Integrand size = 12, antiderivative size = 40 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {1}{6} \text {arctanh}\left (\sin \left (\frac {\pi }{4}+3 x\right )\right )+\frac {1}{6} \sec \left (\frac {\pi }{4}+3 x\right ) \tan \left (\frac {\pi }{4}+3 x\right ) \]
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {1}{6} \text {arctanh}\left (\sin \left (\frac {\pi }{4}+3 x\right )\right )+\frac {1}{6} \sec \left (\frac {\pi }{4}+3 x\right ) \tan \left (\frac {\pi }{4}+3 x\right ) \]
Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sec ^3\left (3 x+\frac {\pi }{4}\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (3 x+\frac {3 \pi }{4}\right )^3dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {1}{2} \int \sec \left (3 x+\frac {\pi }{4}\right )dx+\frac {1}{6} \tan \left (3 x+\frac {\pi }{4}\right ) \sec \left (3 x+\frac {\pi }{4}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \csc \left (3 x+\frac {3 \pi }{4}\right )dx+\frac {1}{6} \tan \left (3 x+\frac {\pi }{4}\right ) \sec \left (3 x+\frac {\pi }{4}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {1}{6} \text {arctanh}\left (\sin \left (3 x+\frac {\pi }{4}\right )\right )+\frac {1}{6} \tan \left (3 x+\frac {\pi }{4}\right ) \sec \left (3 x+\frac {\pi }{4}\right )\) |
3.4.40.3.1 Defintions of rubi rules used
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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 | \(-\frac {i \left (\left (-1\right )^{\frac {3}{4}} {\mathrm e}^{9 i x}-\left (-1\right )^{\frac {1}{4}} {\mathrm e}^{3 i x}\right )}{3 \left (i {\mathrm e}^{6 i x}+1\right )^{2}}-\frac {\ln \left (\left (-1\right )^{\frac {1}{4}} {\mathrm e}^{3 i x}-i\right )}{6}+\frac {\ln \left (\left (-1\right )^{\frac {1}{4}} {\mathrm e}^{3 i x}+i\right )}{6}\) | \(67\) |
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 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {\cos \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2} \log \left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \cos \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2} \log \left (-\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) + 2 \, \sin \left (\frac {1}{4} \, \pi + 3 \, x\right )}{12 \, \cos \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2}} \]
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
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} \]
-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 + p i/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 + p i/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)
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.28 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=-\frac {\sin \left (\frac {1}{4} \, \pi + 3 \, x\right )}{6 \, {\left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2} - 1\right )}} + \frac {1}{12} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \frac {1}{12} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) - 1\right ) \]
-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)
Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=-\frac {\sin \left (\frac {1}{4} \, \pi + 3 \, x\right )}{6 \, {\left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right )^{2} - 1\right )}} + \frac {1}{12} \, \log \left (\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) - \frac {1}{12} \, \log \left (-\sin \left (\frac {1}{4} \, \pi + 3 \, x\right ) + 1\right ) \]
-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)
Time = 0.66 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \sec ^3\left (\frac {\pi }{4}+3 x\right ) \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {\Pi }{8}+\frac {3\,x}{2}+\frac {\pi }{4}\right )\right )}{6}+\frac {\mathrm {tan}\left (\frac {\Pi }{4}+3\,x\right )}{6\,\cos \left (\frac {\Pi }{4}+3\,x\right )} \]