3.1.53 \(\int \sqrt {a \cos ^4(x)} \, dx\) [53]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 8} \begin {gather*} \frac {1}{2} \tan (x) \sqrt {a \cos ^4(x)}+\frac {1}{2} x \sec ^2(x) \sqrt {a \cos ^4(x)} \end {gather*}

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 8

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

Rule 2715

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] && Integ
erQ[2*n]

Rule 3286

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
]])

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*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.69 \begin {gather*} \frac {1}{2} \sqrt {a \cos ^4(x)} \sec ^2(x) (x+\cos (x) \sin (x)) \end {gather*}

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 = 0.08, size = 22, normalized size = 0.61

method	result	size
default	\(\frac {\sqrt {a \left (\cos ^{4}\left (x \right )\right )}\, \left (\cos \left (x \right ) \sin \left (x \right )+x \right )}{2 \cos \left (x \right )^{2}}\)	\(22\)
risch	\(\frac {\sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}\, {\mathrm e}^{2 i x} x}{2 \left ({\mathrm e}^{2 i x}+1\right )^{2}}-\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}\, {\mathrm e}^{4 i x}}{8 \left ({\mathrm e}^{2 i x}+1\right )^{2}}+\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{4} {\mathrm e}^{-4 i x}}}{8 \left ({\mathrm e}^{2 i x}+1\right )^{2}}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 22, normalized size = 0.61 \begin {gather*} \frac {1}{2} \, \sqrt {a} x + \frac {\sqrt {a} \tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}} \end {gather*}

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 = 0.39, size = 21, normalized size = 0.58 \begin {gather*} \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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.44, size = 13, normalized size = 0.36 \begin {gather*} \frac {1}{4} \, \sqrt {a} {\left (2 \, x + \sin \left (2 \, x\right )\right )} \end {gather*}

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {a\,{\cos \left (x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*cos(x)^4)^(1/2), x)

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