\(\int \frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx\) [605]

Optimal result

Integrand size = 25, antiderivative size = 135 \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {\frac {a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt {\frac {a (1+\cos (c+d x))}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {a+b} \sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}\right ),-\frac {a-b}{a+b}\right )}{\sqrt {a+b} d} \]

[Out]

-2*(a+b*cos(d*x+c))*csc(d*x+c)*EllipticPi((a+b)^(1/2)*cos(d*x+c)^(1/2)/(a+b*cos(d*x+c))^(1/2),b/(a+b),((-a+b)/
(a+b))^(1/2))*(a*(1-cos(d*x+c))/(a+b*cos(d*x+c)))^(1/2)*(a*(1+cos(d*x+c))/(a+b*cos(d*x+c)))^(1/2)/d/(a+b)^(1/2
)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2890} \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \csc (c+d x) \sqrt {\frac {a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt {\frac {a (\cos (c+d x)+1)}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {a+b} \sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}\right ),-\frac {a-b}{a+b}\right )}{d \sqrt {a+b}} \]

[In]

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

[Out]

(-2*Sqrt[(a*(1 - Cos[c + d*x]))/(a + b*Cos[c + d*x])]*Sqrt[(a*(1 + Cos[c + d*x]))/(a + b*Cos[c + d*x])]*(a + b
*Cos[c + d*x])*Csc[c + d*x]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[Cos[c + d*x]])/Sqrt[a + b*Cos[c + d
*x]]], -((a - b)/(a + b))])/(Sqrt[a + b]*d)

Rule 2890

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {\frac {a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt {\frac {a (1+\cos (c+d x))}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {a+b} \sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}\right ),-\frac {a-b}{a+b}\right )}{\sqrt {a+b} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \left ((a-b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+2 b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )\right )}{d \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {a+b \cos (c+d x)}} \]

[In]

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

[Out]

(2*Sqrt[Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((a - b)*EllipticF[ArcSin[Tan[(c
+ d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]))/(d*Sqrt[Cos[
c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]])

Maple [A] (verified)

Time = 8.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.34

method	result	size
default	\(-\frac {2 \sqrt {\frac {a +\cos \left (d x +c \right ) b}{\left (1+\cos \left (d x +c \right )\right ) \left (a +b \right )}}\, \left (F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) a -F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) b +2 b \Pi \left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, \sqrt {-\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )\right )}{d \sqrt {a +\cos \left (d x +c \right ) b}\, \sqrt {\cos \left (d x +c \right )}}\)	\(181\)

[In]

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

[Out]

-2/d*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a-El
lipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*b+2*b*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,(-(a-b)/(a+b))^(
1/2)))/(a+cos(d*x+c)*b)^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(1+cos(d*x+c))/cos(d*x+c)^(1/2)

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(b*cos(d*x + c) + a)/sqrt(cos(d*x + c)), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(b*cos(d*x + c) + a)/sqrt(cos(d*x + c)), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(b*cos(d*x + c) + a)/sqrt(cos(d*x + c)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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