3.128 \(\int x^m \sqrt{\cos (a+b \log (c x^n))} \, dx\)

Optimal. Leaf size=129 \[ \frac{2 x^{m+1} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{b n+2 i m+2 i}{4 b n},-\frac{-3 b n+2 i m+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-i b n+2 m+2) \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

[Out]

(2*x^(1 + m)*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[-1/2, -(2*I + (2*I)*m + b*n)/(4*b*n), -(2*I + (2*I)
*m - 3*b*n)/(4*b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/((2 + 2*m - I*b*n)*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)
*b)])

________________________________________________________________________________________

Rubi [A]  time = 0.0990102, antiderivative size = 126, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4494, 4492, 364} \[ \frac{2 x^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} \left (-\frac{2 i (m+1)}{b n}-1\right );-\frac{2 i m-3 b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}}{(-i b n+2 m+2) \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Sqrt[Cos[a + b*Log[c*x^n]]],x]

[Out]

(2*x^(1 + m)*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[-1/2, (-1 - ((2*I)*(1 + m))/(b*n))/4, -(2*I + (2*I)
*m - 3*b*n)/(4*b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/((2 + 2*m - I*b*n)*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)
*b)])

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^m \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\left (x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{n}} \sqrt{\cos (a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x^{1+m} \left (c x^n\right )^{\frac{i b}{2}-\frac{1+m}{n}} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname{Subst}\left (\int x^{-1-\frac{i b}{2}+\frac{1+m}{n}} \sqrt{1+e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}}}\\ &=\frac{2 x^{1+m} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} \left (-1-\frac{2 i (1+m)}{b n}\right );-\frac{2 i+2 i m-3 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-i b n) \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}}}\\ \end{align*}

Mathematica [B]  time = 5.23837, size = 436, normalized size = 3.38 \[ \frac{2 x^{m+1} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{2 (m+1) \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-b n \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}-\frac{2 e^{i a} b n x^{m+1} \left (c x^n\right )^{i b} \sqrt{2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \left ((b n+2 i m+2 i) x^{2 i b n} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{i \left (\frac{3 i b n}{2}+m+1\right )}{2 b n},-\frac{-7 b n+2 i m+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(3 b n-2 i m-2 i) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{b n+2 i m+2 i}{4 b n},-\frac{-3 b n+2 i m+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(-i b n+2 m+2) (3 i b n+2 m+2) \sqrt{e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} \left (e^{2 i a} (i b n+2 m+2) \left (c x^n\right )^{2 i b}+(-i b n+2 m+2) x^{2 i b n}\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*b*E^(I*a)*n*x^(1 + m)*(c*x^n)^(I*b)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*((2*I + (2*I)*m + b*n)*x^((2
*I)*b*n)*Hypergeometric2F1[1/2, ((-I/2)*(1 + m + ((3*I)/2)*b*n))/(b*n), -(2*I + (2*I)*m - 7*b*n)/(4*b*n), -(E^
((2*I)*a)*(c*x^n)^((2*I)*b))] + (-2*I - (2*I)*m + 3*b*n)*Hypergeometric2F1[1/2, -(2*I + (2*I)*m + b*n)/(4*b*n)
, -(2*I + (2*I)*m - 3*b*n)/(4*b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]))/((2 + 2*m - I*b*n)*(2 + 2*m + (3*I)*b*
n)*Sqrt[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*((2 + 2*m - I*b*n)*x^((2*I)*b*n) + E^((2*I)*a)*(2 +
2*m + I*b*n)*(c*x^n)^((2*I)*b))) + (2*x^(1 + m)*Sqrt[Cos[a + b*Log[c*x^n]]]*Cos[a - b*n*Log[x] + b*Log[c*x^n]
])/(2*(1 + m)*Cos[a - b*n*Log[x] + b*Log[c*x^n]] - b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]])

________________________________________________________________________________________

Maple [F]  time = 0.2, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cos(a+b*ln(c*x^n))^(1/2),x)

[Out]

int(x^m*cos(a+b*ln(c*x^n))^(1/2),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cos(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cos(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sqrt{\cos{\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**m*cos(a+b*ln(c*x**n))**(1/2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cos(a+b*log(c*x^n))^(1/2),x, algorithm="giac")

[Out]

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