Integrand size = 17, antiderivative size = 74 \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {i x^3}{3}-\frac {2}{3} i x^3 \operatorname {Hypergeometric2F1}\left (1,-\frac {3 i}{2 b d n},1-\frac {3 i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
1/3*I*x^3-2/3*I*x^3*hypergeom([1, -3/2*I/b/d/n],[1-3/2*I/b/d/n],exp(2*I*a* d)*(c*x^n)^(2*I*b*d))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(229\) vs. \(2(74)=148\).
Time = 4.38 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.09 \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x^3 \left (3 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {3 i}{2 b d n},2-\frac {3 i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-3 i+2 b d n) \left (\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+i \operatorname {Hypergeometric2F1}\left (1,-\frac {3 i}{2 b d n},1-\frac {3 i}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))\right )\right )}{-9 i+6 b d n} \]
-((x^3*(3*E^((2*I)*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - ((3*I)/2 )/(b*d*n), 2 - ((3*I)/2)/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + (-3*I + 2*b*d*n)*(Cot[d*(a + b*Log[c*x^n])] - Cot[d*(a - b*n*Log[x] + b*Log[c*x^ n])] + I*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(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*Log[x]])))/(-9*I + 6*b*d*n))
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.49, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, 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.
\(\displaystyle \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 5009 |
\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \int \left (c x^n\right )^{\frac {3}{n}-1} \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 5007 |
\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \int \frac {\left (c x^n\right )^{\frac {3}{n}-1} \left (-i e^{2 i a d} \left (c x^n\right )^{2 i b d}-i\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \left (\frac {1}{3} i n \left (c x^n\right )^{3/n}-2 i \int \frac {\left (c x^n\right )^{\frac {3}{n}-1}}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}d\left (c x^n\right )\right )}{n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \left (\frac {1}{3} i n \left (c x^n\right )^{3/n}-\frac {2}{3} i n \left (c x^n\right )^{3/n} \operatorname {Hypergeometric2F1}\left (1,-\frac {3 i}{2 b d n},1-\frac {3 i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )\right )}{n}\) |
(x^3*((I/3)*n*(c*x^n)^(3/n) - ((2*I)/3)*n*(c*x^n)^(3/n)*Hypergeometric2F1[ 1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I) *b*d)]))/(n*(c*x^n)^(3/n))
3.3.10.3.1 Defintions of rubi rules used
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])
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]
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]
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])
\[\int x^{2} \cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
\[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{2} \cot {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
\[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^2\,\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]