3.362 \(\int \sec ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \tan ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx\)

Optimal. Leaf size=76 \[ -\frac{1}{4} \tanh ^{-1}\left (\sin \left (\frac{x}{2}+\frac{\pi }{4}\right )\right )+\frac{1}{2} \tan \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec ^3\left (\frac{x}{2}+\frac{\pi }{4}\right )-\frac{1}{4} \tan \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec \left (\frac{x}{2}+\frac{\pi }{4}\right ) \]

[Out]

-ArcTanh[Sin[Pi/4 + x/2]]/4 - (Sec[Pi/4 + x/2]*Tan[Pi/4 + x/2])/4 + (Sec[Pi/4 +
x/2]^3*Tan[Pi/4 + x/2])/2

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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]

[Out]

-ArcTanh[Sin[Pi/4 + x/2]]/4 - (Sec[Pi/4 + x/2]*Tan[Pi/4 + x/2])/4 + (Sec[Pi/4 +
x/2]^3*Tan[Pi/4 + x/2])/2

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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]

Timed 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]

[Out]

(2*Log[Cos[(Pi + 2*x)/8] - Sin[(Pi + 2*x)/8]] - 2*Log[Cos[(Pi + 2*x)/8] + Sin[(P
i + 2*x)/8]] + Sec[(Pi + 2*x)/4]^3*(3 + Sin[x])*Tan[(Pi + 2*x)/4])/8

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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)

[Out]

1/2*sin(1/4*Pi+1/2*x)^3/cos(1/4*Pi+1/2*x)^4+1/4*sin(1/4*Pi+1/2*x)^3/cos(1/4*Pi+1
/2*x)^2+1/4*sin(1/4*Pi+1/2*x)-1/4*ln(sec(1/4*Pi+1/2*x)+tan(1/4*Pi+1/2*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")

[Out]

1/4*(sin(1/4*pi + 1/2*x)^3 + sin(1/4*pi + 1/2*x))/(sin(1/4*pi + 1/2*x)^4 - 2*sin
(1/4*pi + 1/2*x)^2 + 1) - 1/8*log(sin(1/4*pi + 1/2*x) + 1) + 1/8*log(sin(1/4*pi
+ 1/2*x) - 1)

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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]

-1/8*(cos(1/4*pi + 1/2*x)^4*log(sin(1/4*pi + 1/2*x) + 1) - cos(1/4*pi + 1/2*x)^4
*log(-sin(1/4*pi + 1/2*x) + 1) + 2*(cos(1/4*pi + 1/2*x)^2 - 2)*sin(1/4*pi + 1/2*
x))/cos(1/4*pi + 1/2*x)^4

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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]

Integral(tan(x/2 + pi/4)**2*sec(x/2 + pi/4)**3, x)

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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]

1/4*(1/sin(1/4*pi + 1/2*x) + sin(1/4*pi + 1/2*x))/((1/sin(1/4*pi + 1/2*x) + sin(
1/4*pi + 1/2*x))^2 - 4) - 1/16*ln(abs(1/sin(1/4*pi + 1/2*x) + sin(1/4*pi + 1/2*x
) + 2)) + 1/16*ln(abs(1/sin(1/4*pi + 1/2*x) + sin(1/4*pi + 1/2*x) - 2))