Integrand size = 15, antiderivative size = 60 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)} \, dx=\frac {B x}{b}+\frac {2 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b \sqrt {a+b}} \]
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)} \, dx=\frac {B x}{b}+\frac {2 (-A b+a B) \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{b \sqrt {-a^2+b^2}} \]
(B*x)/b + (2*(-(A*b) + a*B)*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/ (b*Sqrt[-a^2 + b^2])
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
\(\displaystyle \int \frac {A+B \cosh (x)}{a+b \cosh (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin \left (\frac {\pi }{2}+i x\right )}{a+b \sin \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle \frac {(A b-a B) \int \frac {1}{a+b \cosh (x)}dx}{b}+\frac {B x}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {B x}{b}+\frac {(A b-a B) \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {2 (A b-a B) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{b}+\frac {B x}{b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b \sqrt {a-b} \sqrt {a+b}}+\frac {B x}{b}\) |
(B*x)/b + (2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sq rt[a - b]*b*Sqrt[a + b])
3.2.10.3.1 Defintions of rubi rules used
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]
Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.22
method | result | size |
default | \(-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}-\frac {2 \left (-b A +B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(73\) |
risch | \(\frac {B x}{b}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B a}{\sqrt {a^{2}-b^{2}}\, b}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A}{\sqrt {a^{2}-b^{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B a}{\sqrt {a^{2}-b^{2}}\, b}\) | \(234\) |
-B/b*ln(tanh(1/2*x)-1)+B/b*ln(tanh(1/2*x)+1)-2/b*(-A*b+B*a)/((a+b)*(a-b))^ (1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))
Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.00 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)} \, dx=\left [-\frac {{\left (B a - A b\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) - {\left (B a^{2} - B b^{2}\right )} x}{a^{2} b - b^{3}}, \frac {2 \, {\left (B a - A b\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (B a^{2} - B b^{2}\right )} x}{a^{2} b - b^{3}}\right ] \]
[-((B*a - A*b)*sqrt(a^2 - b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b* cosh(x) + 2*a^2 - b^2 + 2*(b^2*cosh(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)) - (B*a^2 - B*b^2)*x)/(a^2*b - b^3), (2*(B*a - A*b)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a)/(a^2 - b^2)) + (B*a^2 - B*b^2)*x)/(a^2*b - b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (49) = 98\).
Time = 12.97 (sec) , antiderivative size = 403, normalized size of antiderivative = 6.72 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)} \, dx=\begin {cases} \tilde {\infty } \left (2 A \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )} + B x\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {A \tanh {\left (\frac {x}{2} \right )}}{b} + \frac {B x}{b} - \frac {B \tanh {\left (\frac {x}{2} \right )}}{b} & \text {for}\: a = b \\- \frac {A}{b \tanh {\left (\frac {x}{2} \right )}} + \frac {B x}{b} - \frac {B}{b \tanh {\left (\frac {x}{2} \right )}} & \text {for}\: a = - b \\\frac {A x + B \sinh {\left (x \right )}}{a} & \text {for}\: b = 0 \\- \frac {A b \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {A b \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {B a x \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {B a \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {B a \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {B b x \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(2*A*atan(tanh(x/2)) + B*x), Eq(a, 0) & Eq(b, 0)), (A*tanh( x/2)/b + B*x/b - B*tanh(x/2)/b, Eq(a, b)), (-A/(b*tanh(x/2)) + B*x/b - B/( b*tanh(x/2)), Eq(a, -b)), ((A*x + B*sinh(x))/a, Eq(b, 0)), (-A*b*log(-sqrt (a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b* *2*sqrt(a/(a - b) + b/(a - b))) + A*b*log(sqrt(a/(a - b) + b/(a - b)) + ta nh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b ))) + B*a*x*sqrt(a/(a - b) + b/(a - b))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) + B*a*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) - B*a*log(sqrt(a/(a - b) + b/(a - b)) + tanh(x/2))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b/(a - b))) - B*b*x*sqrt(a/(a - b) + b/(a - b))/(a*b*sqrt(a/(a - b) + b/(a - b)) - b**2*sqrt(a/(a - b) + b /(a - b))), True))
Exception generated. \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \]
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 = 50, normalized size of antiderivative = 0.83 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)} \, dx=\frac {B x}{b} - \frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b} \]
Time = 1.87 (sec) , antiderivative size = 242, normalized size of antiderivative = 4.03 \[ \int \frac {A+B \cosh (x)}{a+b \cosh (x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {b^2\,{\mathrm {e}}^x\,\sqrt {b^4-a^2\,b^2}\,\left (\frac {2\,\left (A\,b\,\sqrt {b^4-a^2\,b^2}-B\,a\,\sqrt {b^4-a^2\,b^2}\right )}{b^4\,\sqrt {b^4-a^2\,b^2}\,\sqrt {{\left (A\,b-B\,a\right )}^2}}+\frac {2\,a^2\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{b^2\,\left (b^4-a^2\,b^2\right )\,\left (A\,b-B\,a\right )}\right )}{2}+\frac {a\,b\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{\sqrt {b^4-a^2\,b^2}\,\left (A\,b-B\,a\right )}\right )\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{\sqrt {b^4-a^2\,b^2}}+\frac {B\,x}{b} \]
(2*atan((b^2*exp(x)*(b^4 - a^2*b^2)^(1/2)*((2*(A*b*(b^4 - a^2*b^2)^(1/2) - B*a*(b^4 - a^2*b^2)^(1/2)))/(b^4*(b^4 - a^2*b^2)^(1/2)*((A*b - B*a)^2)^(1 /2)) + (2*a^2*(A^2*b^2 + B^2*a^2 - 2*A*B*a*b)^(1/2))/(b^2*(b^4 - a^2*b^2)* (A*b - B*a))))/2 + (a*b*(A^2*b^2 + B^2*a^2 - 2*A*B*a*b)^(1/2))/((b^4 - a^2 *b^2)^(1/2)*(A*b - B*a)))*(A^2*b^2 + B^2*a^2 - 2*A*B*a*b)^(1/2))/(b^4 - a^ 2*b^2)^(1/2) + (B*x)/b