Integrand size = 15, antiderivative size = 190 \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
x*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(-I*(1+exp(2*I*a*d)*(c*x^n)^(2*I*b* d))/(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d)))^p*AppellF1(-1/2*I/b/d/n,p,-p,1-1/2 *I/b/d/n,exp(2*I*a*d)*(c*x^n)^(2*I*b*d),-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/( (1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(458\) vs. \(2(190)=380\).
Time = 1.03 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.41 \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(-i+2 b d n) x \left (\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{2 b d e^{2 i a d} n p \left (c x^n\right )^{2 i b d} \operatorname {AppellF1}\left (1-\frac {i}{2 b d n},p,1-p,2-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+2 b d e^{2 i a d} n p \left (c x^n\right )^{2 i b d} \operatorname {AppellF1}\left (1-\frac {i}{2 b d n},1+p,-p,2-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(-i+2 b d n) \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \]
((-I + 2*b*d*n)*x*((I*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(-1 + E^((2 *I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*AppellF1[(-1/2*I)/(b*d*n), p, -p, 1 - (I/ 2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2 *I)*b*d))])/(2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)^((2*I)*b*d)*AppellF1[1 - (I/2 )/(b*d*n), p, 1 - p, 2 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + 2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)^( (2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), 1 + p, -p, 2 - (I/2)/(b*d*n), E^((2 *I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + (-I + 2*b*d*n)*AppellF1[(-1/2*I)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((2*I)*a *d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])
Time = 0.44 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5005, 5007, 2058, 1013, 1012}
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.
\(\displaystyle \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 5005 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 5007 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \left (\frac {-i e^{2 i a d} \left (c x^n\right )^{2 i b d}-i}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^pd\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (-i e^{2 i a d} \left (c x^n\right )^{2 i b d}-i\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \int \left (c x^n\right )^{\frac {1}{n}-1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-i e^{2 i a d} \left (c x^n\right )^{2 i b d}-i\right )^pd\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \int \left (c x^n\right )^{\frac {1}{n}-1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (e^{2 i a d} \left (c x^n\right )^{2 i b d}+1\right )^pd\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},p,-p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )\) |
(x*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p*(((-I)*(1 + E^((2*I)*a*d)*(c* x^n)^((2*I)*b*d)))/(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*AppellF1[(-1 /2*I)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d) , -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/(1 + E^((2*I)*a*d)*(c*x^n)^((2*I) *b*d))^p
3.3.29.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
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])
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]
\[\int {\cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{p}d x\]
\[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]
\[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \cot ^{p}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]
\[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]
Timed out. \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p \,d x \]