Integrand size = 21, antiderivative size = 210 \[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i (1+m)}{2 b d n},p,-p,1-\frac {i (1+m)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m)} \] Output:
(e*x)^(1+m)*(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*(1+m)/ b/d/n,p,-p,1-1/2*I*(1+m)/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))/e/(1+m)/((1+exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^p)
Time = 0.89 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.98 \[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x (e x)^m \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i (1+m)}{2 b d n},p,-p,1-\frac {i (1+m)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1+m} \] Input:
Integrate[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^p,x]
Output:
(x*(e*x)^m*(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*Appell F1[((-1/2*I)*(1 + m))/(b*d*n), p, -p, 1 - ((I/2)*(1 + m))/(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 + m )*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p)
Time = 0.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5009, 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.
\(\displaystyle \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 5009 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 5007 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \left (\frac {-i e^{2 i a d} \left (c x^n\right )^{2 i b d}-i}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^pd\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (-i e^{2 i a d} \left (c x^n\right )^{2 i b d}-i\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \int \left (c x^n\right )^{\frac {m+1}{n}-1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-i e^{2 i a d} \left (c x^n\right )^{2 i b d}-i\right )^pd\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \int \left (c x^n\right )^{\frac {m+1}{n}-1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (e^{2 i a d} \left (c x^n\right )^{2 i b d}+1\right )^pd\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {(e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i (m+1)}{2 b d n},p,-p,1-\frac {i (m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (m+1)}\) |
Input:
Int[(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^p,x]
Output:
((e*x)^(1 + m)*(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)*(1 + m))/(b*d*n), p, -p, 1 - ((I/2)*(1 + m))/(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))])/( e*(1 + m)*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^p)
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[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])
\[\int \left (e x \right )^{m} {\cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{p}d x\]
Input:
int((e*x)^m*cot(d*(a+b*ln(c*x^n)))^p,x)
Output:
int((e*x)^m*cot(d*(a+b*ln(c*x^n)))^p,x)
\[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \] Input:
integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^p,x, algorithm="fricas")
Output:
integral((e*x)^m*cot(b*d*log(c*x^n) + a*d)^p, x)
\[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \cot ^{p}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \] Input:
integrate((e*x)**m*cot(d*(a+b*ln(c*x**n)))**p,x)
Output:
Integral((e*x)**m*cot(a*d + b*d*log(c*x**n))**p, x)
\[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \] Input:
integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^p,x, algorithm="maxima")
Output:
integrate((e*x)^m*cot((b*log(c*x^n) + a)*d)^p, x)
Timed out. \[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \] Input:
integrate((e*x)^m*cot(d*(a+b*log(c*x^n)))^p,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p\,{\left (e\,x\right )}^m \,d x \] Input:
int(cot(d*(a + b*log(c*x^n)))^p*(e*x)^m,x)
Output:
int(cot(d*(a + b*log(c*x^n)))^p*(e*x)^m, x)
\[ \int (e x)^m \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {e^{m} \left (x^{m} {\cot \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}^{p} x +\left (\int \frac {x^{m} {\cot \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}^{p}}{\cot \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}d x \right ) b d n p +\left (\int x^{m} {\cot \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}^{p} \cot \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )d x \right ) b d n p \right )}{m +1} \] Input:
int((e*x)^m*cot(d*(a+b*log(c*x^n)))^p,x)
Output:
(e**m*(x**m*cot(log(x**n*c)*b*d + a*d)**p*x + int((x**m*cot(log(x**n*c)*b* d + a*d)**p)/cot(log(x**n*c)*b*d + a*d),x)*b*d*n*p + int(x**m*cot(log(x**n *c)*b*d + a*d)**p*cot(log(x**n*c)*b*d + a*d),x)*b*d*n*p))/(m + 1)