3.54 \(\int \frac {1}{\sqrt {a \cos ^4(x)}} \, dx\)

Optimal. Leaf size=15 \[ \frac {\sin (x) \cos (x)}{\sqrt {a \cos ^4(x)}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 15, 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 = {3207, 3767, 8} \[ \frac {\sin (x) \cos (x)}{\sqrt {a \cos ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Cos[x]^4],x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a \cos ^4(x)}} \, dx &=\frac {\cos ^2(x) \int \sec ^2(x) \, dx}{\sqrt {a \cos ^4(x)}}\\ &=-\frac {\cos ^2(x) \operatorname {Subst}(\int 1 \, dx,x,-\tan (x))}{\sqrt {a \cos ^4(x)}}\\ &=\frac {\cos (x) \sin (x)}{\sqrt {a \cos ^4(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\sin (x) \cos (x)}{\sqrt {a \cos ^4(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Cos[x]^4],x]

[Out]

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

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fricas [A]  time = 0.49, size = 18, normalized size = 1.20 \[ \frac {\sqrt {a \cos \relax (x)^{4}} \sin \relax (x)}{a \cos \relax (x)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a*cos(x)^4)*sin(x)/(a*cos(x)^3)

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giac [A]  time = 0.28, size = 6, normalized size = 0.40 \[ \frac {\tan \relax (x)}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)/sqrt(a)

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maple [A]  time = 0.08, size = 14, normalized size = 0.93 \[ \frac {\cos \relax (x ) \sin \relax (x )}{\sqrt {a \left (\cos ^{4}\relax (x )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 6, normalized size = 0.40 \[ \frac {\tan \relax (x)}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)/sqrt(a)

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mupad [B]  time = 0.23, size = 6, normalized size = 0.40 \[ \frac {\mathrm {tan}\relax (x)}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)/a^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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