Optimal. Leaf size=73 \[ -\frac{\sin (a+b x) \cos (a+b x)}{4 b^3}+\frac{x \cos ^2(a+b x)}{2 b^2}+\frac{x^2 \sin (a+b x) \cos (a+b x)}{2 b}-\frac{x}{4 b^2}+\frac{x^3}{6} \]
[Out]
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Rubi [A] time = 0.0753563, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\sin (a+b x) \cos (a+b x)}{4 b^3}+\frac{x \cos ^2(a+b x)}{2 b^2}+\frac{x^2 \sin (a+b x) \cos (a+b x)}{2 b}-\frac{x}{4 b^2}+\frac{x^3}{6} \]
Antiderivative was successfully verified.
[In] Int[x^2*Cos[a + b*x]^2,x]
[Out]
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Rubi in Sympy [A] time = 2.71277, size = 65, normalized size = 0.89 \[ \frac{x^{3}}{6} + \frac{x^{2} \sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{2 b} + \frac{x \cos ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac{x}{4 b^{2}} - \frac{\sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{4 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*cos(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.134142, size = 47, normalized size = 0.64 \[ \frac{\left (6 b^2 x^2-3\right ) \sin (2 (a+b x))+6 b x \cos (2 (a+b x))+4 b^3 x^3}{24 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*Cos[a + b*x]^2,x]
[Out]
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Maple [B] time = 0.007, size = 158, normalized size = 2.2 \[{\frac{1}{{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \left ({\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) +{\frac{ \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{2}}-{\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{4}}-{\frac{bx}{4}}-{\frac{a}{4}}-{\frac{ \left ( bx+a \right ) ^{3}}{3}}-2\,a \left ( \left ( bx+a \right ) \left ( 1/2\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) +1/2\,bx+a/2 \right ) -1/4\, \left ( bx+a \right ) ^{2}-1/4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) +{a}^{2} \left ({\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*cos(b*x+a)^2,x)
[Out]
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Maxima [A] time = 1.42097, size = 153, normalized size = 2.1 \[ \frac{4 \,{\left (b x + a\right )}^{3} + 6 \,{\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} - 6 \,{\left (2 \,{\left (b x + a\right )}^{2} + 2 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} a + 6 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + 3 \,{\left (2 \,{\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )}{24 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*cos(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229324, size = 73, normalized size = 1. \[ \frac{2 \, b^{3} x^{3} + 6 \, b x \cos \left (b x + a\right )^{2} + 3 \,{\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 3 \, b x}{12 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*cos(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.6014, size = 105, normalized size = 1.44 \[ \begin{cases} \frac{x^{3} \sin ^{2}{\left (a + b x \right )}}{6} + \frac{x^{3} \cos ^{2}{\left (a + b x \right )}}{6} + \frac{x^{2} \sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{2 b} - \frac{x \sin ^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac{x \cos ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac{\sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{4 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \cos ^{2}{\left (a \right )}}{3} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*cos(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.20154, size = 61, normalized size = 0.84 \[ \frac{1}{6} \, x^{3} + \frac{x \cos \left (2 \, b x + 2 \, a\right )}{4 \, b^{2}} + \frac{{\left (2 \, b^{2} x^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2*cos(b*x + a)^2,x, algorithm="giac")
[Out]