\(\int \frac {\sec ^4(a+b \log (c x^n))}{x} \, dx\) [256]

Optimal result

Integrand size = 17, antiderivative size = 42 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[Out]

tan(a+b*ln(c*x^n))/b/n+1/3*tan(a+b*ln(c*x^n))^3/b/n

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3852} \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\tan \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[In]

Int[Sec[a + b*Log[c*x^n]]^4/x,x]

[Out]

Tan[a + b*Log[c*x^n]]/(b*n) + Tan[a + b*Log[c*x^n]]^3/(3*b*n)

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sec ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\tan \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} \tan ^3\left (a+b \log \left (c x^n\right )\right )}{b n} \]

[In]

Integrate[Sec[a + b*Log[c*x^n]]^4/x,x]

[Out]

(Tan[a + b*Log[c*x^n]] + Tan[a + b*Log[c*x^n]]^3/3)/(b*n)

Maple [A] (verified)

Time = 16.93 (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	\(\frac {4 i \left (3 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}{3 b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}^{3}}\)	\(225\)

[In]

int(sec(a+b*ln(c*x^n))^4/x,x,method=_RETURNVERBOSE)

[Out]

-1/n/b*(-2/3-1/3*sec(a+b*ln(c*x^n))^2)*tan(a+b*ln(c*x^n))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3}} \]

[In]

integrate(sec(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sec ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[In]

integrate(sec(a+b*ln(c*x**n))**4/x,x)

[Out]

Integral(sec(a + b*log(c*x**n))**4/x, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (40) = 80\).

Time = 0.23 (sec) , antiderivative size = 1323, normalized size of antiderivative = 31.50 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

[In]

integrate(sec(a+b*log(c*x^n))^4/x,x, algorithm="maxima")

[Out]

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*log(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)) + sin(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*cos(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(2*b*log(x^n) + 2*a)^2 + b*n + 2*(b*n*cos(6*b*log(c)) + 3*(b*cos
(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) + 3*(b*cos(6*b*log
(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + 3*(b*cos(4*b*log(c))*sin
(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) + 3*(b*cos(2*b*log(c))*sin(6*b*log
(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6*(b*n*cos(4*b*
log(c)) + 3*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a)
+ 3*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(4*b
*log(x^n) + 4*a) - 2*(3*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(
x^n) + 4*a) + 3*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2
*a) + b*n*sin(6*b*log(c)) - 3*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*sin(4*
b*log(x^n) + 4*a) - 3*(b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^
n) + 2*a))*sin(6*b*log(x^n) + 6*a) - 6*(3*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c
)))*n*cos(2*b*log(x^n) + 2*a) + b*n*sin(4*b*log(c)) - 3*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))
*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

Giac [F]

\[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^4/x,x, algorithm="giac")

[Out]

integrate(sec(b*log(c*x^n) + a)^4/x, x)

Mupad [B] (verification not implemented)

Time = 38.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,3{}\mathrm {i}+1{}\mathrm {i}\right )}{3\,b\,n\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}^3} \]

[In]

int(1/(x*cos(a + b*log(c*x^n))^4),x)

[Out]

(4*(exp(a*2i)*(c*x^n)^(b*2i)*3i + 1i))/(3*b*n*(exp(a*2i)*(c*x^n)^(b*2i) + 1)^3)