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

Optimal. Leaf size=117 \[ \frac{2}{13} a^2 \tan (x) \sec ^4(x) \sqrt{a \sec ^3(x)}+\frac{22}{117} a^2 \tan (x) \sec ^2(x) \sqrt{a \sec ^3(x)}+\frac{154}{585} a^2 \tan (x) \sqrt{a \sec ^3(x)}-\frac{154}{195} a^2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \sec ^3(x)}+\frac{154}{195} a^2 \sin (x) \cos (x) \sqrt{a \sec ^3(x)} \]

[Out]

(-154*a^2*Cos[x]^(3/2)*EllipticE[x/2, 2]*Sqrt[a*Sec[x]^3])/195 + (154*a^2*Cos[x]*Sqrt[a*Sec[x]^3]*Sin[x])/195
+ (154*a^2*Sqrt[a*Sec[x]^3]*Tan[x])/585 + (22*a^2*Sec[x]^2*Sqrt[a*Sec[x]^3]*Tan[x])/117 + (2*a^2*Sec[x]^4*Sqrt
[a*Sec[x]^3]*Tan[x])/13

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Rubi [A]  time = 0.0522832, 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, 3768, 3771, 2639} \[ \frac{2}{13} a^2 \tan (x) \sec ^4(x) \sqrt{a \sec ^3(x)}+\frac{22}{117} a^2 \tan (x) \sec ^2(x) \sqrt{a \sec ^3(x)}+\frac{154}{585} a^2 \tan (x) \sqrt{a \sec ^3(x)}-\frac{154}{195} a^2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \sec ^3(x)}+\frac{154}{195} a^2 \sin (x) \cos (x) \sqrt{a \sec ^3(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-154*a^2*Cos[x]^(3/2)*EllipticE[x/2, 2]*Sqrt[a*Sec[x]^3])/195 + (154*a^2*Cos[x]*Sqrt[a*Sec[x]^3]*Sin[x])/195
+ (154*a^2*Sqrt[a*Sec[x]^3]*Tan[x])/585 + (22*a^2*Sec[x]^2*Sqrt[a*Sec[x]^3]*Tan[x])/117 + (2*a^2*Sec[x]^4*Sqrt
[a*Sec[x]^3]*Tan[x])/13

