Integrand size = 15, antiderivative size = 39 \[ \int \frac {\cosh ^2(x)}{a+b \cosh ^2(x)} \, dx=\frac {x}{b}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{b \sqrt {a+b}} \]
Time = 0.45 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh ^2(x)}{a+b \cosh ^2(x)} \, dx=\frac {x-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{\sqrt {a+b}}}{b} \]
Time = 0.29 (sec) , antiderivative size = 39, 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, 3650, 3042, 3660, 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.
\(\displaystyle \int \frac {\cosh ^2(x)}{a+b \cosh ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )^2}{a+b \sin \left (\frac {\pi }{2}+i x\right )^2}dx\) |
\(\Big \downarrow \) 3650 |
\(\displaystyle \frac {x}{b}-\frac {a \int \frac {1}{b \cosh ^2(x)+a}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x}{b}-\frac {a \int \frac {1}{b \sin \left (i x+\frac {\pi }{2}\right )^2+a}dx}{b}\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle \frac {x}{b}-\frac {a \int \frac {1}{a-(a+b) \coth ^2(x)}d\coth (x)}{b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {x}{b}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a}}\right )}{b \sqrt {a+b}}\) |
3.1.25.3.1 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[(e_.) + (f_.)*(x_)]^2)/((a_) + (b_.)*sin[(e_.) + (f_ .)*(x_)]^2), x_Symbol] :> Simp[B*(x/b), x] + Simp[(A*b - a*B)/b Int[1/(a + b*Sin[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, A, B}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(31)=62\).
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.36
method | result | size |
risch | \(\frac {x}{b}+\frac {\sqrt {a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 x}+\frac {2 \sqrt {a \left (a +b \right )}+2 a +b}{b}\right )}{2 \left (a +b \right ) b}-\frac {\sqrt {a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {a \left (a +b \right )}-2 a -b}{b}\right )}{2 \left (a +b \right ) b}\) | \(92\) |
default | \(\frac {2 a \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}-2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}\right )}{b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}\) | \(108\) |
x/b+1/2*(a*(a+b))^(1/2)/(a+b)/b*ln(exp(2*x)+(2*(a*(a+b))^(1/2)+2*a+b)/b)-1 /2*(a*(a+b))^(1/2)/(a+b)/b*ln(exp(2*x)-(2*(a*(a+b))^(1/2)-2*a-b)/b)
Time = 0.26 (sec) , antiderivative size = 317, normalized size of antiderivative = 8.13 \[ \int \frac {\cosh ^2(x)}{a+b \cosh ^2(x)} \, dx=\left [\frac {\sqrt {\frac {a}{a + b}} \log \left (\frac {b^{2} \cosh \left (x\right )^{4} + 4 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{2} \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + 8 \, a^{2} + 8 \, a b + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + {\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, {\left ({\left (a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a b + b^{2}\right )} \sinh \left (x\right )^{2} + 2 \, a^{2} + 3 \, a b + b^{2}\right )} \sqrt {\frac {a}{a + b}}}{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + {\left (2 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}\right ) + 2 \, x}{2 \, b}, -\frac {\sqrt {-\frac {a}{a + b}} \arctan \left (\frac {{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + 2 \, a + b\right )} \sqrt {-\frac {a}{a + b}}}{2 \, a}\right ) - x}{b}\right ] \]
[1/2*(sqrt(a/(a + b))*log((b^2*cosh(x)^4 + 4*b^2*cosh(x)*sinh(x)^3 + b^2*s inh(x)^4 + 2*(2*a*b + b^2)*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + 2*a*b + b^2)*s inh(x)^2 + 8*a^2 + 8*a*b + b^2 + 4*(b^2*cosh(x)^3 + (2*a*b + b^2)*cosh(x)) *sinh(x) + 4*((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 + 2*a^2 + 3*a*b + b^2)*sqrt(a/(a + b)))/(b*cosh(x)^4 + 4 *b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x )^2 + 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b )) + 2*x)/b, -(sqrt(-a/(a + b))*arctan(1/2*(b*cosh(x)^2 + 2*b*cosh(x)*sinh (x) + b*sinh(x)^2 + 2*a + b)*sqrt(-a/(a + b))/a) - x)/b]
Timed out. \[ \int \frac {\cosh ^2(x)}{a+b \cosh ^2(x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (31) = 62\).
