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)*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticE(sin(1/2*x),2^(1/2))*(a*sec(x)^3)^(1/2)+2*cos(x)*sin(
x)*(a*sec(x)^3)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 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]

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 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]

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 = 0.02, 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])

________________________________________________________________________________________

fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \sec \relax (x)^{3}}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sec \relax (x)^{3}}\,{d x} \]

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)

________________________________________________________________________________________

maple [C]  time = 0.71, size = 191, normalized size = 4.55 \[ \frac {2 \left (\cos \relax (x )+1\right )^{2} \left (-1+\cos \relax (x )\right )^{2} \left (i \cos \relax (x ) \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}-i \cos \relax (x ) \sin \relax (x ) \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+i \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}-i \sin \relax (x ) \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}-\cos \relax (x )+1\right ) \cos \relax (x ) \sqrt {\frac {a}{\cos \relax (x )^{3}}}}{\sin \relax (x )^{5}} \]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sec \relax (x)^{3}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\frac {a}{{\cos \relax (x)}^3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/cos(x)^3)^(1/2),x)

[Out]

int((a/cos(x)^3)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sec ^{3}{\relax (x )}}\, dx \]

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)

________________________________________________________________________________________