\(\int \frac {(e x)^m}{\sin ^{\frac {5}{2}}(d (a+b \log (c x^n)))} \, dx\) [78]

Optimal result

Integrand size = 23, antiderivative size = 150 \[ \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {2 (e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {2 i+2 i m-5 b d n}{4 b d n},-\frac {2 i+2 i m-9 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+5 i b d n) \sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]

[Out]

2*(e*x)^(1+m)*(1-exp(2*I*a*d)*(c*x^n)^(2*I*b*d))^(5/2)*hypergeom([5/2, 1/4*(-2*I-2*I*m+5*b*d*n)/b/d/n],[1/4*(-
2*I-2*I*m+9*b*d*n)/b/d/n],exp(2*I*a*d)*(c*x^n)^(2*I*b*d))/e/(2+2*m+5*I*b*d*n)/sin(d*(a+b*ln(c*x^n)))^(5/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4581, 4579, 371} \[ \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {2 (e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i (m+1)}{b d n}\right ),-\frac {2 i m-9 b d n+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (5 i b d n+2 m+2) \sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]

[In]

Int[(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(5/2),x]

[Out]

(2*(e*x)^(1 + m)*(1 - E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^(5/2)*Hypergeometric2F1[5/2, (5 - ((2*I)*(1 + m))/(b*
d*n))/4, -1/4*(2*I + (2*I)*m - 9*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(2 + 2*m + (5*I)*b*d*n
)*Sin[d*(a + b*Log[c*x^n])]^(5/2))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4579

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] :> Dist[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] /
; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rule 4581

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\sin ^{\frac {5}{2}}(d (a+b \log (x)))} \, dx,x,c x^n\right )}{e n} \\ & = \frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {5}{2} i b d-\frac {1+m}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {5 i b d}{2}+\frac {1+m}{n}}}{\left (1-e^{2 i a d} x^{2 i b d}\right )^{5/2}} \, dx,x,c x^n\right )}{e n \sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \\ & = \frac {2 (e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i (1+m)}{b d n}\right ),-\frac {2 i+2 i m-9 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+5 i b d n) \sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.37 \[ \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {x (e x)^m \left (-2 b d n \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )+i e^{-i d \left (a+b \log \left (c x^n\right )\right )} \left (1-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )^{3/2} (2+2 m-i b d n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {-2 i-2 i m+b d n}{4 b d n},-\frac {2 i+2 i m-5 b d n}{4 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-4 (1+m) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{3 b^2 d^2 n^2 \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]

[In]

Integrate[(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(5/2),x]

[Out]

(x*(e*x)^m*(-2*b*d*n*Cos[d*(a + b*Log[c*x^n])] + (I*(1 - E^((2*I)*d*(a + b*Log[c*x^n])))^(3/2)*(2 + 2*m - I*b*
d*n)*Hypergeometric2F1[1/2, (-2*I - (2*I)*m + b*d*n)/(4*b*d*n), -1/4*(2*I + (2*I)*m - 5*b*d*n)/(b*d*n), E^((2*
I)*d*(a + b*Log[c*x^n]))])/E^(I*d*(a + b*Log[c*x^n])) - 4*(1 + m)*Sin[d*(a + b*Log[c*x^n])]))/(3*b^2*d^2*n^2*S
in[d*(a + b*Log[c*x^n])]^(3/2))

Maple [F]

\[\int \frac {\left (e x \right )^{m}}{{\sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{\frac {5}{2}}}d x\]

[In]

int((e*x)^m/sin(d*(a+b*ln(c*x^n)))^(5/2),x)

[Out]

int((e*x)^m/sin(d*(a+b*ln(c*x^n)))^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((e*x)^m/sin(d*(a+b*log(c*x^n)))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate((e*x)**m/sin(d*(a+b*ln(c*x**n)))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\int { \frac {\left (e x\right )^{m}}{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^m/sin(d*(a+b*log(c*x^n)))^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x)^m/sin((b*log(c*x^n) + a)*d)^(5/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate((e*x)^m/sin(d*(a+b*log(c*x^n)))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m}{\sin ^{\frac {5}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^{5/2}} \,d x \]

[In]

int((e*x)^m/sin(d*(a + b*log(c*x^n)))^(5/2),x)

[Out]

int((e*x)^m/sin(d*(a + b*log(c*x^n)))^(5/2), x)