3.266 \(\int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=37 \[ \frac {2 a \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2771} \[ \frac {2 a \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]

[Out]

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

Rule 2771

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[(-2*b^2*Cos[e + f*x])/(f*(b*c + a*d)*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*} \int \frac {\sqrt {a-a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {2 a \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {a-a \cos (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 40, normalized size = 1.08 \[ \frac {2 \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {a-a \cos (c+d x)}}{d \sqrt {\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]

[Out]

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

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fricas [A]  time = 2.32, size = 42, normalized size = 1.14 \[ \frac {2 \, \sqrt {-a \cos \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) + 1\right )}}{d \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.08, size = 62, normalized size = 1.68 \[ -\frac {2 \, \sqrt {2} {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 1\right )} \sqrt {a} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{\sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.12, size = 46, normalized size = 1.24 \[ -\frac {\sqrt {-2 a \left (-1+\cos \left (d x +c \right )\right )}\, \sin \left (d x +c \right ) \sqrt {2}}{d \left (-1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x)

[Out]

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

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maxima [B]  time = 0.63, size = 82, normalized size = 2.22 \[ \frac {2 \, {\left (\sqrt {2} \sqrt {a} - \frac {\sqrt {2} \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

2*(sqrt(2)*sqrt(a) - sqrt(2)*sqrt(a)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/(d*(sin(d*x + c)/(cos(d*x + c) + 1)
+ 1)^(3/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(3/2))

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mupad [B]  time = 0.79, size = 42, normalized size = 1.14 \[ -\frac {2\,\sin \left (c+d\,x\right )\,\sqrt {-a\,\left (\cos \left (c+d\,x\right )-1\right )}}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (\cos \left (c+d\,x\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*cos(c + d*x))^(1/2)/cos(c + d*x)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- a \left (\cos {\left (c + d x \right )} - 1\right )}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cos(d*x+c))**(1/2)/cos(d*x+c)**(3/2),x)

[Out]

Integral(sqrt(-a*(cos(c + d*x) - 1))/cos(c + d*x)**(3/2), x)

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