\(\int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \cos (c+d x)}} \, dx\) [370]

Optimal result

Integrand size = 25, antiderivative size = 56 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d} \]

[Out]

arctan(1/2*sin(d*x+c)*a^(1/2)*sec(d*x+c)^(1/2)*2^(1/2)/(a+a*cos(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4307, 2861, 211} \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d} \]

[In]

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

[Out]

(Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x
]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2861

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[-2*(a/f), Subst[Int[1/(2*b^2 - (a*c - 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]

Rule 4307

Int[(csc[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sin[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {2 \arctan \left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{d \sqrt {a (1+\cos (c+d x))}} \]

[In]

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

[Out]

(2*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]]*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(d*Sqrt
[a*(1 + Cos[c + d*x])])

Maple [A] (verified)

Time = 6.87 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.48

method	result	size
default	\(-\frac {\arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \cos \left (d x +c \right ) \sqrt {2}}{d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, a}\)	\(83\)

[In]

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

[Out]

-1/d*arcsin(cot(d*x+c)-csc(d*x+c))*sec(d*x+c)^(1/2)*(a*(1+cos(d*x+c)))^(1/2)*cos(d*x+c)/(1+cos(d*x+c))/(cos(d*
x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)/a

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.57 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \cos (c+d x)}} \, dx=\left [\frac {\sqrt {2} \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {-\frac {1}{a}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 3 \, \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{2 \, d}, -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{\sqrt {a} d}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 522, normalized size of antiderivative = 9.32 \[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {{\left ({\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{4} + \cos \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{4} + 2 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} {\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2} - 4 \, \cos \left (d x + c\right )^{3} + 2 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2}}, \frac {{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2} + \cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2}}\right )\right ) + \sin \left (d x + c\right )}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}}, \frac {{\left ({\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{4} + \cos \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{4} + 2 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} {\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2} - 4 \, \cos \left (d x + c\right )^{3} + 2 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right ) + 1\right )}^{\frac {1}{4}} \sqrt {a} \cos \left (\frac {1}{2} \, \arctan \left (\frac {2 \, {\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2}}, \frac {{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2} + \cos \left (d x + c\right )^{2} - \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}{{\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}^{2}}\right )\right ) + \sqrt {a} \cos \left (d x + c\right ) - \sqrt {a}}{\sqrt {a} {\left | e^{\left (i \, d x + i \, c\right )} + 1 \right |}}\right )}{\sqrt {a} d} \]

[In]

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

[Out]

sqrt(2)*arctan2(((abs(e^(I*d*x + I*c) + 1)^4 + cos(d*x + c)^4 + sin(d*x + c)^4 + 2*(cos(d*x + c)^2 - sin(d*x +
c)^2 - 2*cos(d*x + c) + 1)*abs(e^(I*d*x + I*c) + 1)^2 - 4*cos(d*x + c)^3 + 2*(cos(d*x + c)^2 - 2*cos(d*x + c)
+ 1)*sin(d*x + c)^2 + 6*cos(d*x + c)^2 - 4*cos(d*x + c) + 1)^(1/4)*sin(1/2*arctan2(2*(cos(d*x + c) - 1)*sin(d
*x + c)/abs(e^(I*d*x + I*c) + 1)^2, (abs(e^(I*d*x + I*c) + 1)^2 + cos(d*x + c)^2 - sin(d*x + c)^2 - 2*cos(d*x
+ c) + 1)/abs(e^(I*d*x + I*c) + 1)^2)) + sin(d*x + c))/abs(e^(I*d*x + I*c) + 1), ((abs(e^(I*d*x + I*c) + 1)^4
+ cos(d*x + c)^4 + sin(d*x + c)^4 + 2*(cos(d*x + c)^2 - sin(d*x + c)^2 - 2*cos(d*x + c) + 1)*abs(e^(I*d*x + I*
c) + 1)^2 - 4*cos(d*x + c)^3 + 2*(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)*sin(d*x + c)^2 + 6*cos(d*x + c)^2 - 4*c
os(d*x + c) + 1)^(1/4)*sqrt(a)*cos(1/2*arctan2(2*(cos(d*x + c) - 1)*sin(d*x + c)/abs(e^(I*d*x + I*c) + 1)^2, (
abs(e^(I*d*x + I*c) + 1)^2 + cos(d*x + c)^2 - sin(d*x + c)^2 - 2*cos(d*x + c) + 1)/abs(e^(I*d*x + I*c) + 1)^2)
) + sqrt(a)*cos(d*x + c) - sqrt(a))/(sqrt(a)*abs(e^(I*d*x + I*c) + 1)))/(sqrt(a)*d)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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