\(\int \frac {\csc ^3(a+b \log (c x^n))}{x} \, dx\) [298]

Optimal result

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

[Out]

-1/2*arctanh(cos(a+b*ln(c*x^n)))/b/n-1/2*cot(a+b*ln(c*x^n))*csc(a+b*ln(c*x^n))/b/n

Rubi [A] (verified)

Time = 0.05 (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} \[ \int \frac {\csc ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\text {arctanh}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[In]

Int[Csc[a + b*Log[c*x^n]]^3/x,x]

[Out]

-1/2*ArcTanh[Cos[a + b*Log[c*x^n]]]/(b*n) - (Cot[a + b*Log[c*x^n]]*Csc[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 \csc ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\text {Subst}\left (\int \csc (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n} \\ & = -\frac {\text {arctanh}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac {\cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right )}{2 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.95 \[ \int \frac {\csc ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\csc ^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac {\log \left (\cos \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}+\frac {\log \left (\sin \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}+\frac {\sec ^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n} \]

[In]

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

[Out]

-1/8*Csc[(a + b*Log[c*x^n])/2]^2/(b*n) - Log[Cos[(a + b*Log[c*x^n])/2]]/(2*b*n) + Log[Sin[(a + b*Log[c*x^n])/2
]]/(2*b*n) + Sec[(a + b*Log[c*x^n])/2]^2/(8*b*n)

Maple [A] (verified)

Time = 1.91 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {-\frac {\csc \left (a +b \ln \left (c \,x^{n}\right )\right ) \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\frac {\ln \left (\csc \left (a +b \ln \left (c \,x^{n}\right )\right )-\cot \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}}{n b}\)	\(61\)
default	\(\frac {-\frac {\csc \left (a +b \ln \left (c \,x^{n}\right )\right ) \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}+\frac {\ln \left (\csc \left (a +b \ln \left (c \,x^{n}\right )\right )-\cot \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2}}{n b}\)	\(61\)
parallelrisch	\(\frac {-{\cot \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}+{\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}+4 \ln \left (\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )\right )}{8 b n}\)	\(64\)
risch	\(\frac {c^{i b} \left (x^{n}\right )^{i b} \left (c^{2 i b} \left (x^{n}\right )^{2 i b} {\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}^{\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}^{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}+1\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}-1\right )}{2 b n}\)	\(552\)

[In]

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

[Out]

1/n/b*(-1/2*csc(a+b*ln(c*x^n))*cot(a+b*ln(c*x^n))+1/2*ln(csc(a+b*ln(c*x^n))-cot(a+b*ln(c*x^n))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (51) = 102\).

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.00 \[ \int \frac {\csc ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {{\left (\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \frac {1}{2}\right ) - {\left (\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \, {\left (b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2168 vs. \(2 (51) = 102\).

Time = 0.30 (sec) , antiderivative size = 2168, normalized size of antiderivative = 39.42 \[ \int \frac {\csc ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(4*((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))*cos(4*b*log(x^n) + 4*a) - 4*(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)))*cos(3*b*log(x^n) + 3*a) - 8*((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))*s
in(b*log(c)))*sin(b*log(x^n) + a))*cos(2*b*log(x^n) + 2*a) + 4*cos(b*log(c))*cos(b*log(x^n) + a) - ((cos(4*b*l
og(c))^2 + sin(4*b*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*lo
g(x^n) + 2*a)^2 + (cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*sin(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + s
in(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 - 2*(2*(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) + 2*(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)))*cos(4*b*log(x^n) + 4*a) - 4*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 2*
(2*(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) - 2*(cos(4*b*lo
g(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)))*sin(4*b*lo
g(x^n) + 4*a) + 4*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 1)*log((cos(a)^2 + sin(a)^2)*cos(b*log(c))^2 + (co
s(a)^2 + sin(a)^2)*sin(b*log(c))^2 + 2*(cos(b*log(c))*cos(a) - sin(b*log(c))*sin(a))*cos(b*log(x^n)) + cos(b*l
og(x^n))^2 - 2*(cos(a)*sin(b*log(c)) + cos(b*log(c))*sin(a))*sin(b*log(x^n)) + sin(b*log(x^n))^2) + ((cos(4*b*
log(c))^2 + sin(4*b*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*l
og(x^n) + 2*a)^2 + (cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*sin(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 +
sin(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 - 2*(2*(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) + 2*(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)))*cos(4*b*log(x^n) + 4*a) - 4*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 2
*(2*(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) - 2*(cos(4*b*l
og(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)))*sin(4*b*l
og(x^n) + 4*a) + 4*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 1)*log((cos(a)^2 + sin(a)^2)*cos(b*log(c))^2 + (c
os(a)^2 + sin(a)^2)*sin(b*log(c))^2 - 2*(cos(b*log(c))*cos(a) - sin(b*log(c))*sin(a))*cos(b*log(x^n)) + cos(b*
log(x^n))^2 + 2*(cos(a)*sin(b*log(c)) + cos(b*log(c))*sin(a))*sin(b*log(x^n)) + sin(b*log(x^n))^2) - 4*((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)) + sin(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))*sin(4*b*log(x^n) + 4*a) + 4*(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)))*si
n(2*b*log(x^n) + 2*a) - sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) + 8*((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))*sin(2*b*log(x^n) + 2*a) - 4*sin(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*s
in(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*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*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]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 32.07 (sec) , antiderivative size = 177, normalized size of antiderivative = 3.22 \[ \int \frac {\csc ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\ln \left (-\frac {1{}\mathrm {i}}{x}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,1{}\mathrm {i}}{x}\right )}{2\,b\,n}+\frac {\ln \left (\frac {1{}\mathrm {i}}{x}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,1{}\mathrm {i}}{x}\right )}{2\,b\,n}+\frac {2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{b\,n\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )} \]

[In]

int(1/(x*sin(a + b*log(c*x^n))^3),x)

[Out]

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