\(\int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\) [175]

Optimal result

Integrand size = 21, antiderivative size = 22 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2}{a d \sqrt {a+a \sin (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 32} \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2}{a d \sqrt {a \sin (c+d x)+a}} \]

[In]

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

[Out]

-2/(a*d*Sqrt[a + a*Sin[c + d*x]])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = -\frac {2}{a d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2}{a d \sqrt {a+a \sin (c+d x)}} \]

[In]

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

[Out]

-2/(a*d*Sqrt[a + a*Sin[c + d*x]])

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(-\frac {2}{a d \sqrt {a +a \sin \left (d x +c \right )}}\)	\(21\)
default	\(-\frac {2}{a d \sqrt {a +a \sin \left (d x +c \right )}}\)	\(21\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\begin {cases} - \frac {2}{a d \sqrt {a \sin {\left (c + d x \right )} + a}} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2/(a*d*sqrt(a*sin(c + d*x) + a)), Ne(d, 0)), (x*cos(c)/(a*sin(c) + a)**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2}{\sqrt {a \sin \left (d x + c\right ) + a} a d} \]

[In]

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

[Out]

-2/(sqrt(a*sin(d*x + c) + a)*a*d)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\sqrt {2}}{a^{\frac {3}{2}} d \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \]

[In]

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

[Out]

-sqrt(2)/(a^(3/2)*d*cos(-1/4*pi + 1/2*d*x + 1/2*c)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))

Mupad [B] (verification not implemented)

Time = 4.61 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.27 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (c+d\,x\right )+1\right )}{a^2\,d\,\left (2\,{\sin \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )+2\right )} \]

[In]

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

[Out]

-(4*(a*(sin(c + d*x) + 1))^(1/2)*(sin(c + d*x) + 1))/(a^2*d*(4*sin(c + d*x) + 2*sin(c + d*x)^2 + 2))