\(\int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) [272]

Optimal result

Integrand size = 25, antiderivative size = 35 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x)}{d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 35, 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 = {2850} \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \sin (c+d x)}{d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \]

[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 2850

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*} \text {integral}& = \frac {2 \sin (c+d x)}{d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \sqrt {1-\cos (c+d x)} \cot \left (\frac {1}{2} (c+d x)\right )}{d \sqrt {\cos (c+d x)}} \]

[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]])

Maple [A] (verified)

Time = 5.50 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17

method	result	size
default	\(\frac {\sqrt {-2 \cos \left (d x +c \right )+2}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sqrt {2}}{d \sqrt {\cos \left (d x +c \right )}}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \, {\left (\cos \left (d x + c\right ) + 1\right )} \sqrt {-\cos \left (d x + c\right ) + 1}}{d \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )} \]

[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))

Sympy [F]

\[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\sqrt {1 - \cos {\left (c + d x \right )}}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (31) = 62\).

Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \, {\left (\sqrt {2} - \frac {\sqrt {2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.61 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \, \sqrt {2} {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 1\right )} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{\sqrt {\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1} d} \]

[In]

integrate((1-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)*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)

Mupad [B] (verification not implemented)

Time = 14.71 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2\,\sin \left (c+d\,x\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {1-\cos \left (c+d\,x\right )}} \]

[In]

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

[Out]

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