Integrand size = 14, antiderivative size = 53 \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}}+\frac {2 \sinh (x)}{\sqrt {a-a \cosh (x)}} \] Output:
-2^(1/2)*arctan(1/2*a^(1/2)*sinh(x)*2^(1/2)/(a-a*cosh(x))^(1/2))/a^(1/2)+2 *sinh(x)/(a-a*cosh(x))^(1/2)
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\frac {2 \left (2 \cosh \left (\frac {x}{2}\right )-\log \left (\cosh \left (\frac {x}{4}\right )\right )+\log \left (\sinh \left (\frac {x}{4}\right )\right )\right ) \sinh \left (\frac {x}{2}\right )}{\sqrt {a-a \cosh (x)}} \] Input:
Integrate[Cosh[x]/Sqrt[a - a*Cosh[x]],x]
Output:
(2*(2*Cosh[x/2] - Log[Cosh[x/4]] + Log[Sinh[x/4]])*Sinh[x/2])/Sqrt[a - a*C osh[x]]
Time = 0.33 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 3230, 3042, 3128, 219}
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.
\(\displaystyle \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )}{\sqrt {a-a \sin \left (\frac {\pi }{2}+i x\right )}}dx\) |
\(\Big \downarrow \) 3230 |
\(\displaystyle \int \frac {1}{\sqrt {a-a \cosh (x)}}dx+\frac {2 \sinh (x)}{\sqrt {a-a \cosh (x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \sinh (x)}{\sqrt {a-a \cosh (x)}}+\int \frac {1}{\sqrt {a-a \sin \left (i x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {2 \sinh (x)}{\sqrt {a-a \cosh (x)}}+2 i \int \frac {1}{\frac {a^2 \sinh ^2(x)}{a-a \cosh (x)}+2 a}d\frac {i a \sinh (x)}{\sqrt {a-a \cosh (x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 \sinh (x)}{\sqrt {a-a \cosh (x)}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}}\) |
Input:
Int[Cosh[x]/Sqrt[a - a*Cosh[x]],x]
Output:
-((Sqrt[2]*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a - a*Cosh[x]])])/Sqrt[a ]) + (2*Sinh[x])/Sqrt[a - a*Cosh[x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Time = 0.49 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\sinh \left (\frac {x}{2}\right ) \left (4 \cosh \left (\frac {x}{2}\right )+\ln \left (\cosh \left (\frac {x}{2}\right )-1\right )-\ln \left (\cosh \left (\frac {x}{2}\right )+1\right )\right )}{\sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\) | \(40\) |
Input:
int(cosh(x)/(a-cosh(x)*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
sinh(1/2*x)*(4*cosh(1/2*x)+ln(cosh(1/2*x)-1)-ln(cosh(1/2*x)+1))/(-2*sinh(1 /2*x)^2*a)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (42) = 84\).
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.74 \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\frac {\sqrt {2} a \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}} \sqrt {-\frac {1}{a}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - \cosh \left (x\right ) - \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - 2 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}}{a} \] Input:
integrate(cosh(x)/(a-a*cosh(x))^(1/2),x, algorithm="fricas")
Output:
(sqrt(2)*a*sqrt(-1/a)*log((2*sqrt(2)*sqrt(1/2)*sqrt(-a/(cosh(x) + sinh(x)) )*sqrt(-1/a)*(cosh(x) + sinh(x)) - cosh(x) - sinh(x) - 1)/(cosh(x) + sinh( x) - 1)) - 2*sqrt(1/2)*sqrt(-a/(cosh(x) + sinh(x)))*(cosh(x) + sinh(x) + 1 ))/a
\[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\int \frac {\cosh {\left (x \right )}}{\sqrt {- a \left (\cosh {\left (x \right )} - 1\right )}}\, dx \] Input:
integrate(cosh(x)/(a-a*cosh(x))**(1/2),x)
Output:
Integral(cosh(x)/sqrt(-a*(cosh(x) - 1)), x)
\[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\int { \frac {\cosh \left (x\right )}{\sqrt {-a \cosh \left (x\right ) + a}} \,d x } \] Input:
integrate(cosh(x)/(a-a*cosh(x))^(1/2),x, algorithm="maxima")
Output:
integrate(cosh(x)/sqrt(-a*cosh(x) + a), x)
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.40 \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=-\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a e^{x}}}{\sqrt {a}}\right )}{\sqrt {a} \mathrm {sgn}\left (-e^{x} + 1\right )} - \frac {\sqrt {2}}{\sqrt {-a e^{x}} \mathrm {sgn}\left (-e^{x} + 1\right )} + \frac {\sqrt {2} \sqrt {-a e^{x}}}{a \mathrm {sgn}\left (-e^{x} + 1\right )} \] Input:
integrate(cosh(x)/(a-a*cosh(x))^(1/2),x, algorithm="giac")
Output:
-2*sqrt(2)*arctan(sqrt(-a*e^x)/sqrt(a))/(sqrt(a)*sgn(-e^x + 1)) - sqrt(2)/ (sqrt(-a*e^x)*sgn(-e^x + 1)) + sqrt(2)*sqrt(-a*e^x)/(a*sgn(-e^x + 1))
Timed out. \[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=\int \frac {\mathrm {cosh}\left (x\right )}{\sqrt {a-a\,\mathrm {cosh}\left (x\right )}} \,d x \] Input:
int(cosh(x)/(a - a*cosh(x))^(1/2),x)
Output:
int(cosh(x)/(a - a*cosh(x))^(1/2), x)
\[ \int \frac {\cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx=-\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\cosh \left (x \right )+1}\, \cosh \left (x \right )}{\cosh \left (x \right )-1}d x \right )}{a} \] Input:
int(cosh(x)/(a-a*cosh(x))^(1/2),x)
Output:
( - sqrt(a)*int((sqrt( - cosh(x) + 1)*cosh(x))/(cosh(x) - 1),x))/a