3.508 \(\int \frac{\cos (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx\)

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

[Out]

(2*Sqrt[a + b*Sin[c + d*x]])/(b*d)

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Rubi [A]  time = 0.0353723, 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} \[ \frac{2 \sqrt{a+b \sin (c+d x)}}{b d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

(2*Sqrt[a + b*Sin[c + d*x]])/(b*d)

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

Mathematica [A]  time = 0.0134846, size = 22, normalized size = 1. \[ \frac{2 \sqrt{a+b \sin (c+d x)}}{b d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

(2*Sqrt[a + b*Sin[c + d*x]])/(b*d)

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Maple [A]  time = 0.005, size = 21, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{a+b\sin \left ( dx+c \right ) }}{bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)/(a+b*sin(d*x+c))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.964226, 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))^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(b*sin(d*x + c) + a)/(b*d)

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Fricas [A]  time = 1.95857, size = 46, normalized size = 2.09 \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))^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(b*sin(d*x + c) + a)/(b*d)

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Sympy [A]  time = 1.02259, size = 54, normalized size = 2.45 \begin{align*} \begin{cases} \frac{x \cos{\left (c \right )}}{\sqrt{a}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \cos{\left (c \right )}}{\sqrt{a + b \sin{\left (c \right )}}} & \text{for}\: d = 0 \\\frac{\sin{\left (c + d x \right )}}{\sqrt{a} d} & \text{for}\: b = 0 \\\frac{2 \sqrt{a + b \sin{\left (c + d x \right )}}}{b d} & \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))**(1/2),x)

[Out]

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

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Giac [A]  time = 1.09504, 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))^(1/2),x, algorithm="giac")

[Out]

2*sqrt(b*sin(d*x + c) + a)/(b*d)