3.116 \(\int \frac{1}{\sqrt{\cos (a+b \log (c x^n))}} \, dx\)

Optimal. Leaf size=109 \[ \frac{2 x \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} \left (1-\frac{2 i}{b n}\right ),\frac{1}{4} \left (5-\frac{2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+i b n) \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

(2*x*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Hypergeometric2F1[1/2, (1 - (2*I)/(b*n))/4, (5 - (2*I)/(b*n))/4,
-(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/((2 + I*b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])

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Rubi [A]  time = 0.0667546, antiderivative size = 109, 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 = {4484, 4492, 364} \[ \frac{2 x \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1-\frac{2 i}{b n}\right );\frac{1}{4} \left (5-\frac{2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+i b n) \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Cos[a + b*Log[c*x^n]]],x]

[Out]

(2*x*Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Hypergeometric2F1[1/2, (1 - (2*I)/(b*n))/4, (5 - (2*I)/(b*n))/4,
-(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/((2 + I*b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])

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]

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 \frac{1}{\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 \frac{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{1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right ) \operatorname{Subst}\left (\int \frac{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{\cos \left (a+b \log \left (c x^n\right )\right )}}\\ &=\frac{2 x \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1-\frac{2 i}{b n}\right );\frac{1}{4} \left (5-\frac{2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+i b n) \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}}\\ \end{align*}

Mathematica [A]  time = 0.380223, size = 99, normalized size = 0.91 \[ -\frac{2 i x \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{4}-\frac{i}{2 b n},\frac{5}{4}-\frac{i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{(b n-2 i) \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[Cos[a + b*Log[c*x^n]]],x]

[Out]

((-2*I)*(1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), -E^((2
*I)*(a + b*Log[c*x^n]))])/((-2*I + b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])

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Maple [F]  time = 0.16, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\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(1/cos(a+b*ln(c*x^n))^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/cos(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(cos(b*log(c*x^n) + a)), x)

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Integral(1/sqrt(cos(a + b*log(c*x**n))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/cos(a+b*log(c*x^n))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(cos(b*log(c*x^n) + a)), x)