Integrand size = 11, antiderivative size = 62 \[ \int \frac {\cosh (x)}{a+b \text {sech}(x)} \, dx=-\frac {b x}{a^2}+\frac {2 b^2 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^2 \sqrt {a-b} \sqrt {a+b}}+\frac {\sinh (x)}{a} \]
-b*x/a^2+sinh(x)/a+2*b^2*arctan((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/a^2/( a-b)^(1/2)/(a+b)^(1/2)
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh (x)}{a+b \text {sech}(x)} \, dx=\frac {b \left (-x-\frac {2 b \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}\right )+a \sinh (x)}{a^2} \]
(b*(-x - (2*b*ArcTan[((-a + b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2 ]) + a*Sinh[x])/a^2
Time = 0.37 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 4340, 25, 27, 3042, 4270, 3042, 3138, 218}
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.
\(\displaystyle \int \frac {\cosh (x)}{a+b \text {sech}(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc \left (\frac {\pi }{2}+i x\right ) \left (a+b \csc \left (\frac {\pi }{2}+i x\right )\right )}dx\) |
\(\Big \downarrow \) 4340 |
\(\displaystyle \frac {\int -\frac {b}{a+b \text {sech}(x)}dx}{a}+\frac {\sinh (x)}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh (x)}{a}-\frac {\int \frac {b}{a+b \text {sech}(x)}dx}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sinh (x)}{a}-\frac {b \int \frac {1}{a+b \text {sech}(x)}dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (x)}{a}-\frac {b \int \frac {1}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx}{a}\) |
\(\Big \downarrow \) 4270 |
\(\displaystyle \frac {\sinh (x)}{a}-\frac {b \left (\frac {x}{a}-\frac {\int \frac {1}{\frac {a \cosh (x)}{b}+1}dx}{a}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh (x)}{a}-\frac {b \left (\frac {x}{a}-\frac {\int \frac {1}{\frac {a \sin \left (i x+\frac {\pi }{2}\right )}{b}+1}dx}{a}\right )}{a}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {\sinh (x)}{a}-\frac {b \left (\frac {x}{a}-\frac {2 \int \frac {1}{\frac {a+b}{b}-\left (1-\frac {a}{b}\right ) \tanh ^2\left (\frac {x}{2}\right )}d\tanh \left (\frac {x}{2}\right )}{a}\right )}{a}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\sinh (x)}{a}-\frac {b \left (\frac {x}{a}-\frac {2 b \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} \sqrt {a+b}}\right )}{a}\) |
-((b*(x/a - (2*b*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b])))/a) + Sinh[x]/a
3.1.98.3.1 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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(-1), x_Symbol] :> Simp[x/a, x] - Simp[1/a Int[1/(1 + (a/b)*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[Cot[e + f*x]*((d*Csc[e + f*x])^n/(a*f*n)), x] - Sim p[1/(a*d*n) Int[((d*Csc[e + f*x])^(n + 1)/(a + b*Csc[e + f*x]))*Simp[b*n - a*(n + 1)*Csc[e + f*x] - b*(n + 1)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.52
method | result | size |
default | \(-\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{2}}-\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{2}}+\frac {2 b^{2} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(94\) |
risch | \(-\frac {b x}{a^{2}}+\frac {{\mathrm e}^{x}}{2 a}-\frac {{\mathrm e}^{-x}}{2 a}-\frac {b^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{2}}+\frac {b^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, a^{2}}\) | \(144\) |
-1/a/(tanh(1/2*x)+1)-1/a^2*b*ln(tanh(1/2*x)+1)-1/a/(tanh(1/2*x)-1)+1/a^2*b *ln(tanh(1/2*x)-1)+2*b^2/a^2/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/ ((a+b)*(a-b))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (52) = 104\).