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 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 \left (a \sec ^3(x)\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a \sec ^3(x)}\right ) \int \sec ^{\frac{15}{2}}(x) \, dx}{\sec ^{\frac{3}{2}}(x)}\\ &=\frac{2}{13} a^2 \sec ^4(x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{\left (11 a^2 \sqrt{a \sec ^3(x)}\right ) \int \sec ^{\frac{11}{2}}(x) \, dx}{13 \sec ^{\frac{3}{2}}(x)}\\ &=\frac{22}{117} a^2 \sec ^2(x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{2}{13} a^2 \sec ^4(x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{\left (77 a^2 \sqrt{a \sec ^3(x)}\right ) \int \sec ^{\frac{7}{2}}(x) \, dx}{117 \sec ^{\frac{3}{2}}(x)}\\ &=\frac{154}{585} a^2 \sqrt{a \sec ^3(x)} \tan (x)+\frac{22}{117} a^2 \sec ^2(x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{2}{13} a^2 \sec ^4(x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{\left (77 a^2 \sqrt{a \sec ^3(x)}\right ) \int \sec ^{\frac{3}{2}}(x) \, dx}{195 \sec ^{\frac{3}{2}}(x)}\\ &=\frac{154}{195} a^2 \cos (x) \sqrt{a \sec ^3(x)} \sin (x)+\frac{154}{585} a^2 \sqrt{a \sec ^3(x)} \tan (x)+\frac{22}{117} a^2 \sec ^2(x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{2}{13} a^2 \sec ^4(x) \sqrt{a \sec ^3(x)} \tan (x)-\frac{\left (77 a^2 \sqrt{a \sec ^3(x)}\right ) \int \frac{1}{\sqrt{\sec (x)}} \, dx}{195 \sec ^{\frac{3}{2}}(x)}\\ &=\frac{154}{195} a^2 \cos (x) \sqrt{a \sec ^3(x)} \sin (x)+\frac{154}{585} a^2 \sqrt{a \sec ^3(x)} \tan (x)+\frac{22}{117} a^2 \sec ^2(x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{2}{13} a^2 \sec ^4(x) \sqrt{a \sec ^3(x)} \tan (x)-\frac{1}{195} \left (77 a^2 \cos ^{\frac{3}{2}}(x) \sqrt{a \sec ^3(x)}\right ) \int \sqrt{\cos (x)} \, dx\\ &=-\frac{154}{195} a^2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \sec ^3(x)}+\frac{154}{195} a^2 \cos (x) \sqrt{a \sec ^3(x)} \sin (x)+\frac{154}{585} a^2 \sqrt{a \sec ^3(x)} \tan (x)+\frac{22}{117} a^2 \sec ^2(x) \sqrt{a \sec ^3(x)} \tan (x)+\frac{2}{13} a^2 \sec ^4(x) \sqrt{a \sec ^3(x)} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0933119, size = 59, normalized size = 0.5 \[ -\frac{2}{585} a \sec (x) \left (a \sec ^3(x)\right )^{3/2} \left (-45 \tan (x)-231 \sin (x) \cos ^5(x)-77 \sin (x) \cos ^3(x)+231 \cos ^{\frac{11}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )-55 \sin (x) \cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Sec[x]*(a*Sec[x]^3)^(3/2)*(231*Cos[x]^(11/2)*EllipticE[x/2, 2] - 55*Cos[x]*Sin[x] - 77*Cos[x]^3*Sin[x] -
231*Cos[x]^5*Sin[x] - 45*Tan[x]))/585

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Maple [C]  time = 0.368, size = 223, normalized size = 1.9 \begin{align*} -{\frac{2\, \left ( \cos \left ( x \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( x \right ) \right ) ^{2}\cos \left ( x \right ) }{585\, \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( 231\,i \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) \sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}\sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -231\,i \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) \sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}\sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) +231\,i \left ( \cos \left ( x \right ) \right ) ^{6}\sin \left ( x \right ) \sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}\sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -231\,i \left ( \cos \left ( x \right ) \right ) ^{6}\sin \left ( x \right ) \sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}\sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) +231\, \left ( \cos \left ( x \right ) \right ) ^{7}-154\, \left ( \cos \left ( x \right ) \right ) ^{6}-22\, \left ( \cos \left ( x \right ) \right ) ^{4}-10\, \left ( \cos \left ( x \right ) \right ) ^{2}-45 \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((a*sec(x)^3)^(5/2),x)

[Out]

-2/585*(cos(x)+1)^2*(-1+cos(x))^2*(231*I*cos(x)^7*sin(x)*(cos(x)/(cos(x)+1))^(1/2)*(1/(cos(x)+1))^(1/2)*Ellipt
icF(I*(-1+cos(x))/sin(x),I)-231*I*cos(x)^7*sin(x)*(cos(x)/(cos(x)+1))^(1/2)*(1/(cos(x)+1))^(1/2)*EllipticE(I*(
-1+cos(x))/sin(x),I)+231*I*cos(x)^6*sin(x)*(cos(x)/(cos(x)+1))^(1/2)*(1/(cos(x)+1))^(1/2)*EllipticF(I*(-1+cos(
x))/sin(x),I)-231*I*cos(x)^6*sin(x)*(cos(x)/(cos(x)+1))^(1/2)*(1/(cos(x)+1))^(1/2)*EllipticE(I*(-1+cos(x))/sin
(x),I)+231*cos(x)^7-154*cos(x)^6-22*cos(x)^4-10*cos(x)^2-45)*cos(x)*(a/cos(x)^3)^(5/2)/sin(x)^5

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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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sec(x)^3)^(5/2), x)