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

Optimal. Leaf size=24 \[ \frac{x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0025662, antiderivative size = 24, 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, 8} \[ \frac{x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[b*Cos[c + d*x]])/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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx &=\frac{\sqrt{b \cos (c+d x)} \int 1 \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0108263, size = 24, normalized size = 1. \[ \frac{x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.188, size = 28, normalized size = 1.2 \begin{align*}{\frac{dx+c}{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((b*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.5425, size = 35, normalized size = 1.46 \begin{align*} \frac{2 \, \sqrt{b} \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.86571, size = 265, normalized size = 11.04 \begin{align*} \left [\frac{\sqrt{-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt{b \cos \left (d x + c\right )} \sqrt{-b} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{2 \, d}, \frac{\sqrt{b} \arctan \left (\frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt{b} \cos \left (d x + c\right )^{\frac{3}{2}}}\right )}{d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(-b)*log(2*b*cos(d*x + c)^2 - 2*sqrt(b*cos(d*x + c))*sqrt(-b)*sqrt(cos(d*x + c))*sin(d*x + c) - b)/d,
sqrt(b)*arctan(sqrt(b*cos(d*x + c))*sin(d*x + c)/(sqrt(b)*cos(d*x + c)^(3/2)))/d]

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Sympy [A]  time = 3.87443, size = 5, normalized size = 0.21 \begin{align*} \sqrt{b} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*x

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \cos \left (d x + c\right )}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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