3.2.62 \(\int \frac {\cos (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\) [162]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {2746, 32} \begin {gather*} \frac {2 \sqrt {a \sin (c+d x)+a}}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + a*Sin[c + d*x]])/(a*d)

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]

Rule 2746

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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a+a \sin (c+d x)}}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + a*Sin[c + d*x]])/(a*d)

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 21, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {2 \sqrt {a +a \sin \left (d x +c \right )}}{d a}\)	\(21\)
default	\(\frac {2 \sqrt {a +a \sin \left (d x +c \right )}}{d a}\)	\(21\)
risch	\(-\frac {i \sqrt {2}\, {\mathrm e}^{-i \left (d x +c \right )} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2}}{\sqrt {-a \left (-2-2 \sin \left (d x +c \right )\right )}\, d}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 20, normalized size = 0.91 \begin {gather*} \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a*sin(d*x + c) + a)/(a*d)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 20, normalized size = 0.91 \begin {gather*} \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a*sin(d*x + c) + a)/(a*d)

________________________________________________________________________________________

Sympy [A]
time = 0.46, size = 32, normalized size = 1.45 \begin {gather*} \begin {cases} \frac {2 \sqrt {a \sin {\left (c + d x \right )} + a}}{a d} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\sqrt {a \sin {\left (c \right )} + a}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 5.00, size = 38, normalized size = 1.73 \begin {gather*} \frac {2 \, \sqrt {2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2)*cos(-1/4*pi + 1/2*d*x + 1/2*c)/(sqrt(a)*d*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))

________________________________________________________________________________________

Mupad [B]
time = 4.82, size = 20, normalized size = 0.91 \begin {gather*} \frac {2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}}{a\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________