\(\int \sec ^3(\frac {\pi }{4}+3 x) \, dx\) [340]

Optimal result

Integrand size = 12, antiderivative size = 40 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {1}{6} \text {arctanh}\left (\sin \left (\frac {\pi }{4}+3 x\right )\right )+\frac {1}{6} \sec \left (\frac {\pi }{4}+3 x\right ) \tan \left (\frac {\pi }{4}+3 x\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3853, 3855} \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {1}{6} \text {arctanh}\left (\sin \left (3 x+\frac {\pi }{4}\right )\right )+\frac {1}{6} \tan \left (3 x+\frac {\pi }{4}\right ) \sec \left (3 x+\frac {\pi }{4}\right ) \]

[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 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = \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} \text {arctanh}\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] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {1}{6} \text {arctanh}\left (\sin \left (\frac {\pi }{4}+3 x\right )\right )+\frac {1}{6} \sec \left (\frac {\pi }{4}+3 x\right ) \tan \left (\frac {\pi }{4}+3 x\right ) \]

[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

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {\sec \left (\frac {\pi }{4}+3 x \right ) \tan \left (\frac {\pi }{4}+3 x \right )}{6}+\frac {\ln \left (\sec \left (\frac {\pi }{4}+3 x \right )+\tan \left (\frac {\pi }{4}+3 x \right )\right )}{6}\)	\(40\)
default	\(\frac {\sec \left (\frac {\pi }{4}+3 x \right ) \tan \left (\frac {\pi }{4}+3 x \right )}{6}+\frac {\ln \left (\sec \left (\frac {\pi }{4}+3 x \right )+\tan \left (\frac {\pi }{4}+3 x \right )\right )}{6}\)	\(40\)
parallelrisch	\(\frac {\left (1-\sin \left (6 x \right )\right ) \ln \left (\tan \left (\frac {\pi }{8}+\frac {3 x}{2}\right )-1\right )+\left (\sin \left (6 x \right )-1\right ) \ln \left (\tan \left (\frac {\pi }{8}+\frac {3 x}{2}\right )+1\right )-2 \sin \left (\frac {\pi }{4}+3 x \right )}{6 \sin \left (6 x \right )-6}\)	\(61\)
norman	\(\frac {\frac {\left (\tan ^{3}\left (\frac {\pi }{8}+\frac {3 x}{2}\right )\right )}{3}+\frac {\tan \left (\frac {\pi }{8}+\frac {3 x}{2}\right )}{3}}{\left (\tan ^{2}\left (\frac {\pi }{8}+\frac {3 x}{2}\right )-1\right )^{2}}-\frac {\ln \left (\tan \left (\frac {\pi }{8}+\frac {3 x}{2}\right )-1\right )}{6}+\frac {\ln \left (\tan \left (\frac {\pi }{8}+\frac {3 x}{2}\right )+1\right )}{6}\)	\(66\)
risch	\(-\frac {i \left (\left (-1\right )^{\frac {3}{4}} {\mathrm e}^{9 i x}-\left (-1\right )^{\frac {1}{4}} {\mathrm e}^{3 i x}\right )}{3 \left (i {\mathrm e}^{6 i x}+1\right )^{2}}-\frac {\ln \left (\left (-1\right )^{\frac {1}{4}} {\mathrm e}^{3 i x}-i\right )}{6}+\frac {\ln \left (\left (-1\right )^{\frac {1}{4}} {\mathrm e}^{3 i x}+i\right )}{6}\)	\(67\)

[In]

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

[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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (30) = 60\).

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.75 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {\cos \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2} \log \left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \cos \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2} \log \left (-\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) + 2 \, \sin \left (\frac {1}{4} \, \pi + 3 \, x\right )}{12 \, \cos \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (29) = 58\).

Time = 0.53 (sec) , antiderivative size = 388, normalized size of antiderivative = 9.70 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=- \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} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.28 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=-\frac {\sin \left (\frac {1}{4} \, \pi + 3 \, x\right )}{6 \, {\left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2} - 1\right )}} + \frac {1}{12} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \frac {1}{12} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) - 1\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=-\frac {\sin \left (\frac {1}{4} \, \pi + 3 \, x\right )}{6 \, {\left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2} - 1\right )}} + \frac {1}{12} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \frac {1}{12} \, \log \left (-\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {\Pi }{8}+\frac {3\,x}{2}+\frac {\pi }{4}\right )\right )}{6}+\frac {\mathrm {tan}\left (\frac {\Pi }{4}+3\,x\right )}{6\,\cos \left (\frac {\Pi }{4}+3\,x\right )} \]

[In]

int(1/cos(Pi/4 + 3*x)^3,x)

[Out]

log(tan(Pi/8 + (3*x)/2 + pi/4))/6 + tan(Pi/4 + 3*x)/(6*cos(Pi/4 + 3*x))