3.272 \(\int \frac{\sqrt{1-\cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=35 \[ \frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}} \]

[Out]

(2*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])

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Rubi [A]  time = 0.0424697, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2771} \[ \frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]

[Out]

(2*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]*Sqrt[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{1-\cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0436007, size = 39, normalized size = 1.11 \[ \frac{2 \sqrt{1-\cos (c+d x)} \cot \left (\frac{1}{2} (c+d x)\right )}{d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]

[Out]

(2*Sqrt[1 - Cos[c + d*x]]*Cot[(c + d*x)/2])/(d*Sqrt[Cos[c + d*x]])

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Maple [A]  time = 0.279, size = 45, normalized size = 1.3 \begin{align*} -{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{2-2\,\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.53573, size = 101, normalized size = 2.89 \begin{align*} \frac{2 \,{\left (\sqrt{2} - \frac{\sqrt{2} \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((1-cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

2*(sqrt(2) - sqrt(2)*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 = 1.85173, size = 111, normalized size = 3.17 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right ) + 1\right )} \sqrt{-\cos \left (d x + c\right ) + 1}}{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((1-cos(d*x+c))^(1/2)/cos(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

2*(cos(d*x + c) + 1)*sqrt(-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{1 - \cos{\left (c + d x \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((1-cos(d*x+c))**(1/2)/cos(d*x+c)**(3/2),x)

[Out]

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

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Giac [A]  time = 2.12484, size = 84, normalized size = 2.4 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{2}{\left (\sqrt{2} - 1\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right ) +{\left (\sqrt{2} - \frac{2}{\sqrt{-\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*(sqrt(2)*(sqrt(2) - 1)*sgn(tan(1/2*d*x + 1/2*c)) + (sqrt(2) - 2/sqrt(-tan(1/2*d*x + 1/2*c)^2 + 1))*sg
n(tan(1/2*d*x + 1/2*c)))/d