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

Optimal. Leaf size=40 \[ \frac{1}{6} \tanh ^{-1}\left (\sin \left (3 x+\frac{\pi }{4}\right )\right )+\frac{1}{6} \tan \left (3 x+\frac{\pi }{4}\right ) \sec \left (3 x+\frac{\pi }{4}\right ) \]

[Out]

ArcTanh[Sin[Pi/4 + 3*x]]/6 + (Sec[Pi/4 + 3*x]*Tan[Pi/4 + 3*x])/6

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Rubi [A]  time = 0.0123502, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3768, 3770} \[ \frac{1}{6} \tanh ^{-1}\left (\sin \left (3 x+\frac{\pi }{4}\right )\right )+\frac{1}{6} \tan \left (3 x+\frac{\pi }{4}\right ) \sec \left (3 x+\frac{\pi }{4}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sec[Pi/4 + 3*x]^3,x]

[Out]

ArcTanh[Sin[Pi/4 + 3*x]]/6 + (Sec[Pi/4 + 3*x]*Tan[Pi/4 + 3*x])/6

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}+3 x\right ) \, dx &=\frac{1}{6} \sec \left (\frac{\pi }{4}+3 x\right ) \tan \left (\frac{\pi }{4}+3 x\right )+\frac{1}{2} \int \csc \left (\frac{\pi }{4}-3 x\right ) \, dx\\ &=\frac{1}{6} \tanh ^{-1}\left (\sin \left (\frac{\pi }{4}+3 x\right )\right )+\frac{1}{6} \sec \left (\frac{\pi }{4}+3 x\right ) \tan \left (\frac{\pi }{4}+3 x\right )\\ \end{align*}

Mathematica [A]  time = 0.0119868, size = 40, normalized size = 1. \[ \frac{1}{6} \tanh ^{-1}\left (\sin \left (3 x+\frac{\pi }{4}\right )\right )+\frac{1}{6} \tan \left (3 x+\frac{\pi }{4}\right ) \sec \left (3 x+\frac{\pi }{4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[Pi/4 + 3*x]^3,x]

[Out]

ArcTanh[Sin[Pi/4 + 3*x]]/6 + (Sec[Pi/4 + 3*x]*Tan[Pi/4 + 3*x])/6

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Maple [A]  time = 0.036, size = 40, normalized size = 1. \begin{align*}{\frac{1}{6}\sec \left ({\frac{\pi }{4}}+3\,x \right ) \tan \left ({\frac{\pi }{4}}+3\,x \right ) }+{\frac{1}{6}\ln \left ( \sec \left ({\frac{\pi }{4}}+3\,x \right ) +\tan \left ({\frac{\pi }{4}}+3\,x \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(1/4*Pi+3*x)^3,x)

[Out]

1/6*sec(1/4*Pi+3*x)*tan(1/4*Pi+3*x)+1/6*ln(sec(1/4*Pi+3*x)+tan(1/4*Pi+3*x))

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Maxima [A]  time = 0.92955, size = 69, normalized size = 1.72 \begin{align*} -\frac{\sin \left (\frac{1}{4} \, \pi + 3 \, x\right )}{6 \,{\left (\sin \left (\frac{1}{4} \, \pi + 3 \, x\right )^{2} - 1\right )}} + \frac{1}{12} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \frac{1}{12} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 3 \, x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(1/4*pi+3*x)^3,x, algorithm="maxima")

[Out]

-1/6*sin(1/4*pi + 3*x)/(sin(1/4*pi + 3*x)^2 - 1) + 1/12*log(sin(1/4*pi + 3*x) + 1) - 1/12*log(sin(1/4*pi + 3*x
) - 1)

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Fricas [B]  time = 1.60991, size = 198, normalized size = 4.95 \begin{align*} \frac{\cos \left (\frac{1}{4} \, \pi + 3 \, x\right )^{2} \log \left (\sin \left (\frac{1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \cos \left (\frac{1}{4} \, \pi + 3 \, x\right )^{2} \log \left (-\sin \left (\frac{1}{4} \, \pi + 3 \, x\right ) + 1\right ) + 2 \, \sin \left (\frac{1}{4} \, \pi + 3 \, x\right )}{12 \, \cos \left (\frac{1}{4} \, \pi + 3 \, x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(1/4*pi+3*x)^3,x, algorithm="fricas")

