3.81 \(\int x \sin ^p(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=114 \[ \frac {x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \, _2F_1\left (\frac {1}{2} \left (-p-\frac {2 i}{b n}\right ),-p;\frac {1}{2} \left (-p-\frac {2 i}{b n}+2\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{2-i b n p} \]

[Out]

x^2*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)/
((1-exp(2*I*a)*(c*x^n)^(2*I*b))^p)

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Rubi [A]  time = 0.08, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4493, 4491, 364} \[ \frac {x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \, _2F_1\left (\frac {1}{2} \left (-p-\frac {2 i}{b n}\right ),-p;\frac {1}{2} \left (-p-\frac {2 i}{b n}+2\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{2-i b n p} \]

Antiderivative was successfully verified.

[In]

Int[x*Sin[a + b*Log[c*x^n]]^p,x]

[Out]

(x^2*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)*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^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])

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 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])

Rubi steps

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

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Mathematica [A]  time = 0.61, size = 98, normalized size = 0.86 \[ \frac {x^2 \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \, _2F_1\left (1,\frac {p}{2}-\frac {i}{b n}+1;-\frac {p}{2}-\frac {i}{b n}+1;e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{-2+i b n p} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1 + E^((2*I)*(a + b*Log[c*x^n])))*x^2*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)

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int x \left (\sin ^{p}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(a + b*log(c*x^n))^p,x)

[Out]

int(x*sin(a + b*log(c*x^n))^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sin ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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