\(\int \frac {\sec ^3(a+b \log (c x^n))}{x} \, dx\) [251]

Optimal result

Integrand size = 17, antiderivative size = 55 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\text {arctanh}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\sec \left (a+b \log \left (c x^n\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3853, 3855} \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\text {arctanh}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[In]

Int[Sec[a + b*Log[c*x^n]]^3/x,x]

[Out]

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)

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 {\text {Subst}\left (\int \sec ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{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}+\frac {\text {Subst}\left (\int \sec (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\text {arctanh}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\sec \left (a+b \log \left (c x^n\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[In]

Integrate[Sec[a + b*Log[c*x^n]]^3/x,x]

[Out]

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)

Maple [A] (verified)

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}\)	\(557\)

[In]

int(sec(a+b*ln(c*x^n))^3/x,x,method=_RETURNVERBOSE)

[Out]

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))))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.82 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) - \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (-\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) + 2 \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}} \]

[In]

integrate(sec(a+b*log(c*x^n))^3/x,x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sec ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[In]

integrate(sec(a+b*ln(c*x**n))**3/x,x)

[Out]

Integral(sec(a + b*log(c*x**n))**3/x, x)

Maxima [F]

\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^3/x,x, algorithm="maxima")

[Out]

-(((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))*sin(b*log(c)))*cos(b*log(x^n) + a) - (cos(4*b*log(c))*cos(3*b*log(c)) + si
n(4*b*log(c))*sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) + (cos(4*b*log(c))*cos(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)
)*sin(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*lo
g(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*log(c))^2*cos(b*log(c)) + b*co
s(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*lo
g(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*log(c)) + b*cos(b*log(c))*si
n(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))*cos(b*log(c)) + 2*(b*cos(4
*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b*cos(b*log(c))*sin(4*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n)
+ 2*a) + 2*(b*cos(2*b*log(c))*cos(b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*cos(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*(b*n*cos(b*log(c))*sin(4*b*log(c)) + 2*(b*cos(2*b*log
(c))*cos(b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a
) - 2*(b*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b*cos(b*log(c))*sin(4*b*log(c))*sin(2*b*log(c)))*n*si
n(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))*integrate(((cos(2*b*log(c))*cos(b*log(x^n) + a) + sin(2*b*log(
c))*sin(b*log(x^n) + a))*cos(2*b*log(x^n) + 2*a) - (cos(b*log(x^n) + a)*sin(2*b*log(c)) - cos(2*b*log(c))*sin(
b*log(x^n) + a))*sin(2*b*log(x^n) + 2*a) + cos(b*log(x^n) + a))/((cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*x*cos
(2*b*log(x^n) + 2*a)^2 + (cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*x*sin(2*b*log(x^n) + 2*a)^2 + 2*x*cos(2*b*log
(c))*cos(2*b*log(x^n) + 2*a) - 2*x*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + x), x) - (4*b*n*cos(2*b*log(c))*c
os(2*b*log(x^n) + 2*a)*sin(b*log(c)) - 4*b*n*sin(2*b*log(c))*sin(b*log(c))*sin(2*b*log(x^n) + 2*a) + (b*cos(4*
b*log(c))^2*sin(b*log(c)) + b*sin(4*b*log(c))^2*sin(b*log(c)))*n*cos(4*b*log(x^n) + 4*a)^2 + 4*(b*cos(2*b*log(
c))^2*sin(b*log(c)) + b*sin(2*b*log(c))^2*sin(b*log(c)))*n*cos(2*b*log(x^n) + 2*a)^2 + (b*cos(4*b*log(c))^2*si
n(b*log(c)) + b*sin(4*b*log(c))^2*sin(b*log(c)))*n*sin(4*b*log(x^n) + 4*a)^2 + 4*(b*cos(2*b*log(c))^2*sin(b*lo
g(c)) + b*sin(2*b*log(c))^2*sin(b*log(c)))*n*sin(2*b*log(x^n) + 2*a)^2 + b*n*sin(b*log(c)) + 2*(b*n*cos(4*b*lo
g(c))*sin(b*log(c)) + 2*(b*cos(4*b*log(c))*cos(2*b*log(c))*sin(b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c))*s
in(b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + 2*(b*cos(2*b*log(c))*sin(4*b*log(c))*sin(b*log(c)) - b*cos(4*b*log(c
))*sin(2*b*log(c))*sin(b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 2*(b*n*sin(4*b*log(c))*
sin(b*log(c)) + 2*(b*cos(2*b*log(c))*sin(4*b*log(c))*sin(b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c))*sin(b*l
og(c)))*n*cos(2*b*log(x^n) + 2*a) - 2*(b*cos(4*b*log(c))*cos(2*b*log(c))*sin(b*log(c)) + b*sin(4*b*log(c))*sin
(2*b*log(c))*sin(b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))*integrate(((cos(b*log(x^n) + a
)*sin(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(x^n) + a))*cos(2*b*log(x^n) + 2*a) + (cos(2*b*log(c))*cos(b*log(
x^n) + a) + sin(2*b*log(c))*sin(b*log(x^n) + a))*sin(2*b*log(x^n) + 2*a) - sin(b*log(x^n) + a))/((cos(2*b*log(
c))^2 + sin(2*b*log(c))^2)*x*cos(2*b*log(x^n) + 2*a)^2 + (cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*x*sin(2*b*log
(x^n) + 2*a)^2 + 2*x*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 2*x*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + x
), x) + cos(b*log(x^n) + a)*sin(b*log(c)) + ((cos(4*b*log(c))*cos(3*b*log(c)) + sin(4*b*log(c))*sin(3*b*log(c)
))*cos(3*b*log(x^n) + 3*a) - (cos(4*b*log(c))*cos(b*log(c)) + sin(4*b*log(c))*sin(b*log(c)))*cos(b*log(x^n) +
a) + (cos(3*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) - (cos(b*log(
c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(b*log(c)))*sin(b*log(x^n) + a))*sin(4*b*log(x^n) + 4*a) - (2*(cos(3*
b*log(c))*cos(2*b*log(c)) + sin(3*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 2*(cos(2*b*log(c))*sin(
3*b*log(c)) - cos(3*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + cos(3*b*log(c)))*sin(3*b*log(x^n) + 3
*a) - 2*((cos(2*b*log(c))*cos(b*log(c)) + sin(2*b*log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) + (cos(b*log(c))*
sin(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(c)))*sin(b*log(x^n) + a))*sin(2*b*log(x^n) + 2*a) + cos(b*log(c))*
sin(b*log(x^n) + a))/((b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 + 4*b*n*cos(2*b*
log(c))*cos(2*b*log(x^n) + 2*a) + 4*(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(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*sin(4*b*log(x^n) + 4*a)^2 - 4*b*n*sin(2*b*log(c))*sin(2*b*log(x^
n) + 2*a) + 4*(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(4*b*l
og(c)) + 2*(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) +
2*(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*(2*(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)) - 2*(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]

\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^3/x,x, algorithm="giac")

[Out]

integrate(sec(b*log(c*x^n) + a)^3/x, x)

Mupad [B] (verification not implemented)

Time = 32.52 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.24 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\ln \left (-\frac {1{}\mathrm {i}}{x}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{x}\right )}{2\,b\,n}-\frac {\ln \left (\frac {1{}\mathrm {i}}{x}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{x}\right )}{2\,b\,n}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,2{}\mathrm {i}}{b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,1{}\mathrm {i}}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )} \]

[In]

int(1/(x*cos(a + b*log(c*x^n))^3),x)

[Out]

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