3.64 \(\int (a^2-4 \cos ^2(x))^{3/4} \sin (2 x) \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{7} \left (a^2-4 \cos ^2(x)\right )^{7/4} \]

[Out]

(a^2 - 4*Cos[x]^2)^(7/4)/7

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Rubi [A]  time = 0.0474616, antiderivative size = 19, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {12, 261} \[ \frac{1}{7} \left (a^2+4 \sin ^2(x)-4\right )^{7/4} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 - 4*Cos[x]^2)^(3/4)*Sin[2*x],x]

[Out]

(-4 + a^2 + 4*Sin[x]^2)^(7/4)/7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx &=\operatorname{Subst}\left (\int 2 x \left (-4+a^2+4 x^2\right )^{3/4} \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int x \left (-4+a^2+4 x^2\right )^{3/4} \, dx,x,\sin (x)\right )\\ &=\frac{1}{7} \left (-4+a^2+4 \sin ^2(x)\right )^{7/4}\\ \end{align*}

Mathematica [A]  time = 0.0168411, size = 19, normalized size = 1.06 \[ \frac{1}{7} \left (a^2+4 \sin ^2(x)-4\right )^{7/4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 - 4*Cos[x]^2)^(3/4)*Sin[2*x],x]

[Out]

(-4 + a^2 + 4*Sin[x]^2)^(7/4)/7

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Maple [A]  time = 0.015, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{7} \left ({a}^{2}-4\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{7}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2-4*cos(x)^2)^(3/4)*sin(2*x),x)

[Out]

1/7*(a^2-4*cos(x)^2)^(7/4)

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Maxima [A]  time = 0.941172, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{7} \,{\left (a^{2} - 4 \, \cos \left (x\right )^{2}\right )}^{\frac{7}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2-4*cos(x)^2)^(3/4)*sin(2*x),x, algorithm="maxima")

[Out]

1/7*(a^2 - 4*cos(x)^2)^(7/4)

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Fricas [A]  time = 2.42498, size = 41, normalized size = 2.28 \begin{align*} \frac{1}{7} \,{\left (a^{2} - 4 \, \cos \left (x\right )^{2}\right )}^{\frac{7}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2-4*cos(x)^2)^(3/4)*sin(2*x),x, algorithm="fricas")

[Out]

1/7*(a^2 - 4*cos(x)^2)^(7/4)

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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**2-4*cos(x)**2)**(3/4)*sin(2*x),x)

[Out]

Timed out

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Giac [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^2-4*cos(x)^2)^(3/4)*sin(2*x),x, algorithm="giac")

[Out]

Timed out