\(\int \csc ^4(a+b \log (c x^n)) \, dx\) [299]

Optimal result

Integrand size = 13, antiderivative size = 84 \[ \int \csc ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {16 e^{4 i a} x \left (c x^n\right )^{4 i b} \operatorname {Hypergeometric2F1}\left (4,\frac {1}{2} \left (4-\frac {i}{b n}\right ),\frac {1}{2} \left (6-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+4 i b n} \]

[Out]

16*exp(4*I*a)*x*(c*x^n)^(4*I*b)*hypergeom([4, 2-1/2*I/b/n],[3-1/2*I/b/n],exp(2*I*a)*(c*x^n)^(2*I*b))/(1+4*I*b*
n)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4600, 4602, 371} \[ \int \csc ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {16 e^{4 i a} x \left (c x^n\right )^{4 i b} \operatorname {Hypergeometric2F1}\left (4,\frac {1}{2} \left (4-\frac {i}{b n}\right ),\frac {1}{2} \left (6-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+4 i b n} \]

[In]

Int[Csc[a + b*Log[c*x^n]]^4,x]

[Out]

(16*E^((4*I)*a)*x*(c*x^n)^((4*I)*b)*Hypergeometric2F1[4, (4 - I/(b*n))/2, (6 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)
^((2*I)*b)])/(1 + (4*I)*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}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \csc ^4(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (16 e^{4 i a} x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+4 i b+\frac {1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^4} \, dx,x,c x^n\right )}{n} \\ & = \frac {16 e^{4 i a} x \left (c x^n\right )^{4 i b} \operatorname {Hypergeometric2F1}\left (4,\frac {1}{2} \left (4-\frac {i}{b n}\right ),\frac {1}{2} \left (6-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+4 i b n} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(221\) vs. \(2(84)=168\).

Time = 11.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.63 \[ \int \csc ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (-4 e^{2 i a} (i+2 b n) \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2 b n},2-\frac {i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-4 i \left (1+4 b^2 n^2\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b n},1-\frac {i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\csc ^3\left (a+b \log \left (c x^n\right )\right ) \left (-\left (\left (1+12 b^2 n^2\right ) \cos \left (a+b \log \left (c x^n\right )\right )\right )+\left (1+4 b^2 n^2\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-4 b n \sin \left (a+b \log \left (c x^n\right )\right )\right )\right )}{24 b^3 n^3} \]

[In]

Integrate[Csc[a + b*Log[c*x^n]]^4,x]

[Out]

(x*(-4*E^((2*I)*a)*(I + 2*b*n)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 - (I/2)/(b*n), 2 - (I/2)/(b*n), E^((2*
I)*(a + b*Log[c*x^n]))] - (4*I)*(1 + 4*b^2*n^2)*Hypergeometric2F1[1, (-1/2*I)/(b*n), 1 - (I/2)/(b*n), E^((2*I)
*(a + b*Log[c*x^n]))] + Csc[a + b*Log[c*x^n]]^3*(-((1 + 12*b^2*n^2)*Cos[a + b*Log[c*x^n]]) + (1 + 4*b^2*n^2)*C
os[3*(a + b*Log[c*x^n])] - 4*b*n*Sin[a + b*Log[c*x^n]])))/(24*b^3*n^3)

Maple [F]

\[\int {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}d x\]

[In]

int(csc(a+b*ln(c*x^n))^4,x)

[Out]

int(csc(a+b*ln(c*x^n))^4,x)

Fricas [F]

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

[In]

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

[Out]

