3.133 \(\int x \cos ^p(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=114 \[ \frac{x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \text{Hypergeometric2F1}\left (\frac{1}{2} \left (-p-\frac{2 i}{b n}\right ),-p,\frac{1}{2} \left (-\frac{2 i}{b n}-p+2\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{2-i b n p} \]

[Out]

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

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Rubi [A]  time = 0.0768798, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4494, 4492, 364} \[ \frac{x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \, _2F_1\left (\frac{1}{2} \left (-p-\frac{2 i}{b n}\right ),-p;\frac{1}{2} \left (-p-\frac{2 i}{b n}+2\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{2-i b n p} \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[a + b*Log[c*x^n]]^p,x]

[Out]

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

Rule 4494

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Cos[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 4492

Int[Cos[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(Cos[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 x \cos ^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}} \cos ^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} \cos ^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} \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, _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 )}{2-i b n p}\\ \end{align*}

Mathematica [A]  time = 0.633202, size = 102, normalized size = 0.89 \[ \frac{i x^2 \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (1,-\frac{i}{b n}+\frac{p}{2}+1,-\frac{i}{b n}-\frac{p}{2}+1,-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{b n p+2 i} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.127, size = 0, normalized size = 0. \begin{align*} \int x \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos ^{p}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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