Integrand size = 17, antiderivative size = 55 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\text {arctanh}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\sec \left (a+b \log \left (c x^n\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\text {arctanh}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\sec \left (a+b \log \left (c x^n\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]
ArcTanh[Sin[a + b*Log[c*x^n]]]/(2*b*n) + (Sec[a + b*Log[c*x^n]]*Tan[a + b* Log[c*x^n]])/(2*b*n)
Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3039, 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.
| \(\displaystyle \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \sec ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )^3d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{2} \int \sec \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )+\frac {\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \int \csc \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )d\log \left (c x^n\right )+\frac {\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\text {arctanh}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b}+\frac {\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
(ArcTanh[Sin[a + b*Log[c*x^n]]]/(2*b) + (Sec[a + b*Log[c*x^n]]*Tan[a + b*L og[c*x^n]])/(2*b))/n
3.3.51.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
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 = 5.48 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(\frac {\frac {\sec \left (a +b \ln \left (c \,x^{n}\right )\right ) \tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\frac {\ln \left (\sec \left (a +b \ln \left (c \,x^{n}\right )\right )+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}}{n b}\) | \(59\) |
| default | \(\frac {\frac {\sec \left (a +b \ln \left (c \,x^{n}\right )\right ) \tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\frac {\ln \left (\sec \left (a +b \ln \left (c \,x^{n}\right )\right )+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}}{n b}\) | \(59\) |
| parallelrisch | \(\frac {\left (-\cos \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )-1\right ) \ln \left (\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )-1\right )+\left (\cos \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+1\right ) \ln \left (\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )+1\right )+2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n \left (\cos \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+1\right )}\) | \(113\) |
| risch | \(-\frac {i \left (x^{n}\right )^{i b} c^{i b} \left (c^{2 i b} \left (x^{n}\right )^{2 i b} {\mathrm e}^{\frac {3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{\frac {3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{3 i a}-{\mathrm e}^{\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{-\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{i a}\right )}{b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}^{2}}-\frac {\ln \left (c^{i b} \left (x^{n}\right )^{i b} {\mathrm e}^{-\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{i a}-i\right )}{2 b n}+\frac {\ln \left (c^{i b} \left (x^{n}\right )^{i b} {\mathrm e}^{-\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}} {\mathrm e}^{\frac {b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}} {\mathrm e}^{-\frac {b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}} {\mathrm e}^{i a}+i\right )}{2 b n}\) | \(557\) |
1/n/b*(1/2*sec(a+b*ln(c*x^n))*tan(a+b*ln(c*x^n))+1/2*ln(sec(a+b*ln(c*x^n)) +tan(a+b*ln(c*x^n))))
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.82 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) - \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (-\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) + 2 \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}} \]
1/4*(cos(b*n*log(x) + b*log(c) + a)^2*log(sin(b*n*log(x) + b*log(c) + a) + 1) - cos(b*n*log(x) + b*log(c) + a)^2*log(-sin(b*n*log(x) + b*log(c) + a) + 1) + 2*sin(b*n*log(x) + b*log(c) + a))/(b*n*cos(b*n*log(x) + b*log(c) + a)^2)
\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sec ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x} \,d x } \]
-(((cos(3*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(3*b*log(c)))*cos (3*b*log(x^n) + 3*a) - (cos(b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*si n(b*log(c)))*cos(b*log(x^n) + a) - (cos(4*b*log(c))*cos(3*b*log(c)) + sin( 4*b*log(c))*sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) + (cos(4*b*log(c))*co s(b*log(c)) + sin(4*b*log(c))*sin(b*log(c)))*sin(b*log(x^n) + a))*cos(4*b* log(x^n) + 4*a) - (2*(cos(2*b*log(c))*sin(3*b*log(c)) - cos(3*b*log(c))*si n(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(cos(3*b*log(c))*cos(2*b*log(c) ) + sin(3*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + sin(3*b*log (c)))*cos(3*b*log(x^n) + 3*a) - 2*((cos(b*log(c))*sin(2*b*log(c)) - cos(2* b*log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) - (cos(2*b*log(c))*cos(b*log( c)) + sin(2*b*log(c))*sin(b*log(c)))*sin(b*log(x^n) + a))*cos(2*b*log(x^n) + 2*a) - (4*b*n*cos(2*b*log(c))*cos(b*log(c))*cos(2*b*log(x^n) + 2*a) - 4 *b*n*cos(b*log(c))*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + (b*cos(4*b*lo g(c))^2*cos(b*log(c)) + b*cos(b*log(c))*sin(4*b*log(c))^2)*n*cos(4*b*log(x ^n) + 4*a)^2 + 4*(b*cos(2*b*log(c))^2*cos(b*log(c)) + b*cos(b*log(c))*sin( 2*b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 + (b*cos(4*b*log(c))^2*cos(b*lo g(c)) + b*cos(b*log(c))*sin(4*b*log(c))^2)*n*sin(4*b*log(x^n) + 4*a)^2 + 4 *(b*cos(2*b*log(c))^2*cos(b*log(c)) + b*cos(b*log(c))*sin(2*b*log(c))^2)*n *sin(2*b*log(x^n) + 2*a)^2 + b*n*cos(b*log(c)) + 2*(b*n*cos(4*b*log(c))*co s(b*log(c)) + 2*(b*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b*co...
\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x} \,d x } \]
Time = 32.52 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.24 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\ln \left (-\frac {1{}\mathrm {i}}{x}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{x}\right )}{2\,b\,n}-\frac {\ln \left (\frac {1{}\mathrm {i}}{x}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{x}\right )}{2\,b\,n}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,2{}\mathrm {i}}{b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,1{}\mathrm {i}}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )} \]