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

Optimal. Leaf size=20 \[ -\frac{2 \sqrt{d \cos (a+b x)}}{b d} \]

[Out]

(-2*Sqrt[d*Cos[a + b*x]])/(b*d)

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Rubi [A]  time = 0.0232669, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2565, 30} \[ -\frac{2 \sqrt{d \cos (a+b x)}}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d*Cos[a + b*x]])/(b*d)

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0145341, size = 20, normalized size = 1. \[ -\frac{2 \sqrt{d \cos (a+b x)}}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d*Cos[a + b*x]])/(b*d)

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Maple [A]  time = 0.008, size = 19, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{d\cos \left ( bx+a \right ) }}{bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.983249, size = 24, normalized size = 1.2 \begin{align*} -\frac{2 \, \sqrt{d \cos \left (b x + a\right )}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

-2*sqrt(d*cos(b*x + a))/(b*d)

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Fricas [A]  time = 1.90865, size = 42, normalized size = 2.1 \begin{align*} -\frac{2 \, \sqrt{d \cos \left (b x + a\right )}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(d*cos(b*x + a))/(b*d)

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Sympy [A]  time = 1.23599, size = 32, normalized size = 1.6 \begin{align*} \begin{cases} - \frac{2 \sqrt{\cos{\left (a + b x \right )}}}{b \sqrt{d}} & \text{for}\: b \neq 0 \\\frac{x \sin{\left (a \right )}}{\sqrt{d \cos{\left (a \right )}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))**(1/2),x)

[Out]

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

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Giac [A]  time = 1.20944, size = 24, normalized size = 1.2 \begin{align*} -\frac{2 \, \sqrt{d \cos \left (b x + a\right )}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)/(d*cos(b*x+a))^(1/2),x, algorithm="giac")

[Out]

-2*sqrt(d*cos(b*x + a))/(b*d)