Integrand size = 17, antiderivative size = 42 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \] Output:
tan(a+b*ln(c*x^n))/b/n+1/3*tan(a+b*ln(c*x^n))^3/b/n
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tan \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} \tan ^3\left (a+b \log \left (c x^n\right )\right )}{b n} \] Input:
Integrate[Sec[a + b*Log[c*x^n]]^4/x,x]
Output:
(Tan[a + b*Log[c*x^n]] + Tan[a + b*Log[c*x^n]]^3/3)/(b*n)
Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3039, 3042, 4254, 2009}
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.
\(\displaystyle \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \sec ^4\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \csc \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )^4d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {\int \left (\tan ^2\left (a+b \log \left (c x^n\right )\right )+1\right )d\left (-\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{3} \tan ^3\left (a+b \log \left (c x^n\right )\right )-\tan \left (a+b \log \left (c x^n\right )\right )}{b n}\) |
Input:
Int[Sec[a + b*Log[c*x^n]]^4/x,x]
Output:
-((-Tan[a + b*Log[c*x^n]] - Tan[a + b*Log[c*x^n]]^3/3)/(b*n))
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_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 17.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(-\frac {\left (-\frac {2}{3}-\frac {{\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3}\right ) \tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(37\) |
default | \(-\frac {\left (-\frac {2}{3}-\frac {{\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3}\right ) \tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(37\) |
parallelrisch | \(\frac {-6 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{5}+4 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}-6 \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{3 b n {\left (\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )-1\right )}^{3} {\left (\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )+1\right )}^{3}}\) | \(103\) |
risch | \(\frac {4 i \left (3 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}{3 b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}^{3}}\) | \(225\) |
Input:
int(sec(a+b*ln(c*x^n))^4/x,x,method=_RETURNVERBOSE)
Output:
-1/n/b*(-2/3-1/3*sec(a+b*ln(c*x^n))^2)*tan(a+b*ln(c*x^n))
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3}} \] Input:
integrate(sec(a+b*log(c*x^n))^4/x,x, algorithm="fricas")
Output:
1/3*(2*cos(b*n*log(x) + b*log(c) + a)^2 + 1)*sin(b*n*log(x) + b*log(c) + a )/(b*n*cos(b*n*log(x) + b*log(c) + a)^3)
\[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sec ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(sec(a+b*ln(c*x**n))**4/x,x)
Output:
Integral(sec(a + b*log(c*x**n))**4/x, x)
Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (40) = 80\).
Time = 0.14 (sec) , antiderivative size = 1323, normalized size of antiderivative = 31.50 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:
integrate(sec(a+b*log(c*x^n))^4/x,x, algorithm="maxima")
Output:
4/3*((3*(cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)) )*cos(2*b*log(x^n) + 2*a) - 3*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*l og(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + sin(6*b*log(c)))*cos(6*b *log(x^n) + 6*a) + 3*(3*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c)) *sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 3*(cos(4*b*log(c))*cos(2*b*log (c)) + sin(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + sin(4*b* log(c)))*cos(4*b*log(x^n) + 4*a) + (3*(cos(6*b*log(c))*cos(2*b*log(c)) + s in(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 3*(cos(2*b*log(c ))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2 *a) + cos(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + 3*(3*(cos(4*b*log(c))*cos (2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 3*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin( 2*b*log(x^n) + 2*a) + cos(4*b*log(c)))*sin(4*b*log(x^n) + 4*a))/((b*cos(6* b*log(c))^2 + b*sin(6*b*log(c))^2)*n*cos(6*b*log(x^n) + 6*a)^2 + 9*(b*cos( 4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 + 6*b*n*c os(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 9*(b*cos(2*b*log(c))^2 + b*sin(2* b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 + (b*cos(6*b*log(c))^2 + b*sin(6* b*log(c))^2)*n*sin(6*b*log(x^n) + 6*a)^2 + 9*(b*cos(4*b*log(c))^2 + b*sin( 4*b*log(c))^2)*n*sin(4*b*log(x^n) + 4*a)^2 - 6*b*n*sin(2*b*log(c))*sin(2*b *log(x^n) + 2*a) + 9*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*sin(...
\[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x} \,d x } \] Input:
integrate(sec(a+b*log(c*x^n))^4/x,x, algorithm="giac")
Output:
integrate(sec(b*log(c*x^n) + a)^4/x, x)
Time = 30.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,3{}\mathrm {i}+1{}\mathrm {i}\right )}{3\,b\,n\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}^3} \] Input:
int(1/(x*cos(a + b*log(c*x^n))^4),x)
Output:
(4*(exp(a*2i)*(c*x^n)^(b*2i)*3i + 1i))/(3*b*n*(exp(a*2i)*(c*x^n)^(b*2i) + 1)^3)
Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.57 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) \left (2 {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2}-3\right )}{3 \cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) b n \left ({\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2}-1\right )} \] Input:
int(sec(a+b*log(c*x^n))^4/x,x)
Output:
(sin(log(x**n*c)*b + a)*(2*sin(log(x**n*c)*b + a)**2 - 3))/(3*cos(log(x**n *c)*b + a)*b*n*(sin(log(x**n*c)*b + a)**2 - 1))