\(\int \frac {\sin ^p(a+b \log (c x^n))}{x^3} \, dx\) [85]

Optimal result

Integrand size = 17, antiderivative size = 115 \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {\left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {2 i}{b n}-p\right ),-p,\frac {1}{2} \left (2+\frac {2 i}{b n}-p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{(2+i b n p) x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4581, 4579, 371} \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {\left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {2 i}{b n}-p\right ),-p,\frac {1}{2} \left (-p+\frac {2 i}{b n}+2\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2 (2+i b n p)} \]

[In]

Int[Sin[a + b*Log[c*x^n]]^p/x^3,x]

[Out]

-((Hypergeometric2F1[((2*I)/(b*n) - p)/2, -p, (2 + (2*I)/(b*n) - p)/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sin[a +
b*Log[c*x^n]]^p)/((2 + I*b*n*p)*x^2*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^p))

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

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.23 \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {\left (2-2 e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \left (-i e^{-i a} \left (c x^n\right )^{-i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )^p \operatorname {Hypergeometric2F1}\left (\frac {i}{b n}-\frac {p}{2},-p,1+\frac {i}{b n}-\frac {p}{2},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-2-i b n p) x^2} \]

[In]

Integrate[Sin[a + b*Log[c*x^n]]^p/x^3,x]

[Out]

((((-I)*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)))/(E^(I*a)*(c*x^n)^(I*b)))^p*Hypergeometric2F1[I/(b*n) - p/2, -p,
1 + I/(b*n) - p/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/((-2 - I*b*n*p)*x^2*(2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b))^p
)

Maple [F]

\[\int \frac {{\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}}{x^{3}}d x\]

[In]

int(sin(a+b*ln(c*x^n))^p/x^3,x)

[Out]

int(sin(a+b*ln(c*x^n))^p/x^3,x)

Fricas [F]

\[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{3}} \,d x } \]

[In]

integrate(sin(a+b*log(c*x^n))^p/x^3,x, algorithm="fricas")

[Out]

integral(sin(b*log(c*x^n) + a)^p/x^3, x)

Sympy [F]

\[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]

[In]

integrate(sin(a+b*ln(c*x**n))**p/x**3,x)

[Out]

Integral(sin(a + b*log(c*x**n))**p/x**3, x)

Maxima [F]

\[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{3}} \,d x } \]

[In]

integrate(sin(a+b*log(c*x^n))^p/x^3,x, algorithm="maxima")

[Out]

integrate(sin(b*log(c*x^n) + a)^p/x^3, x)

Giac [F]

\[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{3}} \,d x } \]

[In]

integrate(sin(a+b*log(c*x^n))^p/x^3,x, algorithm="giac")

[Out]

integrate(sin(b*log(c*x^n) + a)^p/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{x^3} \,d x \]

[In]

int(sin(a + b*log(c*x^n))^p/x^3,x)

[Out]

int(sin(a + b*log(c*x^n))^p/x^3, x)