3.58 \(\int \frac{1}{\sqrt{a \sec ^3(x)}} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 \text{EllipticF}\left (\frac{x}{2},2\right )}{3 \cos ^{\frac{3}{2}}(x) \sqrt{a \sec ^3(x)}}+\frac{2 \tan (x)}{3 \sqrt{a \sec ^3(x)}} \]

[Out]

(2*EllipticF[x/2, 2])/(3*Cos[x]^(3/2)*Sqrt[a*Sec[x]^3]) + (2*Tan[x])/(3*Sqrt[a*Sec[x]^3])

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Rubi [A]  time = 0.0270201, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2641} \[ \frac{2 \tan (x)}{3 \sqrt{a \sec ^3(x)}}+\frac{2 F\left (\left .\frac{x}{2}\right |2\right )}{3 \cos ^{\frac{3}{2}}(x) \sqrt{a \sec ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Sec[x]^3],x]

[Out]

(2*EllipticF[x/2, 2])/(3*Cos[x]^(3/2)*Sqrt[a*Sec[x]^3]) + (2*Tan[x])/(3*Sqrt[a*Sec[x]^3])

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2641

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \sec ^3(x)}} \, dx &=\frac{\sec ^{\frac{3}{2}}(x) \int \frac{1}{\sec ^{\frac{3}{2}}(x)} \, dx}{\sqrt{a \sec ^3(x)}}\\ &=\frac{2 \tan (x)}{3 \sqrt{a \sec ^3(x)}}+\frac{\sec ^{\frac{3}{2}}(x) \int \sqrt{\sec (x)} \, dx}{3 \sqrt{a \sec ^3(x)}}\\ &=\frac{2 \tan (x)}{3 \sqrt{a \sec ^3(x)}}+\frac{\int \frac{1}{\sqrt{\cos (x)}} \, dx}{3 \cos ^{\frac{3}{2}}(x) \sqrt{a \sec ^3(x)}}\\ &=\frac{2 F\left (\left .\frac{x}{2}\right |2\right )}{3 \cos ^{\frac{3}{2}}(x) \sqrt{a \sec ^3(x)}}+\frac{2 \tan (x)}{3 \sqrt{a \sec ^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0398245, size = 31, normalized size = 0.7 \[ \frac{2 \left (\frac{\text{EllipticF}\left (\frac{x}{2},2\right )}{\cos ^{\frac{3}{2}}(x)}+\tan (x)\right )}{3 \sqrt{a \sec ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Sec[x]^3],x]

[Out]

(2*(EllipticF[x/2, 2]/Cos[x]^(3/2) + Tan[x]))/(3*Sqrt[a*Sec[x]^3])

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Maple [C]  time = 0.14, size = 76, normalized size = 1.7 \begin{align*}{\frac{ \left ( -2+2\,\cos \left ( x \right ) \right ) \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{3\, \left ( \cos \left ( x \right ) \right ) ^{2} \left ( \sin \left ( x \right ) \right ) ^{3}} \left ( -i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}\sin \left ( x \right ) + \left ( \cos \left ( x \right ) \right ) ^{2}-\cos \left ( x \right ) \right ){\frac{1}{\sqrt{{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{3}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sec(x)^3)^(1/2),x)

[Out]

2/3*(-1+cos(x))*(-I*EllipticF(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*sin(x)+co
s(x)^2-cos(x))*(cos(x)+1)^2/cos(x)^2/sin(x)^3/(a/cos(x)^3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sec \left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(a*sec(x)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \sec \left (x\right )^{3}}}{a \sec \left (x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^3)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sec(x)^3)/(a*sec(x)^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sec ^{3}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)**3)**(1/2),x)

[Out]

Integral(1/sqrt(a*sec(x)**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sec \left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(a*sec(x)^3), x)