x)+2) \left (1-\tanh ^2(x)\right )}d\tanh (x)}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sinh (x) \int \frac {\tanh ^3(x)+\tanh ^2(x)-\tanh (x)+1}{\tanh ^{\frac {3}{2}}(x) (\tanh (x)+2) \left (1-\tanh ^2(x)\right )}d\tanh (x)}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sinh (x) \int \frac {\coth ^2(x) \left (\tanh ^3(x)+\tanh ^2(x)-\tanh (x)+1\right )}{(\tanh (x)+2) \left (1-\tanh ^2(x)\right )}d\sqrt {\tanh (x)}}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\) |
| \(\Big \downarrow \) 2247 |
| \(\displaystyle -\frac {2 \sinh (x) \int \left (\frac {\coth ^2(x)}{2}+\frac {1}{-\tanh (x)-1}-\frac {1}{3 (\tanh (x)-1)}-\frac {1}{6 (\tanh (x)+2)}\right )d\sqrt {\tanh (x)}}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sinh (x) \left (-\arctan \left (\sqrt {\tanh (x)}\right )-\frac {\arctan \left (\frac {\sqrt {\tanh (x)}}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {1}{3} \text {arctanh}\left (\sqrt {\tanh (x)}\right )-\frac {\coth (x)}{2}\right )}{\sqrt {\sinh (2 x)} \sqrt {\tanh (x)}}\) |
(-2*(-ArcTan[Sqrt[Tanh[x]]] - ArcTan[Sqrt[Tanh[x]]/Sqrt[2]]/(6*Sqrt[2]) + ArcTanh[Sqrt[Tanh[x]]]/3 - Coth[x]/2)*Sinh[x])/(Sqrt[Sinh[2*x]]*Sqrt[Tanh[ x]])
3.6.92.3.1 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[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(f*x)^m*(d + e*x^2)^q*(a + c *x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && PolyQ[Px, x] && Integ erQ[p]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
Int[(u_)*((c_.)*sin[v_])^(m_), x_Symbol] :> With[{w = FunctionOfTrig[u*(Sin [v/2]^(2*m)/(c*Tan[v/2])^m), x]}, Simp[(c*Sin[v])^m*((c*Tan[v/2])^m/Sin[v/2 ]^(2*m)) Int[u*(Sin[v/2]^(2*m)/(c*Tan[v/2])^m), x], x] /; !FalseQ[w] && FunctionOfQ[NonfreeFactors[Tan[w], x], u*(Sin[v/2]^(2*m)/(c*Tan[v/2])^m), x ]] /; FreeQ[c, x] && LinearQ[v, x] && IntegerQ[m + 1/2] && !SumQ[u] && Inv erseFunctionFreeQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs. \(2(53)=106\).
Time = 1.33 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.42
| method | result | size |
| default | \(\frac {\sqrt {\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )-1\right )^{2}}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )-1\right ) \left (2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}\, \sqrt {2}}{2 \tanh \left (\frac {x}{2}\right )}\right ) \tanh \left (\frac {x}{2}\right )+6 \sqrt {2}\, \arctan \left (\frac {\sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}\, \sqrt {2}}{2 \tanh \left (\frac {x}{2}\right )}\right ) \tanh \left (\frac {x}{2}\right )+\arctan \left (\frac {\sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}}{\tanh \left (\frac {x}{2}\right )}\right ) \tanh \left (\frac {x}{2}\right )-3 \sqrt {\tanh ^{3}\left (\frac {x}{2}\right )+\tanh \left (\frac {x}{2}\right )}\right )}{6 \sqrt {\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \tanh \left (\frac {x}{2}\right )}\, \tanh \left (\frac {x}{2}\right )}\) | \(167\) |
int(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/2),x,m ethod=_RETURNVERBOSE)
1/6*((tanh(1/2*x)^2+1)*tanh(1/2*x)/(tanh(1/2*x)^2-1)^2)^(1/2)*(tanh(1/2*x) ^2-1)*(2*2^(1/2)*arctanh(1/2/tanh(1/2*x)*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2) *2^(1/2))*tanh(1/2*x)+6*2^(1/2)*arctan(1/2/tanh(1/2*x)*(tanh(1/2*x)^3+tanh (1/2*x))^(1/2)*2^(1/2))*tanh(1/2*x)+arctan(1/tanh(1/2*x)*(tanh(1/2*x)^3+ta nh(1/2*x))^(1/2))*tanh(1/2*x)-3*(tanh(1/2*x)^3+tanh(1/2*x))^(1/2))/((tanh( 1/2*x)^2+1)*tanh(1/2*x))^(1/2)/tanh(1/2*x)
Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (53) = 106\).
