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

Optimal. Leaf size=115 \[ -\frac{\left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \text{Hypergeometric2F1}\left (\frac{1}{2} \left (-p+\frac{2 i}{b n}\right ),-p,\frac{1}{2} \left (\frac{2 i}{b n}-p+2\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2 (2+i b n p)} \]

[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))

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Rubi [A]  time = 0.0924682, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4493, 4491, 364} \[ -\frac{\left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (\frac{2 i}{b n}-p\right ),-p;\frac{1}{2} \left (-p+\frac{2 i}{b n}+2\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2 (2+i b n p)} \]

Antiderivative was successfully verified.

[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 4493

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])

Rule 4491

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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{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 ) \operatorname{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} \, _2F_1\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]  time = 0.627963, size = 100, normalized size = 0.87 \[ -\frac{i \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (1,\frac{i}{b n}+\frac{p}{2}+1,\frac{i}{b n}-\frac{p}{2}+1,e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{x^2 (b n p-2 i)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{p}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)