3.77 \(\int x \sin ^3\left (x^2\right ) \, dx\)

Optimal. Leaf size=19 \[ \frac{1}{6} \cos ^3\left (x^2\right )-\frac{\cos \left (x^2\right )}{2} \]

[Out]

-Cos[x^2]/2 + Cos[x^2]^3/6

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Rubi [A]  time = 0.0222967, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{1}{6} \cos ^3\left (x^2\right )-\frac{\cos \left (x^2\right )}{2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sin[x^2]^3,x]

[Out]

-Cos[x^2]/2 + Cos[x^2]^3/6

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Rubi in Sympy [A]  time = 1.35506, size = 14, normalized size = 0.74 \[ \frac{\cos ^{3}{\left (x^{2} \right )}}{6} - \frac{\cos{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*sin(x**2)**3,x)

[Out]

cos(x**2)**3/6 - cos(x**2)/2

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Mathematica [A]  time = 0.00472583, size = 19, normalized size = 1. \[ \frac{1}{24} \cos \left (3 x^2\right )-\frac{3 \cos \left (x^2\right )}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sin[x^2]^3,x]

[Out]

(-3*Cos[x^2])/8 + Cos[3*x^2]/24

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Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[ -{\frac{ \left ( 2+ \left ( \sin \left ({x}^{2} \right ) \right ) ^{2} \right ) \cos \left ({x}^{2} \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*sin(x^2)^3,x)

[Out]

-1/6*(2+sin(x^2)^2)*cos(x^2)

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Maxima [A]  time = 1.35989, size = 20, normalized size = 1.05 \[ \frac{1}{24} \, \cos \left (3 \, x^{2}\right ) - \frac{3}{8} \, \cos \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x^2)^3,x, algorithm="maxima")

[Out]

1/24*cos(3*x^2) - 3/8*cos(x^2)

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Fricas [A]  time = 0.227814, size = 20, normalized size = 1.05 \[ \frac{1}{6} \, \cos \left (x^{2}\right )^{3} - \frac{1}{2} \, \cos \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x^2)^3,x, algorithm="fricas")

[Out]

1/6*cos(x^2)^3 - 1/2*cos(x^2)

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Sympy [A]  time = 0.822632, size = 22, normalized size = 1.16 \[ - \frac{\sin ^{2}{\left (x^{2} \right )} \cos{\left (x^{2} \right )}}{2} - \frac{\cos ^{3}{\left (x^{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x**2)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.200703, size = 20, normalized size = 1.05 \[ \frac{1}{6} \, \cos \left (x^{2}\right )^{3} - \frac{1}{2} \, \cos \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*sin(x^2)^3,x, algorithm="giac")

[Out]

1/6*cos(x^2)^3 - 1/2*cos(x^2)