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

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

[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])

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Rubi [A]  time = 0.0367949, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 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 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 \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*}

Mathematica [A]  time = 0.0940153, 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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Maple [C]  time = 0.156, size = 198, normalized size = 2.7 \begin{align*} -{\frac{2}{45\, \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) } \left ( 5\, \left ( \cos \left ( x \right ) \right ) ^{6}-21\,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 ) +21\,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 ) -21\,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 ) +21\,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 ) +2\, \left ( \cos \left ( x \right ) \right ) ^{4}+14\, \left ( \cos \left ( x \right ) \right ) ^{2}-21\,\cos \left ( x \right ) \right ) \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{3}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

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*(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)+21*I*cos(x)*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(-1+cos(x))/sin(x),I)-21
*I*EllipticF(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*sin(x)+21*I*sin(x)*(1/(cos
(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(-1+cos(x))/sin(x),I)+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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sec \left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \sec \left (x\right )^{3}}}{a^{2} \sec \left (x\right )^{6}}, x\right ) \end{align*}

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

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

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)