3.59 \(\int \frac {1}{(a \sec ^3(x))^{3/2}} \, dx\)

Optimal. Leaf size=73 \[ \frac {14 \sin (x)}{45 a \sqrt {a \sec ^3(x)}}+\frac {14 E\left (\left .\frac {x}{2}\right |2\right )}{15 a \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}}+\frac {2 \sin (x) \cos ^2(x)}{9 a \sqrt {a \sec ^3(x)}} \]

[Out]

14/15*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticE(sin(1/2*x),2^(1/2))/a/cos(x)^(3/2)/(a*sec(x)^3)^(1/2)+14/45*si
n(x)/a/(a*sec(x)^3)^(1/2)+2/9*cos(x)^2*sin(x)/a/(a*sec(x)^3)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3769, 3771, 2639} \[ \frac {14 \sin (x)}{45 a \sqrt {a \sec ^3(x)}}+\frac {2 \sin (x) \cos ^2(x)}{9 a \sqrt {a \sec ^3(x)}}+\frac {14 E\left (\left .\frac {x}{2}\right |2\right )}{15 a \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sec[x]^3)^(-3/2),x]

[Out]

(14*EllipticE[x/2, 2])/(15*a*Cos[x]^(3/2)*Sqrt[a*Sec[x]^3]) + (14*Sin[x])/(45*a*Sqrt[a*Sec[x]^3]) + (2*Cos[x]^
2*Sin[x])/(9*a*Sqrt[a*Sec[x]^3])

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 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 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 \frac {1}{\left (a \sec ^3(x)\right )^{3/2}} \, dx &=\frac {\sec ^{\frac {3}{2}}(x) \int \frac {1}{\sec ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \sec ^3(x)}}\\ &=\frac {2 \cos ^2(x) \sin (x)}{9 a \sqrt {a \sec ^3(x)}}+\frac {\left (7 \sec ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(x)} \, dx}{9 a \sqrt {a \sec ^3(x)}}\\ &=\frac {14 \sin (x)}{45 a \sqrt {a \sec ^3(x)}}+\frac {2 \cos ^2(x) \sin (x)}{9 a \sqrt {a \sec ^3(x)}}+\frac {\left (7 \sec ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\sec (x)}} \, dx}{15 a \sqrt {a \sec ^3(x)}}\\ &=\frac {14 \sin (x)}{45 a \sqrt {a \sec ^3(x)}}+\frac {2 \cos ^2(x) \sin (x)}{9 a \sqrt {a \sec ^3(x)}}+\frac {7 \int \sqrt {\cos (x)} \, dx}{15 a \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}}\\ &=\frac {14 E\left (\left .\frac {x}{2}\right |2\right )}{15 a \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}}+\frac {14 \sin (x)}{45 a \sqrt {a \sec ^3(x)}}+\frac {2 \cos ^2(x) \sin (x)}{9 a \sqrt {a \sec ^3(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 43, normalized size = 0.59 \[ \frac {33 \sin (x)+5 \sin (3 x)+\frac {84 E\left (\left .\frac {x}{2}\right |2\right )}{\cos ^{\frac {3}{2}}(x)}}{90 a \sqrt {a \sec ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sec[x]^3)^(-3/2),x]

[Out]

((84*EllipticE[x/2, 2])/Cos[x]^(3/2) + 33*Sin[x] + 5*Sin[3*x])/(90*a*Sqrt[a*Sec[x]^3])

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fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sec \relax (x)^{3}}}{a^{2} \sec \relax (x)^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*sec(x)^3)/(a^2*sec(x)^6), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sec \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sec(x)^3)^(-3/2), x)

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maple [C]  time = 0.55, size = 198, normalized size = 2.71 \[ -\frac {2 \left (5 \left (\cos ^{6}\relax (x )\right )-21 i \cos \relax (x ) \sin \relax (x ) \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+21 i \cos \relax (x ) \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}-21 i \sin \relax (x ) \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+21 i \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+2 \left (\cos ^{4}\relax (x )\right )+14 \left (\cos ^{2}\relax (x )\right )-21 \cos \relax (x )\right )}{45 \cos \relax (x )^{5} \sin \relax (x ) \left (\frac {a}{\cos \relax (x )^{3}}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(5*cos(x)^6-21*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)+21*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)-21
*I*sin(x)*EllipticF(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)+21*I*sin(x)*Ellipti
cE(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)+2*cos(x)^4+14*cos(x)^2-21*cos(x))/co
s(x)^5/sin(x)/(a/cos(x)^3)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sec \relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sec(x)^3)^(-3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {a}{{\cos \relax (x)}^3}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a/cos(x)^3)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sec ^{3}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*sec(x)**3)**(-3/2), x)

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