Integrand size = 13, antiderivative size = 62 \[ \int \frac {\cosh ^2(x)}{a+b \cosh (x)} \, dx=-\frac {a x}{b^2}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b}}+\frac {\sinh (x)}{b} \]
-a*x/b^2+sinh(x)/b+2*a^2*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/b^2/ (a-b)^(1/2)/(a+b)^(1/2)
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh ^2(x)}{a+b \cosh (x)} \, dx=\frac {a \left (-x-\frac {2 a \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}\right )+b \sinh (x)}{b^2} \]
(a*(-x - (2*a*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^ 2]) + b*Sinh[x])/b^2
Time = 0.37 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3042, 3225, 25, 27, 3042, 3214, 3042, 3138, 221}
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.
\(\displaystyle \int \frac {\cosh ^2(x)}{a+b \cosh (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )^2}{a+b \sin \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 3225 |
\(\displaystyle \frac {\int -\frac {a \cosh (x)}{a+b \cosh (x)}dx}{b}+\frac {\sinh (x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {\int \frac {a \cosh (x)}{a+b \cosh (x)}dx}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {a \int \frac {\cosh (x)}{a+b \cosh (x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {a \int \frac {\sin \left (i x+\frac {\pi }{2}\right )}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {a \left (\frac {x}{b}-\frac {a \int \frac {1}{a+b \cosh (x)}dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {a \left (\frac {x}{b}-\frac {a \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {a \left (\frac {x}{b}-\frac {2 a \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{b}\right )}{b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {a \left (\frac {x}{b}-\frac {2 a \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b \sqrt {a-b} \sqrt {a+b}}\right )}{b}\) |
-((a*(x/b - (2*a*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b ]*b*Sqrt[a + b])))/b) + Sinh[x]/b
3.1.56.3.1 Defintions of rubi rules used
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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2/((c_.) + (d_.)*sin[(e_.) + (f _.)*(x_)]), x_Symbol] :> Simp[(-b^2)*(Cos[e + f*x]/(d*f)), x] + Simp[1/d Int[Simp[a^2*d - b*(b*c - 2*a*d)*Sin[e + f*x], x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.52
method | result | size |
default | \(\frac {2 a^{2} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}}\) | \(94\) |
risch | \(-\frac {a x}{b^{2}}+\frac {{\mathrm e}^{x}}{2 b}-\frac {{\mathrm e}^{-x}}{2 b}+\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, b^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, b^{2}}\) | \(144\) |
2*a^2/b^2/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2 ))-1/b/(tanh(1/2*x)-1)+a/b^2*ln(tanh(1/2*x)-1)-1/b/(tanh(1/2*x)+1)-a/b^2*l n(tanh(1/2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (52) = 104\).
Time = 0.26 (sec) , antiderivative size = 449, normalized size of antiderivative = 7.24 \[ \int \frac {\cosh ^2(x)}{a+b \cosh (x)} \, dx=\left [-\frac {a^{2} b - b^{3} + 2 \, {\left (a^{3} - a b^{2}\right )} x \cosh \left (x\right ) - {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2} - {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{2} \cosh \left (x\right ) + a^{2} \sinh \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) + 2 \, {\left ({\left (a^{3} - a b^{2}\right )} x - {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{2 \, {\left ({\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right ) + {\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )\right )}}, -\frac {a^{2} b - b^{3} + 2 \, {\left (a^{3} - a b^{2}\right )} x \cosh \left (x\right ) - {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2} - {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{2} \cosh \left (x\right ) + a^{2} \sinh \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) + 2 \, {\left ({\left (a^{3} - a b^{2}\right )} x - {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{2 \, {\left ({\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right ) + {\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )\right )}}\right ] \]
[-1/2*(a^2*b - b^3 + 2*(a^3 - a*b^2)*x*cosh(x) - (a^2*b - b^3)*cosh(x)^2 - (a^2*b - b^3)*sinh(x)^2 - 2*(a^2*cosh(x) + a^2*sinh(x))*sqrt(a^2 - b^2)*l og((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*c osh(x) + a*b)*sinh(x) - 2*sqrt(a^2 - b^2)*(b*cosh(x) + b*sinh(x) + a))/(b* cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) + b)) + 2*((a^3 - a*b^2)*x - (a^2*b - b^3)*cosh(x))*sinh(x))/((a^2*b^2 - b^4)*cosh (x) + (a^2*b^2 - b^4)*sinh(x)), -1/2*(a^2*b - b^3 + 2*(a^3 - a*b^2)*x*cosh (x) - (a^2*b - b^3)*cosh(x)^2 - (a^2*b - b^3)*sinh(x)^2 + 4*(a^2*cosh(x) + a^2*sinh(x))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sin h(x) + a)/(a^2 - b^2)) + 2*((a^3 - a*b^2)*x - (a^2*b - b^3)*cosh(x))*sinh( x))/((a^2*b^2 - b^4)*cosh(x) + (a^2*b^2 - b^4)*sinh(x))]
Leaf count of result is larger than twice the leaf count of optimal. 1275 vs. \(2 (53) = 106\).
