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

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

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Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.05, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.01, size = 20, normalized size = 0.65 \[ \frac {2 \, {\left (77 \, \cos \relax (x)^{4} - 66 \, \cos \relax (x)^{2} + 21\right )}}{231 \, \cos \relax (x)^{\frac {11}{2}}} \]

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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giac [A]  time = 0.65, size = 20, normalized size = 0.65 \[ \frac {2 \, {\left (77 \, \cos \relax (x)^{4} - 66 \, \cos \relax (x)^{2} + 21\right )}}{231 \, \cos \relax (x)^{\frac {11}{2}}} \]

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)/cos(x)^(11/2)

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maple [B]  time = 0.20, size = 49, normalized size = 1.58

method	result	size
default	\(\frac {\frac {32 \left (\sin ^{8}\left (\frac {x}{2}\right )\right )}{3}-\frac {64 \left (\sin ^{6}\left (\frac {x}{2}\right )\right )}{3}+\frac {96 \left (\sin ^{4}\left (\frac {x}{2}\right )\right )}{7}-\frac {64 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}{21}+\frac {64}{231}}{\left (-2 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )+1\right )^{\frac {11}{2}}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.54, size = 19, normalized size = 0.61 \[ \frac {2}{3 \, \cos \relax (x)^{\frac {3}{2}}} - \frac {4}{7 \, \cos \relax (x)^{\frac {7}{2}}} + \frac {2}{11 \, \cos \relax (x)^{\frac {11}{2}}} \]

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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mupad [B]  time = 1.12, size = 22, normalized size = 0.71 \[ \frac {2\,{\left (\frac {1}{\cos \relax (x)}\right )}^{11/2}\,\left (77\,{\cos \relax (x)}^4-66\,{\cos \relax (x)}^2+21\right )}{231} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)^5*(1/cos(x))^(3/2),x)

[Out]

(2*(1/cos(x))^(11/2)*(77*cos(x)^4 - 66*cos(x)^2 + 21))/231

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sympy [A]  time = 178.37, size = 39, normalized size = 1.26 \[ \frac {2 \tan ^{4}{\relax (x )} \sec ^{\frac {3}{2}}{\relax (x )}}{11} - \frac {16 \tan ^{2}{\relax (x )} \sec ^{\frac {3}{2}}{\relax (x )}}{77} + \frac {64 \sec ^{\frac {3}{2}}{\relax (x )}}{231} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*tan(x)**4*sec(x)**(3/2)/11 - 16*tan(x)**2*sec(x)**(3/2)/77 + 64*sec(x)**(3/2)/231

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