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

Optimal. Leaf size=24 \[ \frac {2 (a \sin (c+d x)+a)^{3/2}}{3 a d} \]

[Out]

2/3*(a+a*sin(d*x+c))^(3/2)/a/d

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Rubi [A]  time = 0.03, antiderivative size = 24, 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 = {2667, 32} \[ \frac {2 (a \sin (c+d x)+a)^{3/2}}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + a*Sin[c + d*x])^(3/2))/(3*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 2667

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

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Mathematica [A]  time = 0.08, size = 44, normalized size = 1.83 \[ \frac {2 \sqrt {a (\sin (c+d x)+1)} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*Sqrt[a*(1 + Sin[c + d*x])])/(3*d)

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fricas [A]  time = 0.63, size = 25, normalized size = 1.04 \[ \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sin \left (d x + c\right ) + 1\right )}}{3 \, d} \]

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]

2/3*sqrt(a*sin(d*x + c) + a)*(sin(d*x + c) + 1)/d

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giac [B]  time = 0.92, size = 68, normalized size = 2.83 \[ \frac {1}{3} \, \sqrt {2} \sqrt {a} {\left (\frac {3 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {\mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d}\right )} \]

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]

1/3*sqrt(2)*sqrt(a)*(3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(1/4*pi + 1/2*d*x + 1/2*c)/d + sgn(cos(-1/4*pi +
1/2*d*x + 1/2*c))*sin(-1/4*pi + 3/2*d*x + 3/2*c)/d)

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maple [A]  time = 0.04, size = 21, normalized size = 0.88 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d a} \]

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)

[Out]

2/3*(a+a*sin(d*x+c))^(3/2)/d/a

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maxima [A]  time = 0.67, size = 20, normalized size = 0.83 \[ \frac {2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{3 \, a d} \]

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]

2/3*(a*sin(d*x + c) + a)^(3/2)/(a*d)

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mupad [B]  time = 4.57, size = 20, normalized size = 0.83 \[ \frac {2\,{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^{3/2}}{3\,a\,d} \]

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))^(3/2))/(3*a*d)

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sympy [A]  time = 0.68, size = 58, normalized size = 2.42 \[ \begin {cases} \frac {2 \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )}}{3 d} + \frac {2 \sqrt {a \sin {\left (c + d x \right )} + a}}{3 d} & \text {for}\: d \neq 0 \\x \sqrt {a \sin {\relax (c )} + a} \cos {\relax (c )} & \text {otherwise} \end {cases} \]

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)*sin(c + d*x)/(3*d) + 2*sqrt(a*sin(c + d*x) + a)/(3*d), Ne(d, 0)), (x*sqr
t(a*sin(c) + a)*cos(c), True))

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