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

Optimal. Leaf size=112 \[ \frac {x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \, _2F_1\left (-p,-\frac {b n p+i}{2 b n};\frac {1}{2} \left (-p-\frac {i}{b n}+2\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{1-i b n p} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4484, 4492, 364} \[ \frac {x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \, _2F_1\left (-p,-\frac {b n p+i}{2 b n};\frac {1}{2} \left (-p-\frac {i}{b n}+2\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{1-i b n p} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*Log[c*x^n]]^p,x]

[Out]

(x*Cos[a + b*Log[c*x^n]]^p*Hypergeometric2F1[-p, -(I + b*n*p)/(2*b*n), (2 - I/(b*n) - p)/2, -(E^((2*I)*a)*(c*x
^n)^((2*I)*b))])/((1 - 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 4484

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

Rubi steps

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

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Mathematica [A]  time = 0.56, size = 102, normalized size = 0.91 \[ \frac {i x \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \, _2F_1\left (1,\frac {1}{2} \left (p-\frac {i}{b n}+2\right );-\frac {p}{2}-\frac {i}{2 b n}+1;-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{b n p+i} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[a + b*Log[c*x^n]]^p,x]

[Out]

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

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b \log \left (c x^{n}\right ) + a\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(b*log(c*x^n) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(b*log(c*x^n) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(b*log(c*x^n) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*log(c*x^n))^p,x)

[Out]

int(cos(a + b*log(c*x^n))^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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