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

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

[Out]

-154/195*a^2*cos(x)^(3/2)*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticE(sin(1/2*x),2^(1/2))*(a*sec(x)^3)^(1/2)+154
/195*a^2*cos(x)*sin(x)*(a*sec(x)^3)^(1/2)+154/585*a^2*(a*sec(x)^3)^(1/2)*tan(x)+22/117*a^2*sec(x)^2*(a*sec(x)^
3)^(1/2)*tan(x)+2/13*a^2*sec(x)^4*(a*sec(x)^3)^(1/2)*tan(x)

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Rubi [A]  time = 0.05, 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, 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 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]

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 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 \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*}

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Mathematica [A]  time = 0.10, size = 59, normalized size = 0.50 \[ -\frac {2}{585} a \sec (x) \left (a \sec ^3(x)\right )^{3/2} \left (-45 \tan (x)+231 \cos ^{\frac {11}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )-231 \sin (x) \cos ^5(x)-77 \sin (x) \cos ^3(x)-55 \sin (x) \cos (x)\right ) \]

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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fricas [F]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \sec \relax (x)^{3}} a^{2} \sec \relax (x)^{6}, x\right ) \]

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

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)

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maple [C]  time = 0.76, size = 223, normalized size = 1.91 \[ \frac {2 \left (\cos \relax (x )+1\right )^{2} \left (-1+\cos \relax (x )\right )^{2} \left (231 i \left (\cos ^{7}\relax (x )\right ) \sin \relax (x ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )-231 i \left (\cos ^{7}\relax (x )\right ) \sin \relax (x ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )+231 i \left (\cos ^{6}\relax (x )\right ) \sin \relax (x ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )-231 i \left (\cos ^{6}\relax (x )\right ) \sin \relax (x ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )-231 \left (\cos ^{7}\relax (x )\right )+154 \left (\cos ^{6}\relax (x )\right )+22 \left (\cos ^{4}\relax (x )\right )+10 \left (\cos ^{2}\relax (x )\right )+45\right ) \cos \relax (x ) \left (\frac {a}{\cos \relax (x )^{3}}\right )^{\frac {5}{2}}}{585 \sin \relax (x )^{5}} \]

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)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*Ellipti
cE(I*(-1+cos(x))/sin(x),I)-231*I*cos(x)^7*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(-
1+cos(x))/sin(x),I)+231*I*cos(x)^6*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(-1+cos(x
))/sin(x),I)-231*I*cos(x)^6*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sec \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a/cos(x)^3)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*sec(x)**3)**(5/2), x)

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