Time = 0.26 (sec) , antiderivative size = 376, normalized size of antiderivative = 5.45 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx =\text {Too large to display} \]
integrate(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/ 2),x, algorithm="fricas")
-1/12*((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*arctan(1/2*(sqrt(2) *cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + 3*sqrt(2))*sq rt(cosh(x)*sinh(x)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sin h(x)^4 - 1)) + 6*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)* sinh(x)^2 - sqrt(2))*arctan(2*sqrt(cosh(x)*sinh(x)/(cosh(x)^2 - 2*cosh(x)* sinh(x) + sinh(x)^2))/(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)) - (sqrt(2)*cosh(x)^2 + 2*sqrt (2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 - sqrt(2))*log(2*cosh(x)^4 + 8*cos h(x)^3*sinh(x) + 12*cosh(x)^2*sinh(x)^2 + 8*cosh(x)*sinh(x)^3 + 2*sinh(x)^ 4 - 4*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(cosh(x)*sinh(x)/(co sh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 1) - 12*sqrt(2)*sqrt(cosh(x)*s inh(x)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x)^2 + 2*cosh(x )*sinh(x) + sinh(x)^2 - 1)
\[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=- \int \frac {\cosh {\left (x \right )} \cosh {\left (2 x \right )}}{\sinh ^{2}{\left (x \right )} \sqrt {\sinh {\left (2 x \right )}} + \sinh ^{\frac {3}{2}}{\left (2 x \right )}}\, dx - \int \left (- \frac {\cosh {\left (x \right )} \tanh {\left (x \right )}}{\sinh ^{2}{\left (x \right )} \sqrt {\sinh {\left (2 x \right )}} + \sinh ^{\frac {3}{2}}{\left (2 x \right )}}\right )\, dx \]
-Integral(cosh(x)*cosh(2*x)/(sinh(x)**2*sqrt(sinh(2*x)) + sinh(2*x)**(3/2) ), x) - Integral(-cosh(x)*tanh(x)/(sinh(x)**2*sqrt(sinh(2*x)) + sinh(2*x)* *(3/2)), x)
\[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\int { -\frac {{\left (\cosh \left (2 \, x\right ) - \tanh \left (x\right )\right )} \cosh \left (x\right )}{{\left (\sinh \left (x\right )^{2} + \sinh \left (2 \, x\right )\right )} \sqrt {\sinh \left (2 \, x\right )}} \,d x } \]
integrate(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/ 2),x, algorithm="maxima")
Time = 0.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.30 \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=\sqrt {2} \arctan \left (\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )}\right ) + \frac {1}{6} \, \sqrt {2} \log \left (-\sqrt {e^{\left (4 \, x\right )} - 1} + e^{\left (2 \, x\right )}\right ) + \frac {\sqrt {2}}{\sqrt {e^{\left (4 \, x\right )} - 1} - e^{\left (2 \, x\right )} + 1} + \frac {1}{6} \, \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, \sqrt {e^{\left (4 \, x\right )} - 1} - 3 \, e^{\left (2 \, x\right )} - 1\right )}\right ) \]
integrate(cosh(x)*(-cosh(2*x)+tanh(x))/(sinh(x)^2+sinh(2*x))/sinh(2*x)^(1/ 2),x, algorithm="giac")
sqrt(2)*arctan(sqrt(e^(4*x) - 1) - e^(2*x)) + 1/6*sqrt(2)*log(-sqrt(e^(4*x ) - 1) + e^(2*x)) + sqrt(2)/(sqrt(e^(4*x) - 1) - e^(2*x) + 1) + 1/6*arctan (1/4*sqrt(2)*(3*sqrt(e^(4*x) - 1) - 3*e^(2*x) - 1))
Timed out. \[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=-\int \frac {\mathrm {cosh}\left (x\right )\,\left (\mathrm {cosh}\left (2\,x\right )-\mathrm {tanh}\left (x\right )\right )}{\sqrt {\mathrm {sinh}\left (2\,x\right )}\,\left ({\mathrm {sinh}\left (x\right )}^2+\mathrm {sinh}\left (2\,x\right )\right )} \,d x \]
\[ \int \frac {\cosh (x) (-\cosh (2 x)+\tanh (x))}{\sqrt {\sinh (2 x)} \left (\sinh ^2(x)+\sinh (2 x)\right )} \, dx=-\left (\int \frac {\sqrt {\sinh \left (2 x \right )}\, \cosh \left (2 x \right ) \cosh \left (x \right )}{\sinh \left (2 x \right )^{2}+\sinh \left (2 x \right ) \sinh \left (x \right )^{2}}d x \right )+\int \frac {\sqrt {\sinh \left (2 x \right )}\, \cosh \left (x \right ) \tanh \left (x \right )}{\sinh \left (2 x \right )^{2}+\sinh \left (2 x \right ) \sinh \left (x \right )^{2}}d x \]