3.95 \(\int x^2 \sin ^2(x) \, dx\)

Optimal. Leaf size=41 \[ \frac{x^3}{6}-\frac{1}{2} x^2 \sin (x) \cos (x)-\frac{x}{4}+\frac{1}{2} x \sin ^2(x)+\frac{1}{4} \sin (x) \cos (x) \]

[Out]

-x/4 + x^3/6 + (Cos[x]*Sin[x])/4 - (x^2*Cos[x]*Sin[x])/2 + (x*Sin[x]^2)/2

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Rubi [A]  time = 0.0477853, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{x^3}{6}-\frac{1}{2} x^2 \sin (x) \cos (x)-\frac{x}{4}+\frac{1}{2} x \sin ^2(x)+\frac{1}{4} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sin[x]^2,x]

[Out]

-x/4 + x^3/6 + (Cos[x]*Sin[x])/4 - (x^2*Cos[x]*Sin[x])/2 + (x*Sin[x]^2)/2

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Rubi in Sympy [A]  time = 1.65059, size = 36, normalized size = 0.88 \[ \frac{x^{3}}{6} - \frac{x^{2} \sin{\left (x \right )} \cos{\left (x \right )}}{2} + \frac{x \sin ^{2}{\left (x \right )}}{2} - \frac{x}{4} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/6 - x**2*sin(x)*cos(x)/2 + x*sin(x)**2/2 - x/4 + sin(x)*cos(x)/4

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Mathematica [A]  time = 0.03695, size = 29, normalized size = 0.71 \[ \frac{1}{24} \left (4 x^3+\left (3-6 x^2\right ) \sin (2 x)-6 x \cos (2 x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sin[x]^2,x]

[Out]

(4*x^3 - 6*x*Cos[2*x] + (3 - 6*x^2)*Sin[2*x])/24

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Maple [A]  time = 0.019, size = 37, normalized size = 0.9 \[{x}^{2} \left ({\frac{x}{2}}-{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \right ) -{\frac{x \left ( \cos \left ( x \right ) \right ) ^{2}}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{4}}+{\frac{x}{4}}-{\frac{{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*sin(x)^2,x)

[Out]

x^2*(1/2*x-1/2*cos(x)*sin(x))-1/2*x*cos(x)^2+1/4*cos(x)*sin(x)+1/4*x-1/3*x^3

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Maxima [A]  time = 1.34507, size = 35, normalized size = 0.85 \[ \frac{1}{6} \, x^{3} - \frac{1}{4} \, x \cos \left (2 \, x\right ) - \frac{1}{8} \,{\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^3 - 1/4*x*cos(2*x) - 1/8*(2*x^2 - 1)*sin(2*x)

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Fricas [A]  time = 0.251267, size = 39, normalized size = 0.95 \[ \frac{1}{6} \, x^{3} - \frac{1}{2} \, x \cos \left (x\right )^{2} - \frac{1}{4} \,{\left (2 \, x^{2} - 1\right )} \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{4} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^3 - 1/2*x*cos(x)^2 - 1/4*(2*x^2 - 1)*cos(x)*sin(x) + 1/4*x

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Sympy [A]  time = 0.840045, size = 56, normalized size = 1.37 \[ \frac{x^{3} \sin ^{2}{\left (x \right )}}{6} + \frac{x^{3} \cos ^{2}{\left (x \right )}}{6} - \frac{x^{2} \sin{\left (x \right )} \cos{\left (x \right )}}{2} + \frac{x \sin ^{2}{\left (x \right )}}{4} - \frac{x \cos ^{2}{\left (x \right )}}{4} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*sin(x)**2,x)

[Out]

x**3*sin(x)**2/6 + x**3*cos(x)**2/6 - x**2*sin(x)*cos(x)/2 + x*sin(x)**2/4 - x*c
os(x)**2/4 + sin(x)*cos(x)/4

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GIAC/XCAS [A]  time = 0.201769, size = 35, normalized size = 0.85 \[ \frac{1}{6} \, x^{3} - \frac{1}{4} \, x \cos \left (2 \, x\right ) - \frac{1}{8} \,{\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^3 - 1/4*x*cos(2*x) - 1/8*(2*x^2 - 1)*sin(2*x)