\(\int \csc ^3(a+b \log (c x^n)) \, dx\) [297]

Optimal result

Integrand size = 13, antiderivative size = 84 \[ \int \csc ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {8 e^{3 i a} x \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right ),\frac {1}{2} \left (5-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i-3 b n} \] Output:

-8*exp(3*I*a)*x*(c*x^n)^(3*I*b)*hypergeom([3, 3/2-1/2*I/b/n],[5/2-1/2*I/b/ 
n],exp(2*I*a)*(c*x^n)^(2*I*b))/(I-3*b*n)
 

Mathematica [A] (verified)

Time = 4.62 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.39 \[ \int \csc ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {x \left (\left (1+b n \cot \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right )+2 e^{i a} (i+b n) \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {i}{2 b n},\frac {3}{2}-\frac {i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{2 b^2 n^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5015, 5017, 888}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \csc ^3\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 5015

\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \csc ^3\left (a+b \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 5017

\(\displaystyle \frac {8 i e^{3 i a} x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{3 i b+\frac {1}{n}-1}}{\left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^3}d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {8 i e^{3 i a} x \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {i}{b n}\right ),\frac {1}{2} \left (5-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{n \left (\frac {1}{n}+3 i b\right )}\)

Input:

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

Output:

((8*I)*E^((3*I)*a)*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)])/(((3*I)*b + n^(-1))*n)
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 5015

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Si 
mp[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 5017

Int[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(-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]
 

Maple [F]

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

Input:

int(csc(a+b*ln(c*x^n))^3,x)
 

Output:

int(csc(a+b*ln(c*x^n))^3,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-((b*n*cos(b*log(c)) - sin(b*log(c)))*x*cos(b*log(x^n) + a) - (b*n*sin(b*l 
og(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 + co 
s(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))*cos(3*b*log(c)) + sin(4*b*log(c))*sin(3*b*lo 
g(c)))*x*sin(3*b*log(x^n) + 3*a) + ((b*cos(b*log(c))*sin(4*b*log(c)) - b*c 
os(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*c 
os(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*lo 
g(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...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \csc ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {1}{{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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