\(\int (-((1+b^2 n^2) \csc (a+b \log (c x^n)))+2 b^2 n^2 \csc ^3(a+b \log (c x^n))) \, dx\) [301]

Optimal result

Integrand size = 44, antiderivative size = 42 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-x \csc \left (a+b \log \left (c x^n\right )\right )-b n x \cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.15 (sec) , antiderivative size = 172, normalized size of antiderivative = 4.10, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {4600, 4602, 371} \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=2 e^{i a} x (b n+i) \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {i}{b n}\right ),\frac {1}{2} \left (3-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )-\frac {16 e^{3 i a} b^2 n^2 x \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right ),\frac {1}{2} \left (5-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-3 b n+i} \]

[In]

Int[-((1 + b^2*n^2)*Csc[a + b*Log[c*x^n]]) + 2*b^2*n^2*Csc[a + b*Log[c*x^n]]^3,x]

[Out]

2*E^(I*a)*(I + b*n)*x*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1 - I/(b*n))/2, (3 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)
^((2*I)*b)] - (16*b^2*E^((3*I)*a)*n^2*x*(c*x^n)^((3*I)*b)*Hypergeometric2F1[3, (3 - I/(b*n))/2, (5 - I/(b*n))/
2, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(I - 3*b*n)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4600

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])

Rule 4602

Int[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(-2*I)^p*E^(I*a*d*p), Int[(
e*x)^m*(x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \left (2 b^2 n^2\right ) \int \csc ^3\left (a+b \log \left (c x^n\right )\right ) \, dx+\left (-1-b^2 n^2\right ) \int \csc \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = \left (2 b^2 n x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \csc ^3(a+b \log (x)) \, dx,x,c x^n\right )+\frac {\left (\left (-1-b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \left (16 i b^2 e^{3 i a} n x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+3 i b+\frac {1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^3} \, dx,x,c x^n\right )-\frac {\left (2 i e^{i a} \left (-1-b^2 n^2\right ) x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+i b+\frac {1}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n} \\ & = 2 e^{i a} (i+b n) x \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {i}{b n}\right ),\frac {1}{2} \left (3-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )-\frac {16 b^2 e^{3 i a} n^2 x \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right ),\frac {1}{2} \left (5-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i-3 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-x \left (1+b n \cot \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right ) \]

[In]

Integrate[-((1 + b^2*n^2)*Csc[a + b*Log[c*x^n]]) + 2*b^2*n^2*Csc[a + b*Log[c*x^n]]^3,x]

[Out]

-(x*(1 + b*n*Cot[a + b*Log[c*x^n]])*Csc[a + b*Log[c*x^n]])

Maple [A] (verified)

Time = 14.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.95

method	result	size
parallelrisch	\(\frac {x \left ({\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4} b n -2 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}-b n -2 \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )\right )}{4 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}}\)	\(82\)
risch	\(\frac {2 c^{i b} \left (x^{n}\right )^{i b} x \left (n b \,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}+b n \,{\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}-i 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}+i {\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 )}{{\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}}\)	\(523\)

[In]

int(-(b^2*n^2+1)*csc(a+b*ln(c*x^n))+2*b^2*n^2*csc(a+b*ln(c*x^n))^3,x,method=_RETURNVERBOSE)

[Out]

1/4*x*(tan(1/2*a+b*ln((c*x^n)^(1/2)))^4*b*n-2*tan(1/2*a+b*ln((c*x^n)^(1/2)))^3-b*n-2*tan(1/2*a+b*ln((c*x^n)^(1
/2))))/tan(1/2*a+b*ln((c*x^n)^(1/2)))^2

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {b n x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + x \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1} \]

[In]

integrate(-(b^2*n^2+1)*csc(a+b*log(c*x^n))+2*b^2*n^2*csc(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (2 b^{2} n^{2} \csc ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} - b^{2} n^{2} - 1\right ) \csc {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]

[In]

integrate(-(b**2*n**2+1)*csc(a+b*ln(c*x**n))+2*b**2*n**2*csc(a+b*ln(c*x**n))**3,x)

[Out]

Integral((2*b**2*n**2*csc(a + b*log(c*x**n))**2 - b**2*n**2 - 1)*csc(a + b*log(c*x**n)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1701 vs. \(2 (42) = 84\).

Time = 0.50 (sec) , antiderivative size = 1701, normalized size of antiderivative = 40.50 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Too large to display} \]

[In]

integrate(-(b^2*n^2+1)*csc(a+b*log(c*x^n))+2*b^2*n^2*csc(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { 2 \, b^{2} n^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right )^{3} - {\left (b^{2} n^{2} + 1\right )} \csc \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]

[In]

integrate(-(b^2*n^2+1)*csc(a+b*log(c*x^n))+2*b^2*n^2*csc(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

integrate(2*b^2*n^2*csc(b*log(c*x^n) + a)^3 - (b^2*n^2 + 1)*csc(b*log(c*x^n) + a), x)

Mupad [B] (verification not implemented)

Time = 28.65 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.02 \[ \int \left (-\left (\left (1+b^2 n^2\right ) \csc \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2 \csc ^3\left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,\left (b\,n+1{}\mathrm {i}\right )+2\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,\left (b\,n-\mathrm {i}\right )}{{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}^2} \]

[In]

int((2*b^2*n^2)/sin(a + b*log(c*x^n))^3 - (b^2*n^2 + 1)/sin(a + b*log(c*x^n)),x)

[Out]

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