Time = 0.26 (sec) , antiderivative size = 430, normalized size of antiderivative = 6.94 \[ \int \frac {\cosh (x)}{a+b \text {sech}(x)} \, dx=\left [-\frac {a^{3} - a b^{2} + 2 \, {\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right ) - {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{2} - {\left (a^{3} - a b^{2}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) + a}\right ) + 2 \, {\left ({\left (a^{2} b - b^{3}\right )} x - {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{2 \, {\left ({\left (a^{4} - a^{2} b^{2}\right )} \cosh \left (x\right ) + {\left (a^{4} - a^{2} b^{2}\right )} \sinh \left (x\right )\right )}}, -\frac {a^{3} - a b^{2} + 2 \, {\left (a^{2} b - b^{3}\right )} x \cosh \left (x\right ) - {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{2} - {\left (a^{3} - a b^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b^{2} \cosh \left (x\right ) + b^{2} \sinh \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cosh \left (x\right ) + a \sinh \left (x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right ) + 2 \, {\left ({\left (a^{2} b - b^{3}\right )} x - {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{2 \, {\left ({\left (a^{4} - a^{2} b^{2}\right )} \cosh \left (x\right ) + {\left (a^{4} - a^{2} b^{2}\right )} \sinh \left (x\right )\right )}}\right ] \]
[-1/2*(a^3 - a*b^2 + 2*(a^2*b - b^3)*x*cosh(x) - (a^3 - a*b^2)*cosh(x)^2 - (a^3 - a*b^2)*sinh(x)^2 + 2*(b^2*cosh(x) + b^2*sinh(x))*sqrt(-a^2 + b^2)* log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) - a^2 + 2*b^2 + 2*(a^2* cosh(x) + a*b)*sinh(x) - 2*sqrt(-a^2 + b^2)*(a*cosh(x) + a*sinh(x) + b))/( a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x) + a)) + 2*((a^2*b - b^3)*x - (a^3 - a*b^2)*cosh(x))*sinh(x))/((a^4 - a^2*b^2)*co sh(x) + (a^4 - a^2*b^2)*sinh(x)), -1/2*(a^3 - a*b^2 + 2*(a^2*b - b^3)*x*co sh(x) - (a^3 - a*b^2)*cosh(x)^2 - (a^3 - a*b^2)*sinh(x)^2 + 4*(b^2*cosh(x) + b^2*sinh(x))*sqrt(a^2 - b^2)*arctan(-(a*cosh(x) + a*sinh(x) + b)/sqrt(a ^2 - b^2)) + 2*((a^2*b - b^3)*x - (a^3 - a*b^2)*cosh(x))*sinh(x))/((a^4 - a^2*b^2)*cosh(x) + (a^4 - a^2*b^2)*sinh(x))]
\[ \int \frac {\cosh (x)}{a+b \text {sech}(x)} \, dx=\int \frac {\cosh {\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\cosh (x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \]
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*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{a+b \text {sech}(x)} \, dx=\frac {2 \, b^{2} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{2}} - \frac {b x}{a^{2}} - \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} \]
2*b^2*arctan((a*e^x + b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a^2) - b*x/a^2 - 1/2*e^(-x)/a + 1/2*e^x/a
Time = 2.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.24 \[ \int \frac {\cosh (x)}{a+b \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^x}{2\,a}-\frac {{\mathrm {e}}^{-x}}{2\,a}-\frac {b\,x}{a^2}+\frac {b^2\,\ln \left (-\frac {2\,b^2\,{\mathrm {e}}^x}{a^3}-\frac {2\,b^2\,\left (a+b\,{\mathrm {e}}^x\right )}{a^3\,\sqrt {a+b}\,\sqrt {b-a}}\right )}{a^2\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {b^2\,\ln \left (\frac {2\,b^2\,\left (a+b\,{\mathrm {e}}^x\right )}{a^3\,\sqrt {a+b}\,\sqrt {b-a}}-\frac {2\,b^2\,{\mathrm {e}}^x}{a^3}\right )}{a^2\,\sqrt {a+b}\,\sqrt {b-a}} \]
exp(x)/(2*a) - exp(-x)/(2*a) - (b*x)/a^2 + (b^2*log(- (2*b^2*exp(x))/a^3 - (2*b^2*(a + b*exp(x)))/(a^3*(a + b)^(1/2)*(b - a)^(1/2))))/(a^2*(a + b)^( 1/2)*(b - a)^(1/2)) - (b^2*log((2*b^2*(a + b*exp(x)))/(a^3*(a + b)^(1/2)*( b - a)^(1/2)) - (2*b^2*exp(x))/a^3))/(a^2*(a + b)^(1/2)*(b - a)^(1/2))