Integrand size = 23, antiderivative size = 150 \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {2 (e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {2 i+2 i m-3 b d n}{4 b d n},-\frac {2 i+2 i m-7 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+3 i b d n) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]
2*(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^(3/2)*hypergeom([3/2, 1/4 *(-2*I-2*I*m+3*b*d*n)/b/d/n],[1/4*(-2*I-2*I*m+7*b*d*n)/b/d/n],exp(2*I*a*d) *(c*x^n)^(2*I*b*d))/e/(2+2*m+3*I*b*d*n)/sin(d*(a+b*ln(c*x^n)))^(3/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(544\) vs. \(2(150)=300\).
Time = 3.99 (sec) , antiderivative size = 544, normalized size of antiderivative = 3.63 \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {\left (4+8 m+4 m^2+b^2 d^2 n^2\right ) x^{1+i b d n} (e x)^m \sqrt {2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {i \left (1+m+\frac {3}{2} i b d n\right )}{2 b d n},-\frac {2 i+2 i m-7 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+\frac {(-2 i-2 i m+3 b d n) x^{1-i b d n} (e x)^m \left (-2 x^{i b d n} \sqrt {-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} (b d n \cos (b d n \log (x))-2 (1+m) \sin (b d n \log (x)))+(-2 i-2 i m+b d n) \sqrt {2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m+b d n}{4 b d n},-\frac {2 i+2 i m-3 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}\right )}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}}{b d n (-2 i-2 i m+3 b d n) \sqrt {-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \left (b d n \cos \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+2 (1+m) \sin \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )} \]
((4 + 8*m + 4*m^2 + b^2*d^2*n^2)*x^(1 + I*b*d*n)*(e*x)^m*Sqrt[2 - 2*E^((2* I)*a*d)*(c*x^n)^((2*I)*b*d)]*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3 *I)/2)*b*d*n))/(b*d*n), -1/4*(2*I + (2*I)*m - 7*b*d*n)/(b*d*n), E^((2*I)*a *d)*(c*x^n)^((2*I)*b*d)] + ((-2*I - (2*I)*m + 3*b*d*n)*x^(1 - I*b*d*n)*(e* x)^m*(-2*x^(I*b*d*n)*Sqrt[((-I)*(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/ (E^(I*a*d)*(c*x^n)^(I*b*d))]*(b*d*n*Cos[b*d*n*Log[x]] - 2*(1 + m)*Sin[b*d* n*Log[x]]) + (-2*I - (2*I)*m + b*d*n)*Sqrt[2 - 2*E^((2*I)*a*d)*(c*x^n)^((2 *I)*b*d)]*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*d*n)/(b*d*n), -1/ 4*(2*I + (2*I)*m - 3*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]*Sq rt[Sin[d*(a + b*Log[c*x^n])]]))/Sqrt[Sin[d*(a + b*Log[c*x^n])]])/(b*d*n*(- 2*I - (2*I)*m + 3*b*d*n)*Sqrt[((-I)*(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d )))/(E^(I*a*d)*(c*x^n)^(I*b*d))]*(b*d*n*Cos[d*(a - b*n*Log[x] + b*Log[c*x^ n])] + 2*(1 + m)*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])]))
Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4996, 4994, 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.
\(\displaystyle \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {(e x)^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \frac {\left (c x^n\right )^{\frac {m+1}{n}-1}}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}d\left (c x^n\right )}{e n}\) |
\(\Big \downarrow \) 4994 |
\(\displaystyle \frac {(e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \left (c x^n\right )^{-\frac {m+1}{n}-\frac {3}{2} i b d} \int \frac {\left (c x^n\right )^{\frac {3 i b d}{2}+\frac {m+1}{n}-1}}{\left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}}d\left (c x^n\right )}{e n \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 (e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (m+1)}{b d n}\right ),-\frac {2 i m-7 b d n+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (3 i b d n+2 m+2) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}\) |
(2*(e*x)^(1 + m)*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^(3/2)*Hypergeomet ric2F1[3/2, (3 - ((2*I)*(1 + m))/(b*d*n))/4, -1/4*(2*I + (2*I)*m - 7*b*d*n )/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(2 + 2*m + (3*I)*b*d*n)* Sin[d*(a + b*Log[c*x^n])]^(3/2))
3.1.77.3.1 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_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] : > Simp[Sin[d*(a + b*Log[x])]^p*(x^(I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p ) Int[(e*x)^m*((1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)), x], x] /; Fr eeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_ .), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)) Subst[Int[x ^((m + 1)/n - 1)*Sin[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])
\[\int \frac {\left (e x \right )^{m}}{{\sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\int \frac {\left (e x\right )^{m}}{\sin ^{\frac {3}{2}}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}\, dx \]
\[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\int { \frac {\left (e x\right )^{m}}{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^{3/2}} \,d x \]