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

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Rubi [A]  time = 0.0140777, antiderivative size = 15, 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, 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 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]

Rule 8

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

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*}

Mathematica [A]  time = 0.0055221, size = 15, normalized size = 1. \[ \frac{\sin (x) \cos (x)}{\sqrt{a \cos ^4(x)}} \]

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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Maple [A]  time = 0.181, size = 14, normalized size = 0.9 \begin{align*}{\cos \left ( x \right ) \sin \left ( x \right ){\frac{1}{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{4}}}}} \end{align*}

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 = 2.05925, size = 8, normalized size = 0.53 \begin{align*} \frac{\tan \left (x\right )}{\sqrt{a}} \end{align*}

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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Fricas [A]  time = 1.02808, size = 51, normalized size = 3.4 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4}} \sin \left (x\right )}{a \cos \left (x\right )^{3}} \end{align*}

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError