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

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

[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])

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Rubi [A]  time = 0.0561162, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2641} \[ \frac{26 \tan (x)}{77 a^2 \sqrt{a \sec ^3(x)}}+\frac{2 \sin (x) \cos ^5(x)}{15 a^2 \sqrt{a \sec ^3(x)}}+\frac{26 \sin (x) \cos ^3(x)}{165 a^2 \sqrt{a \sec ^3(x)}}+\frac{26 F\left (\left .\frac{x}{2}\right |2\right )}{77 a^2 \cos ^{\frac{3}{2}}(x) \sqrt{a \sec ^3(x)}}+\frac{78 \sin (x) \cos (x)}{385 a^2 \sqrt{a \sec ^3(x)}} \]

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 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}{\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*}

Mathematica [A]  time = 0.096753, size = 59, normalized size = 0.5 \[ \frac{\cos (x) \sqrt{a \sec ^3(x)} \left (24960 \sqrt{\cos (x)} \text{EllipticF}\left (\frac{x}{2},2\right )+19122 \sin (2 x)+4406 \sin (4 x)+826 \sin (6 x)+77 \sin (8 x)\right )}{73920 a^3} \]

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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Maple [C]  time = 0.206, size = 114, normalized size = 1. \begin{align*}{\frac{ \left ( -2+2\,\cos \left ( x \right ) \right ) \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{1155\, \left ( \cos \left ( x \right ) \right ) ^{8} \left ( \sin \left ( x \right ) \right ) ^{3}} \left ( 77\, \left ( \cos \left ( x \right ) \right ) ^{8}-77\, \left ( \cos \left ( x \right ) \right ) ^{7}+91\, \left ( \cos \left ( x \right ) \right ) ^{6}-91\, \left ( \cos \left ( x \right ) \right ) ^{5}-195\,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 ) +117\, \left ( \cos \left ( x \right ) \right ) ^{4}-117\, \left ( \cos \left ( x \right ) \right ) ^{3}+195\, \left ( \cos \left ( x \right ) \right ) ^{2}-195\,\cos \left ( x \right ) \right ) \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{3}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

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*EllipticF(I*(-1+cos(x))/sin(x),I)*(1
/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*sin(x)+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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sec \left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \sec \left (x\right )^{3}}}{a^{3} \sec \left (x\right )^{9}}, x\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sec ^{3}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sec \left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

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)