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.08 \[ \int \frac {\cosh ^2(x)}{a+b \cosh ^2(x)} \, dx=-\frac {{\left (2 \, a + b\right )} \log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{4 \, \sqrt {{\left (a + b\right )} a} b} - \frac {\log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{4 \, \sqrt {{\left (a + b\right )} a}} + \frac {x}{b} \]
-1/4*(2*a + b)*log((b*e^(2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(2*x) + 2*a + b + 2*sqrt((a + b)*a)))/(sqrt((a + b)*a)*b) - 1/4*log((b*e^(-2*x) + 2*a + b - 2*sqrt((a + b)*a))/(b*e^(-2*x) + 2*a + b + 2*sqrt((a + b)*a)))/s qrt((a + b)*a) + x/b
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28 \[ \int \frac {\cosh ^2(x)}{a+b \cosh ^2(x)} \, dx=-\frac {a \arctan \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} b} + \frac {x}{b} \]
Time = 2.33 (sec) , antiderivative size = 376, normalized size of antiderivative = 9.64 \[ \int \frac {\cosh ^2(x)}{a+b \cosh ^2(x)} \, dx=\frac {x}{b}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\left (b^5\,\sqrt {-b^3-a\,b^2}+a\,b^4\,\sqrt {-b^3-a\,b^2}\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (\frac {2\,\left (8\,a^{5/2}\,\sqrt {-b^3-a\,b^2}+\sqrt {a}\,b^2\,\sqrt {-b^3-a\,b^2}+8\,a^{3/2}\,b\,\sqrt {-b^3-a\,b^2}\right )\,\left (8\,a^2+8\,a\,b+b^2\right )}{b^8\,{\left (a+b\right )}^2\,\sqrt {-b^3-a\,b^2}}+\frac {4\,\sqrt {a}\,\left (4\,a+2\,b\right )\,\left (8\,a^3\,b+12\,a^2\,b^2+4\,a\,b^3\right )}{b^7\,\left (a+b\right )\,\sqrt {-b^2\,\left (a+b\right )}\,\sqrt {-b^3-a\,b^2}}\right )+\frac {2\,\left (\sqrt {a}\,b^2\,\sqrt {-b^3-a\,b^2}+2\,a^{3/2}\,b\,\sqrt {-b^3-a\,b^2}\right )\,\left (8\,a^2+8\,a\,b+b^2\right )}{b^8\,{\left (a+b\right )}^2\,\sqrt {-b^3-a\,b^2}}+\frac {4\,\sqrt {a}\,\left (2\,a^2\,b^2+2\,a\,b^3\right )\,\left (4\,a+2\,b\right )}{b^7\,\left (a+b\right )\,\sqrt {-b^2\,\left (a+b\right )}\,\sqrt {-b^3-a\,b^2}}\right )}{4\,a}\right )}{\sqrt {-b^3-a\,b^2}} \]
x/b + (a^(1/2)*atan(((b^5*(- a*b^2 - b^3)^(1/2) + a*b^4*(- a*b^2 - b^3)^(1 /2))*(exp(2*x)*((2*(8*a^(5/2)*(- a*b^2 - b^3)^(1/2) + a^(1/2)*b^2*(- a*b^2 - b^3)^(1/2) + 8*a^(3/2)*b*(- a*b^2 - b^3)^(1/2))*(8*a*b + 8*a^2 + b^2))/ (b^8*(a + b)^2*(- a*b^2 - b^3)^(1/2)) + (4*a^(1/2)*(4*a + 2*b)*(4*a*b^3 + 8*a^3*b + 12*a^2*b^2))/(b^7*(a + b)*(-b^2*(a + b))^(1/2)*(- a*b^2 - b^3)^( 1/2))) + (2*(a^(1/2)*b^2*(- a*b^2 - b^3)^(1/2) + 2*a^(3/2)*b*(- a*b^2 - b^ 3)^(1/2))*(8*a*b + 8*a^2 + b^2))/(b^8*(a + b)^2*(- a*b^2 - b^3)^(1/2)) + ( 4*a^(1/2)*(2*a*b^3 + 2*a^2*b^2)*(4*a + 2*b))/(b^7*(a + b)*(-b^2*(a + b))^( 1/2)*(- a*b^2 - b^3)^(1/2))))/(4*a)))/(- a*b^2 - b^3)^(1/2)