\(\int x^2 \csc (a+b \log (c x^n)) \, dx\) [289]

Optimal result

Integrand size = 15, antiderivative size = 86 \[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right ),\frac {3}{2} \left (1-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i-b n} \] Output:

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

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {3 i}{2 b n},\frac {3}{2}-\frac {3 i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-3 i+b n} \] Input:

Integrate[x^2*Csc[a + b*Log[c*x^n]],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5021, 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.

\(\displaystyle \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 5021

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

\(\Big \downarrow \) 5017

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

\(\Big \downarrow \) 888

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

Input:

Int[x^2*Csc[a + b*Log[c*x^n]],x]
 

Output:

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

rule 5021

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

Maple [F]

\[\int x^{2} \csc \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]

Input:

int(x^2*csc(a+b*ln(c*x^n)),x)
 

Output:

int(x^2*csc(a+b*ln(c*x^n)),x)
 

Fricas [F]

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

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

Output:

integral(x^2*csc(b*log(c*x^n) + a), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {x^2}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

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

int(x^2*csc(a+b*log(c*x^n)),x)
 

Output:

int(csc(log(x**n*c)*b + a)*x**2,x)