\(\int \frac {\sqrt {a-a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx\) [265]

Optimal result

Integrand size = 26, antiderivative size = 48 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}}\right )}{d} \]

[Out]

-2*arctanh(sin(d*x+c)*a^(1/2)/cos(d*x+c)^(1/2)/(a-a*cos(d*x+c))^(1/2))*a^(1/2)/d

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2854, 213} \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}}\right )}{d} \]

[In]

Int[Sqrt[a - a*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]

[Out]

(-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/d

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 2854

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=-\frac {e^{\frac {1}{2} i (c+d x)} \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (\text {arcsinh}\left (e^{i (c+d x)}\right )+\text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right ) \sqrt {a-a \cos (c+d x)} \csc \left (\frac {1}{2} (c+d x)\right )}{\sqrt {2} d \sqrt {1+e^{2 i (c+d x)}}} \]

[In]

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

[Out]

-((E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(ArcSinh[E^(I*(c + d*x))] + ArcTanh[Sqr
t[1 + E^((2*I)*(c + d*x))]])*Sqrt[a - a*Cos[c + d*x]]*Csc[(c + d*x)/2])/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x)
)]))

Maple [A] (verified)

Time = 4.55 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.65

method	result	size
default	\(-\frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-a \left (\cos \left (d x +c \right )-1\right )}\, \operatorname {arctanh}\left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d \sqrt {\cos \left (d x +c \right )}}\)	\(79\)

[In]

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

[Out]

-2/d*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(-a*(cos(d*x+c)-1))^(1/2)*arctanh((cos(d*x+c)/(1+cos(d*x+c)))^(1/2))/co
s(d*x+c)^(1/2)*(cot(d*x+c)+csc(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.23 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {4 \, \sqrt {-a \cos \left (d x + c\right ) + a} {\left (2 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \sqrt {\cos \left (d x + c\right )} - {\left (8 \, a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right )}{2 \, d}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a \cos \left (d x + c\right ) + a} \sqrt {-a} {\left (2 \, \cos \left (d x + c\right ) + 1\right )}}{2 \, a \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}\right )}{d}\right ] \]

[In]

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

[Out]

[1/2*sqrt(a)*log((4*sqrt(-a*cos(d*x + c) + a)*(2*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*sqrt(cos(d*x + c
)) - (8*a*cos(d*x + c)^2 + 8*a*cos(d*x + c) + a)*sin(d*x + c))/sin(d*x + c))/d, sqrt(-a)*arctan(1/2*sqrt(-a*co
s(d*x + c) + a)*sqrt(-a)*(2*cos(d*x + c) + 1)/(a*sqrt(cos(d*x + c))*sin(d*x + c)))/d]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (40) = 80\).

Time = 0.36 (sec) , antiderivative size = 148, normalized size of antiderivative = 3.08 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {\sqrt {-a} \arctan \left ({\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + \sin \left (d x + c\right ), {\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, d x + 2 \, c\right ), \cos \left (2 \, d x + 2 \, c\right ) + 1\right )\right ) + \cos \left (d x + c\right )\right )}{d} \]

[In]

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

[Out]

sqrt(-a)*arctan2((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(
2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + sin(d*x + c), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x +
2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + cos(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (40) = 80\).

Time = 0.45 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.54 \[ \int \frac {\sqrt {a-a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \, \sqrt {a} \log \left (\frac {2 \, {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 2 \, \sqrt {2} - \sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} + 1\right )}}{{\left | -2 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 4 \, \sqrt {2} + 2 \, \sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} - 2 \right |}}\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]

[In]

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

[Out]

2*sqrt(a)*log(2*(tan(1/4*d*x + 1/4*c)^2 + 2*sqrt(2) - sqrt(tan(1/4*d*x + 1/4*c)^4 - 6*tan(1/4*d*x + 1/4*c)^2 +
1) + 1)/abs(-2*tan(1/4*d*x + 1/4*c)^2 + 4*sqrt(2) + 2*sqrt(tan(1/4*d*x + 1/4*c)^4 - 6*tan(1/4*d*x + 1/4*c)^2
+ 1) - 2))*sgn(sin(1/2*d*x + 1/2*c))/d

Mupad [F(-1)]

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

[In]

int((a - a*cos(c + d*x))^(1/2)/cos(c + d*x)^(1/2),x)

[Out]

int((a - a*cos(c + d*x))^(1/2)/cos(c + d*x)^(1/2), x)