\(\int \cot (d (a+b \log (c x^n))) \, dx\) [212]

Optimal result

Integrand size = 13, antiderivative size = 66 \[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=i x-2 i x \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b d n},1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \] Output:

I*x-2*I*x*hypergeom([1, -1/2*I/b/d/n],[1-1/2*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. \(141\) vs. \(2(66)=132\).

Time = 7.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.14 \[ \int \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \left (-\frac {e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2 b d n},2-\frac {i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )}{-i+2 b d n}-i \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{2 b d n},1-\frac {i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.52, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5005, 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.

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

\(\Big \downarrow \) 5005

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

\(\Big \downarrow \) 5007

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

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 5005

Int[Cot[((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)*Cot[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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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