Optimal. Leaf size=16 \[ \frac{1}{2} x \sec ^2(x)-\frac{\tan (x)}{2} \]
[Out]
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Rubi [A] time = 0.0288477, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{1}{2} x \sec ^2(x)-\frac{\tan (x)}{2} \]
Antiderivative was successfully verified.
[In] Int[x*Sec[x]^2*Tan[x],x]
[Out]
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Rubi in Sympy [A] time = 1.4015, size = 15, normalized size = 0.94 \[ \frac{x}{2 \cos ^{2}{\left (x \right )}} - \frac{\sin{\left (x \right )}}{2 \cos{\left (x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*sin(x)/cos(x)**3,x)
[Out]
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Mathematica [A] time = 0.00750552, size = 16, normalized size = 1. \[ \frac{1}{2} x \sec ^2(x)-\frac{\tan (x)}{2} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sec[x]^2*Tan[x],x]
[Out]
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Maple [A] time = 0.009, size = 13, normalized size = 0.8 \[{\frac{x}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}}}-{\frac{\tan \left ( x \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*sin(x)/cos(x)^3,x)
[Out]
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Maxima [A] time = 1.33755, size = 178, normalized size = 11.12 \[ \frac{4 \, x \cos \left (2 \, x\right )^{2} + 4 \, x \sin \left (2 \, x\right )^{2} +{\left (2 \, x \cos \left (2 \, x\right ) + \sin \left (2 \, x\right )\right )} \cos \left (4 \, x\right ) + 2 \, x \cos \left (2 \, x\right ) +{\left (2 \, x \sin \left (2 \, x\right ) - \cos \left (2 \, x\right ) - 1\right )} \sin \left (4 \, x\right ) - \sin \left (2 \, x\right )}{2 \,{\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sin(x)/cos(x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236553, size = 20, normalized size = 1.25 \[ -\frac{\cos \left (x\right ) \sin \left (x\right ) - x}{2 \, \cos \left (x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sin(x)/cos(x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.00223, size = 128, normalized size = 8. \[ \frac{x \tan ^{4}{\left (\frac{x}{2} \right )}}{2 \tan ^{4}{\left (\frac{x}{2} \right )} - 4 \tan ^{2}{\left (\frac{x}{2} \right )} + 2} + \frac{2 x \tan ^{2}{\left (\frac{x}{2} \right )}}{2 \tan ^{4}{\left (\frac{x}{2} \right )} - 4 \tan ^{2}{\left (\frac{x}{2} \right )} + 2} + \frac{x}{2 \tan ^{4}{\left (\frac{x}{2} \right )} - 4 \tan ^{2}{\left (\frac{x}{2} \right )} + 2} + \frac{2 \tan ^{3}{\left (\frac{x}{2} \right )}}{2 \tan ^{4}{\left (\frac{x}{2} \right )} - 4 \tan ^{2}{\left (\frac{x}{2} \right )} + 2} - \frac{2 \tan{\left (\frac{x}{2} \right )}}{2 \tan ^{4}{\left (\frac{x}{2} \right )} - 4 \tan ^{2}{\left (\frac{x}{2} \right )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sin(x)/cos(x)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.204393, size = 72, normalized size = 4.5 \[ \frac{x \tan \left (\frac{1}{2} \, x\right )^{4} + 2 \, x \tan \left (\frac{1}{2} \, x\right )^{2} + 2 \, \tan \left (\frac{1}{2} \, x\right )^{3} + x - 2 \, \tan \left (\frac{1}{2} \, x\right )}{2 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{4} - 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*sin(x)/cos(x)^3,x, algorithm="giac")
[Out]