Time = 51.49 (sec) , antiderivative size = 1275, normalized size of antiderivative = 20.56 \[ \int \frac {\cosh ^2(x)}{a+b \cosh (x)} \, dx=\text {Too large to display} \]
Piecewise((zoo*sinh(x), Eq(a, 0) & Eq(b, 0)), (-x*tanh(x/2)**2/(b*tanh(x/2 )**2 - b) + x/(b*tanh(x/2)**2 - b) + tanh(x/2)**3/(b*tanh(x/2)**2 - b) - 3 *tanh(x/2)/(b*tanh(x/2)**2 - b), Eq(a, b)), (x*tanh(x/2)**3/(b*tanh(x/2)** 3 - b*tanh(x/2)) - x*tanh(x/2)/(b*tanh(x/2)**3 - b*tanh(x/2)) - 3*tanh(x/2 )**2/(b*tanh(x/2)**3 - b*tanh(x/2)) + 1/(b*tanh(x/2)**3 - b*tanh(x/2)), Eq (a, -b)), ((-x*sinh(x)**2/2 + x*cosh(x)**2/2 + sinh(x)*cosh(x)/2)/a, Eq(b, 0)), (-a**2*x*sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2/(a*b**2*sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2 - a*b**2*sqrt(a/(a - b) + b/(a - b)) - b**3 *sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2 + b**3*sqrt(a/(a - b) + b/(a - b ))) + a**2*x*sqrt(a/(a - b) + b/(a - b))/(a*b**2*sqrt(a/(a - b) + b/(a - b ))*tanh(x/2)**2 - a*b**2*sqrt(a/(a - b) + b/(a - b)) - b**3*sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2 + b**3*sqrt(a/(a - b) + b/(a - b))) - a**2*log( -sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))*tanh(x/2)**2/(a*b**2*sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2 - a*b**2*sqrt(a/(a - b) + b/(a - b)) - b**3* sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2 + b**3*sqrt(a/(a - b) + b/(a - b) )) + a**2*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b**2*sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2 - a*b**2*sqrt(a/(a - b) + b/(a - b)) - b**3 *sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2 + b**3*sqrt(a/(a - b) + b/(a - b ))) + a**2*log(sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))*tanh(x/2)**2/(a*b* *2*sqrt(a/(a - b) + b/(a - b))*tanh(x/2)**2 - a*b**2*sqrt(a/(a - b) + b...
Exception generated. \[ \int \frac {\cosh ^2(x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2(x)}{a+b \cosh (x)} \, dx=\frac {2 \, a^{2} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{2}} - \frac {a x}{b^{2}} - \frac {e^{\left (-x\right )}}{2 \, b} + \frac {e^{x}}{2 \, b} \]
2*a^2*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/(sqrt(-a^2 + b^2)*b^2) - a*x/b^ 2 - 1/2*e^(-x)/b + 1/2*e^x/b
Time = 1.87 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.24 \[ \int \frac {\cosh ^2(x)}{a+b \cosh (x)} \, dx=\frac {{\mathrm {e}}^x}{2\,b}-\frac {{\mathrm {e}}^{-x}}{2\,b}-\frac {a\,x}{b^2}+\frac {a^2\,\ln \left (-\frac {2\,a^2\,{\mathrm {e}}^x}{b^3}-\frac {2\,a^2\,\left (b+a\,{\mathrm {e}}^x\right )}{b^3\,\sqrt {a+b}\,\sqrt {a-b}}\right )}{b^2\,\sqrt {a+b}\,\sqrt {a-b}}-\frac {a^2\,\ln \left (\frac {2\,a^2\,\left (b+a\,{\mathrm {e}}^x\right )}{b^3\,\sqrt {a+b}\,\sqrt {a-b}}-\frac {2\,a^2\,{\mathrm {e}}^x}{b^3}\right )}{b^2\,\sqrt {a+b}\,\sqrt {a-b}} \]
exp(x)/(2*b) - exp(-x)/(2*b) - (a*x)/b^2 + (a^2*log(- (2*a^2*exp(x))/b^3 - (2*a^2*(b + a*exp(x)))/(b^3*(a + b)^(1/2)*(a - b)^(1/2))))/(b^2*(a + b)^( 1/2)*(a - b)^(1/2)) - (a^2*log((2*a^2*(b + a*exp(x)))/(b^3*(a + b)^(1/2)*( a - b)^(1/2)) - (2*a^2*exp(x))/b^3))/(b^2*(a + b)^(1/2)*(a - b)^(1/2))