3.3.72 \(\int \frac {1}{\sqrt {\sec (a+b \log (c x^n))}} \, dx\) [272]

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

[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))/(2-I*b*n)/(1+exp(2*I*a)*
(c*x^n)^(2*I*b))^(1/2)/sec(a+b*ln(c*x^n))^(1/2)

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Rubi [A]
time = 0.05, 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 = {4599, 4603, 371} \begin {gather*} \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 )}{(2-i b n) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Sec[a + b*Log[c*x^n]]],x]

[Out]

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

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 4599

Int[Sec[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sec[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 4603

Int[((e_.)*(x_))^(m_.)*Sec[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[Sec[d*(a + b*Log[x])]^p*((1
+ E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)), Int[(e*x)^m*(x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p), x], x]
/; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}} \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1}{n}}}{\sqrt {\sec (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}}\right ) \text {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}} \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}\\ &=\frac {2 x \, _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}} \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(364\) vs. \(2(110)=220\).
time = 4.88, size = 364, normalized size = 3.31 \begin {gather*} \frac {2 i \sqrt {2} b e^{-i a} n x \left (c x^n\right )^{-i b} \sqrt {\frac {e^{i a} \left (c x^n\right )^{i b}}{1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \left ((2 i+b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )+\sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \left (-2 i-b n+e^{2 i a} (-2 i+b n) x^{-2 i b n} \left (c x^n\right )^{2 i b}\right ) \, _2F_1\left (\frac {1}{2},-\frac {2 i+b n}{4 b n};\frac {3}{4}-\frac {i}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{\left (4+b^2 n^2\right ) \left (-2 i-b n+e^{2 i a} (-2 i+b n) x^{-2 i b n} \left (c x^n\right )^{2 i b}\right )}-\frac {2 x \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \left (-2 \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 )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*Sqrt[2]*b*n*x*Sqrt[(E^(I*a)*(c*x^n)^(I*b))/(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*((2*I + b*n)*(1 + E^((2
*I)*a)*(c*x^n)^((2*I)*b)) + Sqrt[1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)]*(-2*I - b*n + (E^((2*I)*a)*(-2*I + b*n)*(c
*x^n)^((2*I)*b))/x^((2*I)*b*n))*Hypergeometric2F1[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))]))/(E^(I*a)*(4 + b^2*n^2)*(c*x^n)^(I*b)*(-2*I - b*n + (E^((2*I)*a)*(-2*I + b*n)*(c*x^n)^(
(2*I)*b))/x^((2*I)*b*n))) - (2*x*Cos[a - b*n*Log[x] + b*Log[c*x^n]])/(Sqrt[Sec[a + 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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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\sec {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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