Integrand size = 21, antiderivative size = 119 \[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {4 e^{2 i a d} (e x)^{1+m} \left (c x^n\right )^{2 i b d} \operatorname {Hypergeometric2F1}\left (2,-\frac {i (1+m)-2 b d n}{2 b d n},-\frac {i (1+m)-4 b d n}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m+2 i b d n)} \]
-4*exp(2*I*a*d)*(e*x)^(1+m)*(c*x^n)^(2*I*b*d)*hypergeom([2, 1/2*(-I*(1+m)+ 2*b*d*n)/b/d/n],[1/2*(-I*(1+m)+4*b*d*n)/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b *d))/e/(1+m+2*I*b*d*n)
Time = 13.66 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.89 \[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x (e x)^m \left ((1+m+2 i b d n) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+i (1+m+2 i b d n) \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m)}{2 b d n},1-\frac {i (1+m)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+i e^{2 i a d} (1+m) \left (c x^n\right )^{2 i b d} \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m+2 i b d n)}{2 b d n},-\frac {i (1+m+4 i b d n)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )\right )}{b d n (1+m+2 i b d n)} \]
-((x*(e*x)^m*((1 + m + (2*I)*b*d*n)*Cot[d*(a + b*Log[c*x^n])] + I*(1 + m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m))/(b*d*n), 1 - ((I/2)* (1 + m))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] + I*E^((2*I)*a*d)*(1 + m )*(c*x^n)^((2*I)*b*d)*Hypergeometric2F1[1, ((-1/2*I)*(1 + m + (2*I)*b*d*n) )/(b*d*n), ((-1/2*I)*(1 + m + (4*I)*b*d*n))/(b*d*n), E^((2*I)*a*d)*(c*x^n) ^((2*I)*b*d)]))/(b*d*n*(1 + m + (2*I)*b*d*n)))
Time = 0.34 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5021, 5017, 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.
\(\displaystyle \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 5021 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 5017 |
\(\displaystyle -\frac {4 e^{2 i a d} (e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \frac {\left (c x^n\right )^{2 i b d+\frac {m+1}{n}-1}}{\left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^2}d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {4 e^{2 i a d} (e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}+\frac {2 i b d n+m+1}{n}} \operatorname {Hypergeometric2F1}\left (2,-\frac {i (m+1)-2 b d n}{2 b d n},-\frac {i (m+1)-4 b d n}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2 i b d n+m+1)}\) |
(-4*E^((2*I)*a*d)*(e*x)^(1 + m)*(c*x^n)^(-((1 + m)/n) + (1 + m + (2*I)*b*d *n)/n)*Hypergeometric2F1[2, -1/2*(I*(1 + m) - 2*b*d*n)/(b*d*n), -1/2*(I*(1 + m) - 4*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(1 + m + (2*I)*b*d*n))
3.4.21.3.1 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[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*I)^p*E^(I*a*d*p) Int[(e*x)^m*(x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^ (2*I*b*d))^p), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]
Int[Csc[((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)*Csc[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])
\[\int \left (e x \right )^{m} {\csc \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}d x\]
\[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
\[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \csc ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
\[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
(2*e^m*x*x^m*cos(2*b*d*log(x^n) + 2*a*d)*sin(2*b*d*log(c)) + 2*e^m*x*x^m*c os(2*b*d*log(c))*sin(2*b*d*log(x^n) + 2*a*d) + (((b^2*d^2*cos(2*b*d*log(c) )^2 + b^2*d^2*sin(2*b*d*log(c))^2)*e^m*m + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*e^m)*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)*e^m*m + (b^2*d^2* cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*e^m)*n^2*sin(2*b*d*log( x^n) + 2*a*d)^2 - 2*(b^2*d^2*e^m*m*cos(2*b*d*log(c)) + b^2*d^2*e^m*cos(2*b *d*log(c)))*n^2*cos(2*b*d*log(x^n) + 2*a*d) + 2*(b^2*d^2*e^m*m*sin(2*b*d*l og(c)) + b^2*d^2*e^m*sin(2*b*d*log(c)))*n^2*sin(2*b*d*log(x^n) + 2*a*d) + (b^2*d^2*e^m*m + b^2*d^2*e^m)*n^2)*integrate((x^m*cos(b*d*log(x^n) + a*d)* sin(b*d*log(c)) + x^m*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*l og(c))^2 + b^2*d^2*sin(b*d*log(c))^2)*n^2*sin(b*d*log(x^n) + a*d)^2), x) - (((b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*e^m*m + (b^ 2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*e^m)*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)*e^m*m + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c) )^2)*e^m)*n^2*sin(2*b*d*log(x^n) + 2*a*d)^2 - 2*(b^2*d^2*e^m*m*cos(2*b*...
\[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
Timed out. \[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \frac {{\left (e\,x\right )}^m}{{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2} \,d x \]