3.41 \(\int \sqrt{a \cos ^2(x)} \, dx\)

Optimal. Leaf size=13 \[ \tan (x) \sqrt{a \cos ^2(x)} \]

[Out]

Sqrt[a*Cos[x]^2]*Tan[x]

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Rubi [A]  time = 0.0112618, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 2637} \[ \tan (x) \sqrt{a \cos ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Cos[x]^2],x]

[Out]

Sqrt[a*Cos[x]^2]*Tan[x]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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{a \cos ^2(x)} \, dx &=\left (\sqrt{a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx\\ &=\sqrt{a \cos ^2(x)} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0041028, size = 13, normalized size = 1. \[ \tan (x) \sqrt{a \cos ^2(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Cos[x]^2],x]

[Out]

Sqrt[a*Cos[x]^2]*Tan[x]

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Maple [A]  time = 0.337, size = 15, normalized size = 1.2 \begin{align*}{a\cos \left ( x \right ) \sin \left ( x \right ){\frac{1}{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cos(x)^2)^(1/2),x)

[Out]

1/(a*cos(x)^2)^(1/2)*a*cos(x)*sin(x)

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Maxima [A]  time = 2.32193, size = 8, normalized size = 0.62 \begin{align*} \sqrt{a} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a)*sin(x)

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Fricas [A]  time = 1.0715, size = 43, normalized size = 3.31 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{2}} \sin \left (x\right )}{\cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a*cos(x)^2)*sin(x)/cos(x)

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Sympy [A]  time = 0.576638, size = 19, normalized size = 1.46 \begin{align*} \frac{\sqrt{a} \sqrt{\cos ^{2}{\left (x \right )}} \sin{\left (x \right )}}{\cos{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(x)**2)**(1/2),x)

[Out]

sqrt(a)*sqrt(cos(x)**2)*sin(x)/cos(x)

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Giac [A]  time = 1.23228, size = 12, normalized size = 0.92 \begin{align*} \sqrt{a} \mathrm{sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a)*sgn(cos(x))*sin(x)