\(\int x \cot (d (a+b \log (c x^n))) \, dx\) [211]

Optimal result

Integrand size = 15, antiderivative size = 68 \[ \int x \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {i x^2}{2}-i x^2 \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(219\) vs. \(2(68)=136\).

Time = 4.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.22 \[ \int x \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x^2 \left (e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{b d n},2-\frac {i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-i+b d n) \left (\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+i \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))\right )\right )}{-2 i+2 b d n} \] Input:

Integrate[x*Cot[d*(a + b*Log[c*x^n])],x]
 

Output:

-((x^2*(E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - I/(b*d*n), 
 2 - I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-I + b*d*n)*(Cot[d*(a + 
 b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^n])] + I*Hypergeometri 
c2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + Csc 
[d*(a + b*Log[c*x^n])]*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sin[b*d*n*Lo 
g[x]])))/(-2*I + 2*b*d*n))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.53, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5009, 5007, 959, 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.

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

\(\Big \downarrow \) 5009

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

\(\Big \downarrow \) 5007

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 888

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

Input:

Int[x*Cot[d*(a + b*Log[c*x^n])],x]
 

Output:

(x^2*((I/2)*n*(c*x^n)^(2/n) - I*n*(c*x^n)^(2/n)*Hypergeometric2F1[1, (-I)/ 
(b*d*n), 1 - I/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]))/(n*(c*x^n)^(2 
/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 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 5007

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

rule 5009

Int[Cot[((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)*Cot[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]

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

Input:

int(x*cot(d*(a+b*ln(c*x^n))),x)
 

Output:

int(x*cot(d*(a+b*ln(c*x^n))),x)
 

Fricas [F]

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

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

Output:

integral(x*cot(b*d*log(c*x^n) + a*d), x)
 

Sympy [F]

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

integrate(x*cot(d*(a+b*ln(c*x**n))),x)
 

Output:

Integral(x*cot(a*d + b*d*log(c*x**n)), x)
 

Maxima [F]

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

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

Output:

integrate(x*cot((b*log(c*x^n) + a)*d), x)
 

Giac [F]

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

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

Output:

integrate(x*cot((b*log(c*x^n) + a)*d), x)
 

Mupad [F(-1)]

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

int(x*cot(d*(a + b*log(c*x^n))),x)
 

Output:

int(x*cot(d*(a + b*log(c*x^n))), x)
 

Reduce [F]

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

int(x*cot(d*(a+b*log(c*x^n))),x)
 

Output:

int(cot(log(x**n*c)*b*d + a*d)*x,x)