3.143 \(\int \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \, dx\)

Optimal. Leaf size=32 \[ \frac{\sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]

[Out]

(Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])

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Rubi [A]  time = 0.0067261, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {17, 2637} \[ \frac{\sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])

Rule 17

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + 1/2)*b^(n - 1/2)*Sqrt[b*v])/Sqrt[a*v]
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] &&  !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \, dx &=\frac{\sqrt{b \cos (c+d x)} \int \cos (c+d x) \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{\sqrt{b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0281434, size = 32, normalized size = 1. \[ \frac{\sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])

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Maple [A]  time = 0.269, size = 29, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d}\sqrt{b\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.7803, size = 18, normalized size = 0.56 \begin{align*} \frac{\sqrt{b} \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*sin(d*x + c)/d

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Fricas [A]  time = 1.61199, size = 78, normalized size = 2.44 \begin{align*} \frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{d \sqrt{\cos \left (d x + c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*cos(d*x + c))*sin(d*x + c)/(d*sqrt(cos(d*x + c)))

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Sympy [A]  time = 20.8331, size = 29, normalized size = 0.91 \begin{align*} \begin{cases} \frac{\sqrt{b} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \sqrt{b \cos{\left (c \right )}} \sqrt{\cos{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 3.37655, size = 42, normalized size = 1.31 \begin{align*} \frac{2 \, \sqrt{b} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{d \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b)*tan(1/2*d*x + 1/2*c)/(d*tan(1/2*d*x + 1/2*c)^2 + d)