3.57 \(\int \sqrt{a \sec ^3(x)} \, dx\)

Optimal. Leaf size=42 \[ 2 \sin (x) \cos (x) \sqrt{a \sec ^3(x)}-2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \sec ^3(x)} \]

[Out]

-2*Cos[x]^(3/2)*EllipticE[x/2, 2]*Sqrt[a*Sec[x]^3] + 2*Cos[x]*Sqrt[a*Sec[x]^3]*Sin[x]

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Rubi [A]  time = 0.0256077, antiderivative size = 42, 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, 3768, 3771, 2639} \[ 2 \sin (x) \cos (x) \sqrt{a \sec ^3(x)}-2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \sec ^3(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Cos[x]^(3/2)*EllipticE[x/2, 2]*Sqrt[a*Sec[x]^3] + 2*Cos[x]*Sqrt[a*Sec[x]^3]*Sin[x]

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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[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 2639

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

Rubi steps

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

Mathematica [A]  time = 0.0171513, size = 32, normalized size = 0.76 \[ 2 \cos (x) \sqrt{a \sec ^3(x)} \left (\sin (x)-\sqrt{\cos (x)} E\left (\left .\frac{x}{2}\right |2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Cos[x]*Sqrt[a*Sec[x]^3]*(-(Sqrt[Cos[x]]*EllipticE[x/2, 2]) + Sin[x])

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Maple [C]  time = 0.205, size = 191, normalized size = 4.6 \begin{align*} 2\,{\frac{ \left ( \cos \left ( x \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( x \right ) \right ) ^{2}\cos \left ( x \right ) }{ \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( i\cos \left ( x \right ) \sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -i\sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \cos \left ( x \right ) \sin \left ( x \right ) +i\sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -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 ) -\cos \left ( x \right ) +1 \right ) \sqrt{{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(cos(x)+1)^2*(-1+cos(x))^2*(I*cos(x)*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(-1+c
os(x))/sin(x),I)-I*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(-1+cos(x))/sin(x),I)*cos(x)*sin
(x)+I*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(-1+cos(x))/sin(x),I)-I*EllipticF(I*(-
1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*sin(x)-cos(x)+1)*cos(x)*(a/cos(x)^3)^(1/2)/
sin(x)^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*sec(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*sec(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*sec(x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*sec(x)^3), x)