integral(csc(b*log(c*x^n) + a)^4, x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/3*(6*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*x*cos(4*b*log(x^n) + 4*a)^2 + 6*(b*cos(2*b*log(c))^2 + b*
sin(2*b*log(c))^2)*n*x*cos(2*b*log(x^n) + 2*a)^2 + 6*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*x*sin(4*b*l
og(x^n) + 4*a)^2 + 6*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*x*sin(2*b*log(x^n) + 2*a)^2 - (2*b*n*cos(2*
b*log(c)) - sin(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) + (2*b*n*sin(2*b*log(c)) + cos(2*b*log(c)))*x*sin(2*b*l
og(x^n) + 2*a) - ((2*(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(c
))*sin(6*b*log(c)) + cos(6*b*log(c))*sin(4*b*log(c)))*x*cos(4*b*log(x^n) + 4*a) + 2*(6*(b^2*cos(2*b*log(c))*si
n(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(2*b*log(c)))*n^2 - (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(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n)
+ 2*a) + (2*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n + cos(6*b*log(c))*cos(4
*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*x*sin(4*b*log(x^n) + 4*a) - 2*(6*(b^2*cos(6*b*log(c))*cos(2*b*lo
g(c)) + b^2*sin(6*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(
2*b*log(c)))*n + cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a)
- (4*b^2*n^2*sin(6*b*log(c)) + sin(6*b*log(c)))*x)*cos(6*b*log(x^n) + 6*a) + (3*(12*(b^2*cos(2*b*log(c))*sin(4
*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2 - 4*(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(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n)
+ 2*a) - 3*(12*(b^2*cos(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2 + 4*(b*cos(2*b*
log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n + cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log
(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) - 2*(6*b^2*n^2*sin(4*b*log(c)) - b*n*cos(4*b*log(c)) + sin(4*b
*log(c)))*x)*cos(4*b*log(x^n) + 4*a) + 18*(4*b^8*n^8 + b^6*n^6 + (4*(b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c
))^2)*n^8 + (b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*cos(6*b*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b
*log(c))^2 + b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*cos(4*b*log(x^n
) + 4*a)^2 + 9*(4*(b^8*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*l
og(c))^2)*n^6)*cos(2*b*log(x^n) + 2*a)^2 + (4*(b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8 + (b^6*cos(6
*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*sin(6*b*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b*log(c))^2 + b^8*sin(4
*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*sin(4*b*log(x^n) + 4*a)^2 + 9*(4*(b^8
*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6)*sin(2*b
*log(x^n) + 2*a)^2 - 2*(4*b^8*n^8*cos(6*b*log(c)) + b^6*n^6*cos(6*b*log(c)) + 3*(4*(b^8*cos(6*b*log(c))*cos(4*
b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c)) + b^6*sin(6*b*log(
c))*sin(4*b*log(c)))*n^6)*cos(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(6*b*log(c))*cos(2*b*log(c)) + b^8*sin(6*b*lo
g(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*sin(6*b*log(c))*sin(2*b*log(c)))*n^6)*
cos(2*b*log(x^n) + 2*a) + 3*(4*(b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(4*b*log(c)))*n^8
+ (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) -
3*(4*(b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*si
n(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6
*(4*b^8*n^8*cos(4*b*log(c)) + b^6*n^6*cos(4*b*log(c)) - 3*(4*(b^8*cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*
b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n
^6)*cos(2*b*log(x^n) + 2*a) - 3*(4*(b^8*cos(2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))
*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c)) - b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a
))*cos(4*b*log(x^n) + 4*a) - 6*(4*b^8*n^8*cos(2*b*log(c)) + b^6*n^6*cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) +
2*(4*b^8*n^8*sin(6*b*log(c)) + b^6*n^6*sin(6*b*log(c)) + 3*(4*(b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(
6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))
*n^6)*cos(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)
))*n^8 + (b^6*cos(2*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2
*a) - 3*(4*(b^8*cos(6*b*log(c))*cos(4*b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(
