Optimal. Leaf size=76 \[ -\frac{1}{4} \tanh ^{-1}\left (\sin \left (\frac{x}{2}+\frac{\pi }{4}\right )\right )+\frac{1}{2} \tan \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec ^3\left (\frac{x}{2}+\frac{\pi }{4}\right )-\frac{1}{4} \tan \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec \left (\frac{x}{2}+\frac{\pi }{4}\right ) \]
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Rubi [A] time = 0.0631573, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{1}{4} \tanh ^{-1}\left (\sin \left (\frac{x}{2}+\frac{\pi }{4}\right )\right )+\frac{1}{2} \tan \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec ^3\left (\frac{x}{2}+\frac{\pi }{4}\right )-\frac{1}{4} \tan \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec \left (\frac{x}{2}+\frac{\pi }{4}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sec[Pi/4 + x/2]^3*Tan[Pi/4 + x/2]^2,x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(sec(1/4*pi+1/2*x)**3*tan(1/4*pi+1/2*x)**2,x)
[Out]
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Mathematica [A] time = 1.25496, size = 82, normalized size = 1.08 \[ \frac{1}{8} \left (2 \log \left (\cos \left (\frac{1}{8} (2 x+\pi )\right )-\sin \left (\frac{1}{8} (2 x+\pi )\right )\right )-2 \log \left (\sin \left (\frac{1}{8} (2 x+\pi )\right )+\cos \left (\frac{1}{8} (2 x+\pi )\right )\right )+(\sin (x)+3) \tan \left (\frac{1}{4} (2 x+\pi )\right ) \sec ^3\left (\frac{1}{4} (2 x+\pi )\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sec[Pi/4 + x/2]^3*Tan[Pi/4 + x/2]^2,x]
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Maple [A] time = 0.032, size = 76, normalized size = 1. \[{\frac{1}{2} \left ( \sin \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) \right ) ^{3} \left ( \cos \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) \right ) ^{-4}}+{\frac{1}{4} \left ( \sin \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) \right ) ^{3} \left ( \cos \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{4}\sin \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) }-{\frac{1}{4}\ln \left ( \sec \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) +\tan \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(sec(1/4*Pi+1/2*x)^3*tan(1/4*Pi+1/2*x)^2,x)
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Maxima [A] time = 1.48451, size = 100, normalized size = 1.32 \[ \frac{\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )^{3} + \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )}{4 \,{\left (\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )^{4} - 2 \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )^{2} + 1\right )}} - \frac{1}{8} \, \log \left (\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right ) + 1\right ) + \frac{1}{8} \, \log \left (\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right ) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sec(1/4*pi + 1/2*x)^3*tan(1/4*pi + 1/2*x)^2,x, algorithm="maxima")
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Fricas [A] time = 0.239254, size = 111, normalized size = 1.46 \[ -\frac{\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )^{4} \log \left (\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right ) + 1\right ) - \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )^{4} \log \left (-\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right ) + 1\right ) + 2 \,{\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )^{2} - 2\right )} \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )}{8 \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sec(1/4*pi + 1/2*x)^3*tan(1/4*pi + 1/2*x)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \tan ^{2}{\left (\frac{x}{2} + \frac{\pi }{4} \right )} \sec ^{3}{\left (\frac{x}{2} + \frac{\pi }{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sec(1/4*pi+1/2*x)**3*tan(1/4*pi+1/2*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.24285, size = 128, normalized size = 1.68 \[ \frac{\frac{1}{\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )} + \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )}{4 \,{\left ({\left (\frac{1}{\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )} + \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )\right )}^{2} - 4\right )}} - \frac{1}{16} \,{\rm ln}\left ({\left | \frac{1}{\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )} + \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right ) + 2 \right |}\right ) + \frac{1}{16} \,{\rm ln}\left ({\left | \frac{1}{\sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )} + \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right ) - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sec(1/4*pi + 1/2*x)^3*tan(1/4*pi + 1/2*x)^2,x, algorithm="giac")
[Out]