3.53 \(\int \sqrt{a \cos ^4(x)} \, dx\)

Optimal. Leaf size=36 \[ \frac{1}{2} \tan (x) \sqrt{a \cos ^4(x)}+\frac{1}{2} x \sec ^2(x) \sqrt{a \cos ^4(x)} \]

[Out]

(x*Sqrt[a*Cos[x]^4]*Sec[x]^2)/2 + (Sqrt[a*Cos[x]^4]*Tan[x])/2

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Rubi [A]  time = 0.0153305, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 8} \[ \frac{1}{2} \tan (x) \sqrt{a \cos ^4(x)}+\frac{1}{2} x \sec ^2(x) \sqrt{a \cos ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Cos[x]^4],x]

[Out]

(x*Sqrt[a*Cos[x]^4]*Sec[x]^2)/2 + (Sqrt[a*Cos[x]^4]*Tan[x])/2

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \sqrt{a \cos ^4(x)} \, dx &=\left (\sqrt{a \cos ^4(x)} \sec ^2(x)\right ) \int \cos ^2(x) \, dx\\ &=\frac{1}{2} \sqrt{a \cos ^4(x)} \tan (x)+\frac{1}{2} \left (\sqrt{a \cos ^4(x)} \sec ^2(x)\right ) \int 1 \, dx\\ &=\frac{1}{2} x \sqrt{a \cos ^4(x)} \sec ^2(x)+\frac{1}{2} \sqrt{a \cos ^4(x)} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0137499, size = 25, normalized size = 0.69 \[ \frac{1}{2} \sec ^2(x) \sqrt{a \cos ^4(x)} (x+\sin (x) \cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Cos[x]^4],x]

[Out]

(Sqrt[a*Cos[x]^4]*Sec[x]^2*(x + Cos[x]*Sin[x]))/2

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Maple [A]  time = 0.192, size = 22, normalized size = 0.6 \begin{align*}{\frac{\cos \left ( x \right ) \sin \left ( x \right ) +x}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}}\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cos(x)^4)^(1/2),x)

[Out]

1/2*(a*cos(x)^4)^(1/2)*(cos(x)*sin(x)+x)/cos(x)^2

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Maxima [A]  time = 2.4277, size = 30, normalized size = 0.83 \begin{align*} \frac{1}{2} \, \sqrt{a} x + \frac{\sqrt{a} \tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(x)^4)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(a)*x + 1/2*sqrt(a)*tan(x)/(tan(x)^2 + 1)

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Fricas [A]  time = 1.04887, size = 69, normalized size = 1.92 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4}}{\left (\cos \left (x\right ) \sin \left (x\right ) + x\right )}}{2 \, \cos \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(x)^4)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(a*cos(x)^4)*(cos(x)*sin(x) + x)/cos(x)^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(x)**4)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.55724, size = 18, normalized size = 0.5 \begin{align*} \frac{1}{4} \, \sqrt{a}{\left (2 \, x + \sin \left (2 \, x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(x)^4)^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(a)*(2*x + sin(2*x))