3.359 \(\int \sec ^{\frac{13}{2}}(x) \sin ^5(x) \, dx\)

Optimal. Leaf size=31 \[ \frac{2}{11} \sec ^{\frac{11}{2}}(x)-\frac{4}{7} \sec ^{\frac{7}{2}}(x)+\frac{2}{3} \sec ^{\frac{3}{2}}(x) \]

[Out]

(2*Sec[x]^(3/2))/3 - (4*Sec[x]^(7/2))/7 + (2*Sec[x]^(11/2))/11

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Rubi [A]  time = 0.0258051, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2622, 270} \[ \frac{2}{11} \sec ^{\frac{11}{2}}(x)-\frac{4}{7} \sec ^{\frac{7}{2}}(x)+\frac{2}{3} \sec ^{\frac{3}{2}}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^(13/2)*Sin[x]^5,x]

[Out]

(2*Sec[x]^(3/2))/3 - (4*Sec[x]^(7/2))/7 + (2*Sec[x]^(11/2))/11

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \sec ^{\frac{13}{2}}(x) \sin ^5(x) \, dx &=\operatorname{Subst}\left (\int \sqrt{x} \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\sqrt{x}-2 x^{5/2}+x^{9/2}\right ) \, dx,x,\sec (x)\right )\\ &=\frac{2}{3} \sec ^{\frac{3}{2}}(x)-\frac{4}{7} \sec ^{\frac{7}{2}}(x)+\frac{2}{11} \sec ^{\frac{11}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.0501065, size = 24, normalized size = 0.77 \[ \frac{1}{924} (44 \cos (2 x)+77 \cos (4 x)+135) \sec ^{\frac{11}{2}}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^(13/2)*Sin[x]^5,x]

[Out]

((135 + 44*Cos[2*x] + 77*Cos[4*x])*Sec[x]^(11/2))/924

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Maple [B]  time = 0.092, size = 49, normalized size = 1.6 \begin{align*}{\frac{32}{231} \left ( 77\, \left ( \sin \left ( x/2 \right ) \right ) ^{8}-154\, \left ( \sin \left ( x/2 \right ) \right ) ^{6}+99\, \left ( \sin \left ( x/2 \right ) \right ) ^{4}-22\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+2 \right ) \left ( -2\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^(3/2)*tan(x)^5,x)

[Out]

32/231/(-2*sin(1/2*x)^2+1)^(11/2)*(77*sin(1/2*x)^8-154*sin(1/2*x)^6+99*sin(1/2*x)^4-22*sin(1/2*x)^2+2)

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Maxima [A]  time = 0.938604, size = 26, normalized size = 0.84 \begin{align*} \frac{2}{3 \, \cos \left (x\right )^{\frac{3}{2}}} - \frac{4}{7 \, \cos \left (x\right )^{\frac{7}{2}}} + \frac{2}{11 \, \cos \left (x\right )^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^(3/2)*tan(x)^5,x, algorithm="maxima")

[Out]

2/3/cos(x)^(3/2) - 4/7/cos(x)^(7/2) + 2/11/cos(x)^(11/2)

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Fricas [A]  time = 1.87227, size = 73, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (77 \, \cos \left (x\right )^{4} - 66 \, \cos \left (x\right )^{2} + 21\right )}}{231 \, \cos \left (x\right )^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(77*cos(x)^4 - 66*cos(x)^2 + 21)/cos(x)^(11/2)

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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(sec(x)**(3/2)*tan(x)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.10041, size = 31, normalized size = 1. \begin{align*} \frac{2 \,{\left (77 \, \cos \left (x\right )^{4} - 66 \, \cos \left (x\right )^{2} + 21\right )} \mathrm{sgn}\left (\cos \left (x\right )\right )}{231 \, \cos \left (x\right )^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(77*cos(x)^4 - 66*cos(x)^2 + 21)*sgn(cos(x))/cos(x)^(11/2)