3.923 \(\int x \cos (2 x^2) \sin ^{\frac{3}{4}}(2 x^2) \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{7} \sin ^{\frac{7}{4}}\left (2 x^2\right ) \]

[Out]

Sin[2*x^2]^(7/4)/7

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Rubi [A]  time = 0.0131242, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {3441} \[ \frac{1}{7} \sin ^{\frac{7}{4}}\left (2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[2*x^2]^(7/4)/7

Rule 3441

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[Sin[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 x \cos \left (2 x^2\right ) \sin ^{\frac{3}{4}}\left (2 x^2\right ) \, dx &=\frac{1}{7} \sin ^{\frac{7}{4}}\left (2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0065822, size = 14, normalized size = 1. \[ \frac{1}{7} \sin ^{\frac{7}{4}}\left (2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[2*x^2]^(7/4)/7

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Maple [A]  time = 0.004, size = 11, normalized size = 0.8 \begin{align*}{\frac{1}{7} \left ( \sin \left ( 2\,{x}^{2} \right ) \right ) ^{{\frac{7}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*sin(2*x^2)^(7/4)

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Maxima [A]  time = 0.953281, size = 14, normalized size = 1. \begin{align*} \frac{1}{7} \, \sin \left (2 \, x^{2}\right )^{\frac{7}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*sin(2*x^2)^(7/4)

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Fricas [A]  time = 2.15446, size = 30, normalized size = 2.14 \begin{align*} \frac{1}{7} \, \sin \left (2 \, x^{2}\right )^{\frac{7}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*sin(2*x^2)^(7/4)

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Sympy [A]  time = 120.493, size = 10, normalized size = 0.71 \begin{align*} \frac{\sin ^{\frac{7}{4}}{\left (2 x^{2} \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(2*x**2)**(7/4)/7

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Giac [A]  time = 1.09325, size = 14, normalized size = 1. \begin{align*} \frac{1}{7} \, \sin \left (2 \, x^{2}\right )^{\frac{7}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*sin(2*x^2)^(7/4)