Integrand size = 17, antiderivative size = 158 \[ \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(2 i-b d n) x^2}{2 b d n}+\frac {i x^2 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {2 i x^2 \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n} \]
1/2*(2*I-b*d*n)*x^2/b/d/n+I*x^2*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/b/d/n/( 1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))-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))/b/d/n
Time = 3.89 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11 \[ \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x^2 \left (2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{b d n},2-\frac {i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-i+b d n) \left (b d n+2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+2 i \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )\right )}{2 b d n (-i+b d n)} \]
-1/2*(x^2*(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)*(b*d*n + 2*Cot[d*(a + b*Log[c*x^n])] + (2*I)*Hypergeometric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))])))/(b*d*n*(-I + b*d*n))
Time = 0.43 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5009, 5007, 1004, 27, 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.
\(\displaystyle \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 5009 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \int \left (c x^n\right )^{\frac {2}{n}-1} \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 5007 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \int \frac {\left (c x^n\right )^{\frac {2}{n}-1} \left (-i e^{2 i a d} \left (c x^n\right )^{2 i b d}-i\right )^2}{\left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 1004 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {i \left (c x^n\right )^{2/n} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i e^{-2 i a d} \int \frac {2 \left (c x^n\right )^{\frac {2}{n}-1} \left (\frac {e^{4 i a d} (i b d n+2) \left (c x^n\right )^{2 i b d}}{n}+\frac {e^{2 i a d} (2-i b d n)}{n}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}d\left (c x^n\right )}{2 b d}\right )}{n}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {i \left (c x^n\right )^{2/n} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i e^{-2 i a d} \int \frac {\left (c x^n\right )^{\frac {2}{n}-1} \left (\frac {e^{4 i a d} (i b d n+2) \left (c x^n\right )^{2 i b d}}{n}+\frac {e^{2 i a d} (2-i b d n)}{n}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}d\left (c x^n\right )}{b d}\right )}{n}\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {i \left (c x^n\right )^{2/n} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i e^{-2 i a d} \left (\frac {4 e^{2 i a d} \int \frac {\left (c x^n\right )^{\frac {2}{n}-1}}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}d\left (c x^n\right )}{n}-\frac {1}{2} e^{2 i a d} (2+i b d n) \left (c x^n\right )^{2/n}\right )}{b d}\right )}{n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {i \left (c x^n\right )^{2/n} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {i e^{-2 i a d} \left (2 e^{2 i a d} \left (c x^n\right )^{2/n} \operatorname {Hypergeometric2F1}\left (1,-\frac {i}{b d n},1-\frac {i}{b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-\frac {1}{2} e^{2 i a d} (2+i b d n) \left (c x^n\right )^{2/n}\right )}{b d}\right )}{n}\) |
(x^2*((I*(c*x^n)^(2/n)*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(b*d*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))) - (I*(-1/2*(E^((2*I)*a*d)*(2 + I*b*d*n )*(c*x^n)^(2/n)) + 2*E^((2*I)*a*d)*(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)]))/(b*d*E^((2*I )*a*d))))/(n*(c*x^n)^(2/n))
3.3.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-(c*b - a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1) *((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Simp[1/(a*b*n*(p + 1)) Int [(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c *b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && Lt Q[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
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])
\[\int x {\cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}d x\]
\[ \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
\[ \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x \cot ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
\[ \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
1/2*((b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*n*x^2*cos(2*b*d*l og(x^n) + 2*a*d)^2 + (b*d*cos(2*b*d*log(c))^2 + b*d*sin(2*b*d*log(c))^2)*n *x^2*sin(2*b*d*log(x^n) + 2*a*d)^2 + b*d*n*x^2 - 2*(b*d*n*cos(2*b*d*log(c) ) - 2*sin(2*b*d*log(c)))*x^2*cos(2*b*d*log(x^n) + 2*a*d) + 2*(b*d*n*sin(2* b*d*log(c)) + 2*cos(2*b*d*log(c)))*x^2*sin(2*b*d*log(x^n) + 2*a*d) - 4*(2* b^2*d^2*n^2*cos(2*b*d*log(c))*cos(2*b*d*log(x^n) + 2*a*d) - 2*b^2*d^2*n^2* sin(2*b*d*log(c))*sin(2*b*d*log(x^n) + 2*a*d) - b^2*d^2*n^2 - (b^2*d^2*cos (2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2*cos(2*b*d*log(x^n) + 2 *a*d)^2 - (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2* sin(2*b*d*log(x^n) + 2*a*d)^2)*integrate((x*cos(b*d*log(x^n) + a*d)*sin(b* d*log(c)) + x*cos(b*d*log(c))*sin(b*d*log(x^n) + a*d))/(2*b^2*d^2*n^2*cos( b*d*log(c))*cos(b*d*log(x^n) + a*d) - 2*b^2*d^2*n^2*sin(b*d*log(c))*sin(b* d*log(x^n) + a*d) + b^2*d^2*n^2 + (b^2*d^2*cos(b*d*log(c))^2 + b^2*d^2*sin (b*d*log(c))^2)*n^2*cos(b*d*log(x^n) + a*d)^2 + (b^2*d^2*cos(b*d*log(c))^2 + b^2*d^2*sin(b*d*log(c))^2)*n^2*sin(b*d*log(x^n) + a*d)^2), x) + 4*(2*b^ 2*d^2*n^2*cos(2*b*d*log(c))*cos(2*b*d*log(x^n) + 2*a*d) - 2*b^2*d^2*n^2*si n(2*b*d*log(c))*sin(2*b*d*log(x^n) + 2*a*d) - b^2*d^2*n^2 - (b^2*d^2*cos(2 *b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2*cos(2*b*d*log(x^n) + 2*a *d)^2 - (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2*si n(2*b*d*log(x^n) + 2*a*d)^2)*integrate(-(x*cos(b*d*log(x^n) + a*d)*sin(...
Timed out. \[ \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int x \cot ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x\,{\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2 \,d x \]