3.60 \(\int \frac {1}{(a \sec ^3(x))^{5/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac {26 \tan (x)}{77 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 F\left (\left .\frac {x}{2}\right |2\right )}{77 a^2 \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}}+\frac {2 \sin (x) \cos ^5(x)}{15 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \sin (x) \cos ^3(x)}{165 a^2 \sqrt {a \sec ^3(x)}}+\frac {78 \sin (x) \cos (x)}{385 a^2 \sqrt {a \sec ^3(x)}} \]

[Out]

26/77*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticF(sin(1/2*x),2^(1/2))/a^2/cos(x)^(3/2)/(a*sec(x)^3)^(1/2)+78/385
*cos(x)*sin(x)/a^2/(a*sec(x)^3)^(1/2)+26/165*cos(x)^3*sin(x)/a^2/(a*sec(x)^3)^(1/2)+2/15*cos(x)^5*sin(x)/a^2/(
a*sec(x)^3)^(1/2)+26/77*tan(x)/a^2/(a*sec(x)^3)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3769, 3771, 2641} \[ \frac {26 \tan (x)}{77 a^2 \sqrt {a \sec ^3(x)}}+\frac {2 \sin (x) \cos ^5(x)}{15 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \sin (x) \cos ^3(x)}{165 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 F\left (\left .\frac {x}{2}\right |2\right )}{77 a^2 \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}}+\frac {78 \sin (x) \cos (x)}{385 a^2 \sqrt {a \sec ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sec[x]^3)^(-5/2),x]

[Out]

(26*EllipticF[x/2, 2])/(77*a^2*Cos[x]^(3/2)*Sqrt[a*Sec[x]^3]) + (78*Cos[x]*Sin[x])/(385*a^2*Sqrt[a*Sec[x]^3])
+ (26*Cos[x]^3*Sin[x])/(165*a^2*Sqrt[a*Sec[x]^3]) + (2*Cos[x]^5*Sin[x])/(15*a^2*Sqrt[a*Sec[x]^3]) + (26*Tan[x]
)/(77*a^2*Sqrt[a*Sec[x]^3])

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]

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 )^{5/2}} \, dx &=\frac {\sec ^{\frac {3}{2}}(x) \int \frac {1}{\sec ^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \sec ^3(x)}}\\ &=\frac {2 \cos ^5(x) \sin (x)}{15 a^2 \sqrt {a \sec ^3(x)}}+\frac {\left (13 \sec ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sec ^{\frac {11}{2}}(x)} \, dx}{15 a^2 \sqrt {a \sec ^3(x)}}\\ &=\frac {26 \cos ^3(x) \sin (x)}{165 a^2 \sqrt {a \sec ^3(x)}}+\frac {2 \cos ^5(x) \sin (x)}{15 a^2 \sqrt {a \sec ^3(x)}}+\frac {\left (39 \sec ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sec ^{\frac {7}{2}}(x)} \, dx}{55 a^2 \sqrt {a \sec ^3(x)}}\\ &=\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \cos ^3(x) \sin (x)}{165 a^2 \sqrt {a \sec ^3(x)}}+\frac {2 \cos ^5(x) \sin (x)}{15 a^2 \sqrt {a \sec ^3(x)}}+\frac {\left (39 \sec ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(x)} \, dx}{77 a^2 \sqrt {a \sec ^3(x)}}\\ &=\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \cos ^3(x) \sin (x)}{165 a^2 \sqrt {a \sec ^3(x)}}+\frac {2 \cos ^5(x) \sin (x)}{15 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \tan (x)}{77 a^2 \sqrt {a \sec ^3(x)}}+\frac {\left (13 \sec ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sec (x)} \, dx}{77 a^2 \sqrt {a \sec ^3(x)}}\\ &=\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \cos ^3(x) \sin (x)}{165 a^2 \sqrt {a \sec ^3(x)}}+\frac {2 \cos ^5(x) \sin (x)}{15 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \tan (x)}{77 a^2 \sqrt {a \sec ^3(x)}}+\frac {13 \int \frac {1}{\sqrt {\cos (x)}} \, dx}{77 a^2 \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}}\\ &=\frac {26 F\left (\left .\frac {x}{2}\right |2\right )}{77 a^2 \cos ^{\frac {3}{2}}(x) \sqrt {a \sec ^3(x)}}+\frac {78 \cos (x) \sin (x)}{385 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \cos ^3(x) \sin (x)}{165 a^2 \sqrt {a \sec ^3(x)}}+\frac {2 \cos ^5(x) \sin (x)}{15 a^2 \sqrt {a \sec ^3(x)}}+\frac {26 \tan (x)}{77 a^2 \sqrt {a \sec ^3(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 59, normalized size = 0.50 \[ \frac {\cos (x) \sqrt {a \sec ^3(x)} \left (19122 \sin (2 x)+4406 \sin (4 x)+826 \sin (6 x)+77 \sin (8 x)+24960 \sqrt {\cos (x)} F\left (\left .\frac {x}{2}\right |2\right )\right )}{73920 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sec[x]^3)^(-5/2),x]

[Out]

(Cos[x]*Sqrt[a*Sec[x]^3]*(24960*Sqrt[Cos[x]]*EllipticF[x/2, 2] + 19122*Sin[2*x] + 4406*Sin[4*x] + 826*Sin[6*x]
+ 77*Sin[8*x]))/(73920*a^3)

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sec \relax (x)^{3}}}{a^{3} \sec \relax (x)^{9}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^3)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sec(x)^3)/(a^3*sec(x)^9), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sec \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*sec(x)^3)^(-5/2), x)

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maple [C]  time = 0.52, size = 114, normalized size = 0.97 \[ \frac {2 \left (-1+\cos \relax (x )\right ) \left (77 \left (\cos ^{8}\relax (x )\right )-77 \left (\cos ^{7}\relax (x )\right )+91 \left (\cos ^{6}\relax (x )\right )-91 \left (\cos ^{5}\relax (x )\right )-195 i \sin \relax (x ) \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+117 \left (\cos ^{4}\relax (x )\right )-117 \left (\cos ^{3}\relax (x )\right )+195 \left (\cos ^{2}\relax (x )\right )-195 \cos \relax (x )\right ) \left (\cos \relax (x )+1\right )^{2}}{1155 \cos \relax (x )^{8} \sin \relax (x )^{3} \left (\frac {a}{\cos \relax (x )^{3}}\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sec(x)^3)^(5/2),x)

[Out]

2/1155*(-1+cos(x))*(77*cos(x)^8-77*cos(x)^7+91*cos(x)^6-91*cos(x)^5-195*I*sin(x)*EllipticF(I*(-1+cos(x))/sin(x
),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)+117*cos(x)^4-117*cos(x)^3+195*cos(x)^2-195*cos(x))*(cos(x)
+1)^2/cos(x)^8/sin(x)^3/(a/cos(x)^3)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sec \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sec(x)^3)^(-5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {a}{{\cos \relax (x)}^3}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/cos(x)^3)^(5/2),x)

[Out]

int(1/(a/cos(x)^3)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \sec ^{3}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)**3)**(5/2),x)

[Out]

Integral((a*sec(x)**3)**(-5/2), x)

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