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

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 = 0.039172, 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, Rules used = {2611, 3768, 3770} \[ -\frac{1}{4} \tanh ^{-1}\left (\sin \left (\frac{x}{2}+\frac{\pi }{4}\right )\right )+\frac{1}{2} \tan \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec ^3\left (\frac{x}{2}+\frac{\pi }{4}\right )-\frac{1}{4} \tan \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec \left (\frac{x}{2}+\frac{\pi }{4}\right ) \]

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

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sec ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \tan ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx &=\frac{1}{2} \sec ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \tan \left (\frac{\pi }{4}+\frac{x}{2}\right )-\frac{1}{4} \int \sec ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx\\ &=-\frac{1}{4} \sec \left (\frac{\pi }{4}+\frac{x}{2}\right ) \tan \left (\frac{\pi }{4}+\frac{x}{2}\right )+\frac{1}{2} \sec ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \tan \left (\frac{\pi }{4}+\frac{x}{2}\right )-\frac{1}{8} \int \csc \left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx\\ &=-\frac{1}{4} \tanh ^{-1}\left (\sin \left (\frac{\pi }{4}+\frac{x}{2}\right )\right )-\frac{1}{4} \sec \left (\frac{\pi }{4}+\frac{x}{2}\right ) \tan \left (\frac{\pi }{4}+\frac{x}{2}\right )+\frac{1}{2} \sec ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \tan \left (\frac{\pi }{4}+\frac{x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0735997, size = 74, normalized size = 0.97 \[ -\frac{1}{4} \tanh ^{-1}\left (\sin \left (\frac{x}{2}+\frac{\pi }{4}\right )\right )+\frac{1}{2} \sin \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec ^4\left (\frac{1}{4} (2 x+\pi )\right )-\frac{1}{4} \sin \left (\frac{x}{2}+\frac{\pi }{4}\right ) \sec ^2\left (\frac{1}{4} (2 x+\pi )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[Pi/4 + x/2]^3*Tan[Pi/4 + x/2]^2,x]

[Out]

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

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Maple [A]  time = 0.023, size = 76, normalized size = 1. \begin{align*}{\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 ) } \end{align*}

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 = 0.947992, size = 100, normalized size = 1.32 \begin{align*} \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 ) \end{align*}

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 = 2.6527, size = 252, normalized size = 3.32 \begin{align*} -\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}} \end{align*}

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. \begin{align*} \int \tan ^{2}{\left (\frac{x}{2} + \frac{\pi }{4} \right )} \sec ^{3}{\left (\frac{x}{2} + \frac{\pi }{4} \right )}\, dx \end{align*}

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 [A]  time = 1.18049, size = 128, normalized size = 1.68 \begin{align*} \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} \, \log \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} \, \log \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 ) \end{align*}

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