3.2.17 \(\int \frac {1}{x \sqrt {\cos (a+b \log (c x^n))}} \, dx\) [117]

Optimal. Leaf size=24 \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2720} \begin {gather*} \frac {2 F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticF[(a + b*Log[c*x^n])/2, 2])/(b*n)

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\cos (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 24, normalized size = 1.00 \begin {gather*} \frac {2 F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticF[(a + b*Log[c*x^n])/2, 2])/(b*n)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.06, size = 26, normalized size = 1.08

method	result	size
derivativedivides	\(\frac {2 \,\mathrm {am}^{-1}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}| \sqrt {2}\right )}{n b}\)	\(26\)
default	\(\frac {2 \,\mathrm {am}^{-1}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}| \sqrt {2}\right )}{n b}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.75, size = 78, normalized size = 3.25 \begin {gather*} \frac {-i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*sqrt(2)*weierstrassPInverse(-4, 0, cos(b*n*log(x) + b*log(c) + a) + I*sin(b*n*log(x) + b*log(c) + a)) + I*
sqrt(2)*weierstrassPInverse(-4, 0, cos(b*n*log(x) + b*log(c) + a) - I*sin(b*n*log(x) + b*log(c) + a)))/(b*n)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {\cos {\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/x/cos(a+b*ln(c*x**n))**(1/2),x)

[Out]

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

[Out]

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

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Mupad [B]
time = 2.37, size = 23, normalized size = 0.96 \begin {gather*} \frac {2\,\mathrm {F}\left (\frac {a}{2}+\frac {b\,\ln \left (c\,x^n\right )}{2}\middle |2\right )}{b\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*ellipticF(a/2 + (b*log(c*x^n))/2, 2))/(b*n)

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