\(\int x^m \sqrt {\cos (a+b \log (c x^n))} \, dx\) [128]

Optimal result

Integrand size = 19, antiderivative size = 129 \[ \int x^m \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{1+m} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2 i+2 i m+b n}{4 b n},-\frac {2 i+2 i m-3 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-i b n) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

[Out]

2*x^(1+m)*hypergeom([-1/2, 1/4*(-2*I-2*I*m-b*n)/b/n],[1/4*(-2*I-2*I*m+3*b*n)/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))
*cos(a+b*ln(c*x^n))^(1/2)/(2+2*m-I*b*n)/(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4582, 4580, 371} \[ \int x^m \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b n}-1\right ),-\frac {2 i m-3 b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{(-i b n+2 m+2) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

[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, -1/4*(2*I + (
2*I)*m - 3*b*n)/(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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4580

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 4582

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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {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 ) \text {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 )} \operatorname {Hypergeometric2F1}\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] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(436\) vs. \(2(129)=258\).

Time = 5.76 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.38 \[ \int x^m \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 b e^{i a} n x^{1+m} \left (c x^n\right )^{i b} \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \left ((2 i+2 i m+b n) x^{2 i b n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {i \left (1+m+\frac {3 i b n}{2}\right )}{2 b n},-\frac {2 i+2 i m-7 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(-2 i-2 i m+3 b n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m+b n}{4 b n},-\frac {2 i+2 i m-3 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(2+2 m-i b n) (2+2 m+3 i b n) \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} \left ((2+2 m-i b n) x^{2 i b n}+e^{2 i a} (2+2 m+i b n) \left (c x^n\right )^{2 i b}\right )}+\frac {2 x^{1+m} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{2 (1+m) \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-b n \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )} \]

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

\[\int x^{m} \sqrt {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int x^m \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int x^m \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\int x^{m} \sqrt {\cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

[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)

Maxima [F]

\[ \int x^m \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { x^{m} \sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]

[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)

Giac [F]

\[ \int x^m \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { x^{m} \sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x^m \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\int x^m\,\sqrt {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]

[In]

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

[Out]

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