c))*cos(4*b*log(c)) + b^6*sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) + 3*(4*(b^8*cos(6*b*lo
g(c))*cos(2*b*log(c)) + b^8*sin(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*
sin(6*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) - 6*(4*b^8*n^8*sin(4*b*
log(c)) + b^6*n^6*sin(4*b*log(c)) - 3*(4*(b^8*cos(2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*lo
g(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c)) - b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n)
+ 2*a) + 3*(4*(b^8*cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*
log(c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n)
+ 4*a) + 6*(4*b^8*n^8*sin(2*b*log(c)) + b^6*n^6*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*integrate(1/36*(cos(
b*log(x^n) + a)*sin(b*log(c)) + cos(b*log(c))*sin(b*log(x^n) + a))/(2*b^6*n^6*cos(b*log(c))*cos(b*log(x^n) + a
) - 2*b^6*n^6*sin(b*log(c))*sin(b*log(x^n) + a) + b^6*n^6 + (b^6*cos(b*log(c))^2 + b^6*sin(b*log(c))^2)*n^6*co
s(b*log(x^n) + a)^2 + (b^6*cos(b*log(c))^2 + b^6*sin(b*log(c))^2)*n^6*sin(b*log(x^n) + a)^2), x) - 18*(4*b^8*n
^8 + b^6*n^6 + (4*(b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8 + (b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*l
og(c))^2)*n^6)*cos(6*b*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b*log(c))^2 + b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos
(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*cos(4*b*log(x^n) + 4*a)^2 + 9*(4*(b^8*cos(2*b*log(c))^2 + b^8*sin
(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6)*cos(2*b*log(x^n) + 2*a)^2 + (4*(b^8
*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8 + (b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*sin(6*b
*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b*log(c))^2 + b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*s
in(4*b*log(c))^2)*n^6)*sin(4*b*log(x^n) + 4*a)^2 + 9*(4*(b^8*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 +
(b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6)*sin(2*b*log(x^n) + 2*a)^2 - 2*(4*b^8*n^8*cos(6*b*log(c))
+ b^6*n^6*cos(6*b*log(c)) + 3*(4*(b^8*cos(6*b*log(c))*cos(4*b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n
^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c)) + b^6*sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*cos(4*b*log(x^n) + 4*a)
- 3*(4*(b^8*cos(6*b*log(c))*cos(2*b*log(c)) + b^8*sin(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*
cos(2*b*log(c)) + b^6*sin(6*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) + 3*(4*(b^8*cos(4*b*log(c)
)*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(
6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*co
s(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)
))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6*(4*b^8*n^8*cos(4*b*log(c)) + b^6*n^6*cos(4*b*log(
c)) - 3*(4*(b^8*cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(
c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) - 3*(4*(b^8*cos(2*b*lo
g(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c)) - b^6*
cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 6*(4*b^8*n^8*cos(2*b*
log(c)) + b^6*n^6*cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 2*(4*b^8*n^8*sin(6*b*log(c)) + b^6*n^6*sin(6*b*lo
g(c)) + 3*(4*(b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(4*b*lo
g(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))*n^6)*cos(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(2*b*
log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(6*b*log(c)) - b^
6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) - 3*(4*(b^8*cos(6*b*log(c))*cos(4*b*log(c)) +
b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c)) + b^6*sin(6*b*log(c))*sin(4*b*
log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) + 3*(4*(b^8*cos(6*b*log(c))*cos(2*b*log(c)) + b^8*sin(6*b*log(c))*sin(2*
b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*sin(6*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(
x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) - 6*(4*b^8*n^8*sin(4*b*log(c)) + b^6*n^6*sin(4*b*log(c)) - 3*(4*(b^8*cos(
2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c))
- b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) + 3*(4*(b^8*cos(4*b*log(c))*cos(2*b*log(c)
) + b^8*sin(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(
2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a) + 6*(4*b^8*n^8*sin(2*b*log(c)) + b^6*n^6*si
