\(\int \frac {1}{\sin ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [67]

Optimal result

Integrand size = 15, antiderivative size = 109 \[ \int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),\frac {1}{4} \left (9-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+5 i b n) \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4571, 4579, 371} \[ \int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),\frac {1}{4} \left (9-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+5 i b n) \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[Sin[a + b*Log[c*x^n]]^(-5/2),x]

[Out]

(2*x*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^(5/2)*Hypergeometric2F1[5/2, (5 - (2*I)/(b*n))/4, (9 - (2*I)/(b*n))/4
, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/((2 + (5*I)*b*n)*Sin[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 4571

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sin[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])

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]

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

Mathematica [A] (verified)

Time = 2.43 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \left (\frac {(2-i b n) \sqrt {2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4}-\frac {i}{2 b n},\frac {5}{4}-\frac {i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {-i e^{-i a} \left (c x^n\right )^{-i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )}}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right )+2 \sin \left (a+b \log \left (c x^n\right )\right )}{\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\right )}{3 b^2 n^2} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {1}{{\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {5}{2}}}d x\]

[In]

int(1/sin(a+b*ln(c*x^n))^(5/2),x)

[Out]

int(1/sin(a+b*ln(c*x^n))^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/sin(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. \[ \int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/sin(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/sin(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/sin(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2}} \,d x \]

[In]

int(1/sin(a + b*log(c*x^n))^(5/2),x)

[Out]

int(1/sin(a + b*log(c*x^n))^(5/2), x)