3.56 \(\int (a \sec ^3(x))^{3/2} \, dx\)

Optimal. Leaf size=65 \[ \frac{10}{21} a \cos ^{\frac{3}{2}}(x) \text{EllipticF}\left (\frac{x}{2},2\right ) \sqrt{a \sec ^3(x)}+\frac{10}{21} a \sin (x) \sqrt{a \sec ^3(x)}+\frac{2}{7} a \tan (x) \sec (x) \sqrt{a \sec ^3(x)} \]

[Out]

(10*a*Cos[x]^(3/2)*EllipticF[x/2, 2]*Sqrt[a*Sec[x]^3])/21 + (10*a*Sqrt[a*Sec[x]^3]*Sin[x])/21 + (2*a*Sec[x]*Sq
rt[a*Sec[x]^3]*Tan[x])/7

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Rubi [A]  time = 0.0353327, antiderivative size = 65, 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, 3768, 3771, 2641} \[ \frac{10}{21} a \sin (x) \sqrt{a \sec ^3(x)}+\frac{2}{7} a \tan (x) \sec (x) \sqrt{a \sec ^3(x)}+\frac{10}{21} a \cos ^{\frac{3}{2}}(x) F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \sec ^3(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*a*Cos[x]^(3/2)*EllipticF[x/2, 2]*Sqrt[a*Sec[x]^3])/21 + (10*a*Sqrt[a*Sec[x]^3]*Sin[x])/21 + (2*a*Sec[x]*Sq
rt[a*Sec[x]^3]*Tan[x])/7

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 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 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 \left (a \sec ^3(x)\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a \sec ^3(x)}\right ) \int \sec ^{\frac{9}{2}}(x) \, dx}{\sec ^{\frac{3}{2}}(x)}\\ &=\frac{2}{7} a \sec (x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{\left (5 a \sqrt{a \sec ^3(x)}\right ) \int \sec ^{\frac{5}{2}}(x) \, dx}{7 \sec ^{\frac{3}{2}}(x)}\\ &=\frac{10}{21} a \sqrt{a \sec ^3(x)} \sin (x)+\frac{2}{7} a \sec (x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{\left (5 a \sqrt{a \sec ^3(x)}\right ) \int \sqrt{\sec (x)} \, dx}{21 \sec ^{\frac{3}{2}}(x)}\\ &=\frac{10}{21} a \sqrt{a \sec ^3(x)} \sin (x)+\frac{2}{7} a \sec (x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{1}{21} \left (5 a \cos ^{\frac{3}{2}}(x) \sqrt{a \sec ^3(x)}\right ) \int \frac{1}{\sqrt{\cos (x)}} \, dx\\ &=\frac{10}{21} a \cos ^{\frac{3}{2}}(x) F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \sec ^3(x)}+\frac{10}{21} a \sqrt{a \sec ^3(x)} \sin (x)+\frac{2}{7} a \sec (x) \sqrt{a \sec ^3(x)} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0346709, size = 43, normalized size = 0.66 \[ \frac{2}{21} a \sec (x) \sqrt{a \sec ^3(x)} \left (5 \cos ^{\frac{5}{2}}(x) \text{EllipticF}\left (\frac{x}{2},2\right )+3 \tan (x)+5 \sin (x) \cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sec[x]*Sqrt[a*Sec[x]^3]*(5*Cos[x]^(5/2)*EllipticF[x/2, 2] + 5*Cos[x]*Sin[x] + 3*Tan[x]))/21

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Maple [C]  time = 0.182, size = 87, normalized size = 1.3 \begin{align*} -{\frac{2\, \left ( \cos \left ( x \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( x \right ) \right ) \cos \left ( x \right ) }{21\, \left ( \sin \left ( x \right ) \right ) ^{3}} \left ( 5\,i \left ( \cos \left ( x \right ) \right ) ^{3}\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 EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -5\, \left ( \cos \left ( x \right ) \right ) ^{3}+5\, \left ( \cos \left ( x \right ) \right ) ^{2}-3\,\cos \left ( x \right ) +3 \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((a*sec(x)^3)^(3/2),x)

[Out]

-2/21*(cos(x)+1)^2*(-1+cos(x))*(5*I*cos(x)^3*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I
*(-1+cos(x))/sin(x),I)-5*cos(x)^3+5*cos(x)^2-3*cos(x)+3)*cos(x)*(a/cos(x)^3)^(3/2)/sin(x)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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 (\sqrt{a \sec \left (x\right )^{3}} a \sec \left (x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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 \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((a*sec(x)^3)^(3/2),x, algorithm="giac")

[Out]

integrate((a*sec(x)^3)^(3/2), x)