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

Optimal. Leaf size=22 \[ -\frac{2}{a d \sqrt{a \sin (c+d x)+a}} \]

[Out]

-2/(a*d*Sqrt[a + a*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0335983, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 32} \[ -\frac{2}{a d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[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 2667

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])

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]

Rubi steps

\begin{align*} \int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{\operatorname{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]  time = 0.0299117, size = 22, normalized size = 1. \[ -\frac{2}{a d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[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]  time = 0.004, size = 21, normalized size = 1. \begin{align*} -2\,{\frac{1}{da\sqrt{a+a\sin \left ( dx+c \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.948987, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\sqrt{a \sin \left (d x + c\right ) + a} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 2.23214, size = 78, normalized size = 3.55 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 2.86371, size = 46, normalized size = 2.09 \begin{align*} \begin{cases} \text{NaN} & \text{for}\: c = \frac{3 \pi }{2} \wedge d = 0 \\\frac{x \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{\frac{3}{2}}} & \text{for}\: d = 0 \\\text{NaN} & \text{for}\: c = - d x + \frac{3 \pi }{2} \\- \frac{2}{a d \sqrt{a \sin{\left (c + d x \right )} + a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((nan, Eq(d, 0) & Eq(c, 3*pi/2)), (x*cos(c)/(a*sin(c) + a)**(3/2), Eq(d, 0)), (nan, Eq(c, -d*x + 3*pi
/2)), (-2/(a*d*sqrt(a*sin(c + d*x) + a)), True))

________________________________________________________________________________________

Giac [A]  time = 1.13427, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\sqrt{a \sin \left (d x + c\right ) + a} a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(a*sin(d*x + c) + a)*a*d)