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

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Rubi [A]  time = 0.0566217, antiderivative size = 37, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0444338, 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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Maple [A]  time = 0.332, size = 46, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{-2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) }{\frac{1}{\sqrt{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.57563, size = 111, normalized size = 3. \begin{align*} \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}}} \end{align*}

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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Fricas [A]  time = 1.84738, size = 113, normalized size = 3.05 \begin{align*} \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 )} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- a \left (\cos{\left (c + d x \right )} - 1\right )}}{\cos ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \end{align*}

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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Giac [B]  time = 2.41281, size = 126, normalized size = 3.41 \begin{align*} -\frac{\sqrt{2}{\left (a^{2}{\left (\frac{\sqrt{2}}{\sqrt{a}{\left | a \right |}} - \frac{2}{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\left | a \right |}}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right ) + \frac{\sqrt{2}{\left (\sqrt{2} a^{2} - a^{2}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{\sqrt{a}{\left | a \right |}}\right )}}{d} \end{align*}

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]

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