\(\int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx\) [51]

Optimal result

Integrand size = 15, antiderivative size = 48 \[ \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{\sqrt {a} d} \]

[Out]

-arctan(1/2*sinh(d*x+c)*a^(1/2)*2^(1/2)/(a-a*cosh(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2728, 212} \[ \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{\sqrt {a} d} \]

[In]

Int[1/Sqrt[a - a*Cosh[c + d*x]],x]

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a - a*Cosh[c + d*x]])])/(Sqrt[a]*d))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(2 i) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {i a \sinh (c+d x)}{\sqrt {a-a \cosh (c+d x)}}\right )}{d} \\ & = -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx=\frac {2 \left (-\log \left (\cosh \left (\frac {1}{4} (c+d x)\right )\right )+\log \left (\sinh \left (\frac {1}{4} (c+d x)\right )\right )\right ) \sinh \left (\frac {1}{2} (c+d x)\right )}{d \sqrt {a-a \cosh (c+d x)}} \]

[In]

Integrate[1/Sqrt[a - a*Cosh[c + d*x]],x]

[Out]

(2*(-Log[Cosh[(c + d*x)/4]] + Log[Sinh[(c + d*x)/4]])*Sinh[(c + d*x)/2])/(d*Sqrt[a - a*Cosh[c + d*x]])

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {2 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \operatorname {arctanh}\left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\sqrt {-2 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}\, d}\)	\(41\)

[In]

int(1/(a-a*cosh(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*sinh(1/2*d*x+1/2*c)*arctanh(cosh(1/2*d*x+1/2*c))/(-2*sinh(1/2*d*x+1/2*c)^2*a)^(1/2)/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 3.21 \[ \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx=\left [\frac {\sqrt {2} \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} \sqrt {-\frac {1}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} - \cosh \left (d x + c\right ) - \sinh \left (d x + c\right ) - 1}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}\right )}{d}, \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{\sqrt {a}}\right )}{\sqrt {a} d}\right ] \]

[In]

integrate(1/(a-a*cosh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[sqrt(2)*sqrt(-1/a)*log((2*sqrt(2)*sqrt(1/2)*sqrt(-a/(cosh(d*x + c) + sinh(d*x + c)))*sqrt(-1/a)*(cosh(d*x + c
) + sinh(d*x + c)) - cosh(d*x + c) - sinh(d*x + c) - 1)/(cosh(d*x + c) + sinh(d*x + c) - 1))/d, 2*sqrt(2)*arct
an(sqrt(2)*sqrt(1/2)*sqrt(-a/(cosh(d*x + c) + sinh(d*x + c)))*(cosh(d*x + c) + sinh(d*x + c))/sqrt(a))/(sqrt(a
)*d)]

Sympy [F]

\[ \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx=\int \frac {1}{\sqrt {- a \cosh {\left (c + d x \right )} + a}}\, dx \]

[In]

integrate(1/(a-a*cosh(d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt(-a*cosh(c + d*x) + a), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-a \cosh \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(1/(a-a*cosh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-a*cosh(d*x + c) + a), x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx=-\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a e^{\left (d x + c\right )}}}{\sqrt {a}}\right )}{\sqrt {a} d \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} \]

[In]

integrate(1/(a-a*cosh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-2*sqrt(2)*arctan(sqrt(-a*e^(d*x + c))/sqrt(a))/(sqrt(a)*d*sgn(-e^(d*x + c) + 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a-a \cosh (c+d x)}} \, dx=\int \frac {1}{\sqrt {a-a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]

[In]

int(1/(a - a*cosh(c + d*x))^(1/2),x)

[Out]

int(1/(a - a*cosh(c + d*x))^(1/2), x)