3.2.89 \(\int \frac {\sin (a+b x)}{\sqrt {d \cos (a+b x)}} \, dx\) [189]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {2645, 30} \begin {gather*} -\frac {2 \sqrt {d \cos (a+b x)}}{b d} \end {gather*}

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 30

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

Rule 2645

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])

Rubi steps

\begin {align*} \int \frac {\sin (a+b x)}{\sqrt {d \cos (a+b x)}} \, dx &=-\frac {\text {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 = 0.01, size = 20, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {d \cos (a+b x)}}{b d} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 19, normalized size = 0.95

method	result	size
derivativedivides	\(-\frac {2 \sqrt {d \cos \left (b x +a \right )}}{b d}\)	\(19\)
default	\(-\frac {2 \sqrt {d \cos \left (b x +a \right )}}{b d}\)	\(19\)
risch	\(-\frac {2 \cos \left (b x +a \right )}{\sqrt {d \cos \left (b x +a \right )}\, b}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{b d} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{b d} \end {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 0.70, size = 36, normalized size = 1.80 \begin {gather*} \begin {cases} - \frac {2 \cos {\left (a + b x \right )}}{b \sqrt {d \cos {\left (a + b x \right )}}} & \text {for}\: b \neq 0 \\\frac {x \sin {\left (a \right )}}{\sqrt {d \cos {\left (a \right )}}} & \text {otherwise} \end {cases} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 4.62, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{b d} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.53, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2\,\sqrt {d\,\cos \left (a+b\,x\right )}}{b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(d*cos(a + b*x))^(1/2))/(b*d)

________________________________________________________________________________________