[Out]

1/12*(cos(1/4*pi + 3*x)^2*log(sin(1/4*pi + 3*x) + 1) - cos(1/4*pi + 3*x)^2*log(-sin(1/4*pi + 3*x) + 1) + 2*sin
(1/4*pi + 3*x))/cos(1/4*pi + 3*x)^2

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Sympy [B]  time = 1.29094, size = 388, normalized size = 9.7 \begin{align*} - \frac{\log{\left (\tan{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 1 \right )} \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )}}{6 \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 12 \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 6} + \frac{2 \log{\left (\tan{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 1 \right )} \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )}}{6 \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 12 \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 6} - \frac{\log{\left (\tan{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 1 \right )}}{6 \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 12 \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 6} + \frac{\log{\left (\tan{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 1 \right )} \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )}}{6 \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 12 \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 6} - \frac{2 \log{\left (\tan{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 1 \right )} \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )}}{6 \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 12 \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 6} + \frac{\log{\left (\tan{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 1 \right )}}{6 \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 12 \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 6} + \frac{2 \tan ^{3}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )}}{6 \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 12 \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 6} + \frac{2 \tan{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )}}{6 \tan ^{4}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} - 12 \tan ^{2}{\left (\frac{3 x}{2} + \frac{\pi }{8} \right )} + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(1/4*pi+3*x)**3,x)

[Out]

-log(tan(3*x/2 + pi/8) - 1)*tan(3*x/2 + pi/8)**4/(6*tan(3*x/2 + pi/8)**4 - 12*tan(3*x/2 + pi/8)**2 + 6) + 2*lo
g(tan(3*x/2 + pi/8) - 1)*tan(3*x/2 + pi/8)**2/(6*tan(3*x/2 + pi/8)**4 - 12*tan(3*x/2 + pi/8)**2 + 6) - log(tan
(3*x/2 + pi/8) - 1)/(6*tan(3*x/2 + pi/8)**4 - 12*tan(3*x/2 + pi/8)**2 + 6) + log(tan(3*x/2 + pi/8) + 1)*tan(3*
x/2 + pi/8)**4/(6*tan(3*x/2 + pi/8)**4 - 12*tan(3*x/2 + pi/8)**2 + 6) - 2*log(tan(3*x/2 + pi/8) + 1)*tan(3*x/2
+ pi/8)**2/(6*tan(3*x/2 + pi/8)**4 - 12*tan(3*x/2 + pi/8)**2 + 6) + log(tan(3*x/2 + pi/8) + 1)/(6*tan(3*x/2 +
pi/8)**4 - 12*tan(3*x/2 + pi/8)**2 + 6) + 2*tan(3*x/2 + pi/8)**3/(6*tan(3*x/2 + pi/8)**4 - 12*tan(3*x/2 + pi/
8)**2 + 6) + 2*tan(3*x/2 + pi/8)/(6*tan(3*x/2 + pi/8)**4 - 12*tan(3*x/2 + pi/8)**2 + 6)

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Giac [A]  time = 1.06233, size = 72, normalized size = 1.8 \begin{align*} -\frac{\sin \left (\frac{1}{4} \, \pi + 3 \, x\right )}{6 \,{\left (\sin \left (\frac{1}{4} \, \pi + 3 \, x\right )^{2} - 1\right )}} + \frac{1}{12} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \frac{1}{12} \, \log \left (-\sin \left (\frac{1}{4} \, \pi + 3 \, x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cos(1/4*pi+3*x)^3,x, algorithm="giac")

[Out]

-1/6*sin(1/4*pi + 3*x)/(sin(1/4*pi + 3*x)^2 - 1) + 1/12*log(sin(1/4*pi + 3*x) + 1) - 1/12*log(-sin(1/4*pi + 3*
x) + 1)