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

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

[Out]

-2/(b*d*Sqrt[a + b*Sin[c + d*x]])

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Rubi [A]  time = 0.0377284, 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 = {2668, 32} \[ -\frac{2}{b d \sqrt{a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(b*d*Sqrt[a + b*Sin[c + d*x]])

Rule 2668

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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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

Mathematica [A]  time = 0.0140742, size = 22, normalized size = 1. \[ -\frac{2}{b d \sqrt{a+b \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(b*d*Sqrt[a + b*Sin[c + d*x]])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.945956, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\sqrt{b \sin \left (d x + c\right ) + a} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(b*sin(d*x + c) + a)*b*d)

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Fricas [A]  time = 1.93253, size = 78, normalized size = 3.55 \begin{align*} -\frac{2 \, \sqrt{b \sin \left (d x + c\right ) + a}}{b^{2} d \sin \left (d x + c\right ) + a b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.5954, size = 56, normalized size = 2.55 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{a^{\frac{3}{2}}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \cos{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{\frac{3}{2}}} & \text{for}\: d = 0 \\\frac{\sin{\left (c + d x \right )}}{a^{\frac{3}{2}} d} & \text{for}\: b = 0 \\- \frac{2}{b d \sqrt{a + b \sin{\left (c + d x \right )}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08248, size = 27, normalized size = 1.23 \begin{align*} -\frac{2}{\sqrt{b \sin \left (d x + c\right ) + a} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(b*sin(d*x + c) + a)*b*d)