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

Optimal. Leaf size=110 \[ \frac {2 x \, _2F_1\left (-\frac {1}{2},-\frac {b n+2 i}{4 b n};\frac {1}{4} \left (3-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{(2-i b n) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 110, 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 = {4484, 4492, 364} \[ \frac {2 x \, _2F_1\left (-\frac {1}{2},-\frac {b n+2 i}{4 b n};\frac {1}{4} \left (3-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{(2-i b n) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cos[a + b*Log[c*x^n]]],x]

[Out]

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

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

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Mathematica [B]  time = 3.56, size = 377, normalized size = 3.43 \[ -\frac {2 x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-2 \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}+\frac {2 e^{i a} b n x \left (c x^n\right )^{i b} \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \left ((3 b n-2 i) \, _2F_1\left (\frac {1}{2},-\frac {b n+2 i}{4 b n};\frac {3}{4}-\frac {i}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(b n+2 i) x^{2 i b n} \, _2F_1\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n};\frac {7}{4}-\frac {i}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(b n+2 i) (3 b n-2 i) \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} \left ((-2+i b n) x^{2 i b n}-i e^{2 i a} (b n-2 i) \left (c x^n\right )^{2 i b}\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*b*E^(I*a)*n*x*(c*x^n)^(I*b)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*((2*I + b*n)*x^((2*I)*b*n)*Hypergeome
tric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))] + (-2*I + 3*b*n)*Hypergeo
metric2F1[1/2, -1/4*(2*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]))/((2*I + b*n)*(-2
*I + 3*b*n)*Sqrt[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*((-2 + I*b*n)*x^((2*I)*b*n) - I*E^((2*I)*a
)*(-2*I + b*n)*(c*x^n)^((2*I)*b))) - (2*x*Sqrt[Cos[a + b*Log[c*x^n]]]*Cos[a - b*n*Log[x] + b*Log[c*x^n]])/(-2*
Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \sqrt {\cos }\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*log(c*x^n))^(1/2),x)

[Out]

int(cos(a + b*log(c*x^n))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*ln(c*x**n))**(1/2),x)

[Out]

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

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