n(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*integrate(-1/36*(cos(b*log(x^n) + a)*sin(b*log(c)) + cos(b*log(c))*sin
(b*log(x^n) + a))/(2*b^6*n^6*cos(b*log(c))*cos(b*log(x^n) + a) - 2*b^6*n^6*sin(b*log(c))*sin(b*log(x^n) + a) -
b^6*n^6 - (b^6*cos(b*log(c))^2 + b^6*sin(b*log(c))^2)*n^6*cos(b*log(x^n) + a)^2 - (b^6*cos(b*log(c))^2 + b^6*
sin(b*log(c))^2)*n^6*sin(b*log(x^n) + a)^2), x) + ((2*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*s
in(4*b*log(c)))*n + cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*x*cos(4*b*log(x^n) + 4*
a) - 2*(6*(b^2*cos(6*b*log(c))*cos(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(2*b*log(c))
*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n + cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*si
n(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) - (2*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*l
og(c)))*n - cos(4*b*log(c))*sin(6*b*log(c)) + cos(6*b*log(c))*sin(4*b*log(c)))*x*sin(4*b*log(x^n) + 4*a) - 2*(
6*(b^2*cos(2*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(2*b*log(c)))*n^2 - (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(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*lo
g(c)))*x*sin(2*b*log(x^n) + 2*a) + (4*b^2*n^2*cos(6*b*log(c)) + cos(6*b*log(c)))*x)*sin(6*b*log(x^n) + 6*a) +
(3*(12*(b^2*cos(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2 + 4*(b*cos(2*b*log(c))*
sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n + cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin
(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) + 3*(12*(b^2*cos(2*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin
(2*b*log(c)))*n^2 - 4*(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(
c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) - 2*(6*b^2*n^2*cos(4*b*log(c)
) + b*n*sin(4*b*log(c)) + cos(4*b*log(c)))*x)*sin(4*b*log(x^n) + 4*a))/(6*b^3*n^3*cos(2*b*log(c))*cos(2*b*log(
x^n) + 2*a) - 6*b^3*n^3*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) - b^3*n^3 - (b^3*cos(6*b*log(c))^2 + b^3*sin(6
*b*log(c))^2)*n^3*cos(6*b*log(x^n) + 6*a)^2 - 9*(b^3*cos(4*b*log(c))^2 + b^3*sin(4*b*log(c))^2)*n^3*cos(4*b*lo
g(x^n) + 4*a)^2 - 9*(b^3*cos(2*b*log(c))^2 + b^3*sin(2*b*log(c))^2)*n^3*cos(2*b*log(x^n) + 2*a)^2 - (b^3*cos(6
*b*log(c))^2 + b^3*sin(6*b*log(c))^2)*n^3*sin(6*b*log(x^n) + 6*a)^2 - 9*(b^3*cos(4*b*log(c))^2 + b^3*sin(4*b*l
og(c))^2)*n^3*sin(4*b*log(x^n) + 4*a)^2 - 9*(b^3*cos(2*b*log(c))^2 + b^3*sin(2*b*log(c))^2)*n^3*sin(2*b*log(x^
n) + 2*a)^2 + 2*(b^3*n^3*cos(6*b*log(c)) + 3*(b^3*cos(6*b*log(c))*cos(4*b*log(c)) + b^3*sin(6*b*log(c))*sin(4*
b*log(c)))*n^3*cos(4*b*log(x^n) + 4*a) - 3*(b^3*cos(6*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(2*b*
log(c)))*n^3*cos(2*b*log(x^n) + 2*a) + 3*(b^3*cos(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*lo
g(c)))*n^3*sin(4*b*log(x^n) + 4*a) - 3*(b^3*cos(2*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(2*b*log(
c)))*n^3*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) - 6*(b^3*n^3*cos(4*b*log(c)) - 3*(b^3*cos(4*b*log(c)
)*cos(2*b*log(c)) + b^3*sin(4*b*log(c))*sin(2*b*log(c)))*n^3*cos(2*b*log(x^n) + 2*a) - 3*(b^3*cos(2*b*log(c))*
sin(4*b*log(c)) - b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) -
2*(b^3*n^3*sin(6*b*log(c)) + 3*(b^3*cos(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)))*n^3
*cos(4*b*log(x^n) + 4*a) - 3*(b^3*cos(2*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(2*b*log(c)))*n^3*c
os(2*b*log(x^n) + 2*a) - 3*(b^3*cos(6*b*log(c))*cos(4*b*log(c)) + b^3*sin(6*b*log(c))*sin(4*b*log(c)))*n^3*sin
(4*b*log(x^n) + 4*a) + 3*(b^3*cos(6*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(2*b*log(c)))*n^3*sin(2
*b*log(x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) + 6*(b^3*n^3*sin(4*b*log(c)) - 3*(b^3*cos(2*b*log(c))*sin(4*b*log(
c)) - b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3*cos(2*b*log(x^n) + 2*a) + 3*(b^3*cos(4*b*log(c))*cos(2*b*log(c)
) + b^3*sin(4*b*log(c))*sin(2*b*log(c)))*n^3*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

Giac [F]

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

[In]

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

[Out]

integrate(csc(b*log(c*x^n) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {1}{{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^4} \,d x \]

[In]

int(1/sin(a + b*log(c*x^n))^4,x)

[Out]

int(1/sin(a + b*log(c*x^n))^4, x)