3.658 \(\int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx\)

Optimal. Leaf size=71 \[ \frac {2 (1-5 \cos (x))^{9/2}}{28125}-\frac {8 (1-5 \cos (x))^{7/2}}{21875}-\frac {88 (1-5 \cos (x))^{5/2}}{15625}+\frac {64 (1-5 \cos (x))^{3/2}}{3125}+\frac {1152 \sqrt {1-5 \cos (x)}}{3125} \]

[Out]

64/3125*(1-5*cos(x))^(3/2)-88/15625*(1-5*cos(x))^(5/2)-8/21875*(1-5*cos(x))^(7/2)+2/28125*(1-5*cos(x))^(9/2)+1
152/3125*(1-5*cos(x))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2668, 697} \[ \frac {2 (1-5 \cos (x))^{9/2}}{28125}-\frac {8 (1-5 \cos (x))^{7/2}}{21875}-\frac {88 (1-5 \cos (x))^{5/2}}{15625}+\frac {64 (1-5 \cos (x))^{3/2}}{3125}+\frac {1152 \sqrt {1-5 \cos (x)}}{3125} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^5/Sqrt[1 - 5*Cos[x]],x]

[Out]

(1152*Sqrt[1 - 5*Cos[x]])/3125 + (64*(1 - 5*Cos[x])^(3/2))/3125 - (88*(1 - 5*Cos[x])^(5/2))/15625 - (8*(1 - 5*
Cos[x])^(7/2))/21875 + (2*(1 - 5*Cos[x])^(9/2))/28125

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (25-x^2\right )^2}{\sqrt {1+x}} \, dx,x,-5 \cos (x)\right )}{3125}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {576}{\sqrt {1+x}}+96 \sqrt {1+x}-44 (1+x)^{3/2}-4 (1+x)^{5/2}+(1+x)^{7/2}\right ) \, dx,x,-5 \cos (x)\right )}{3125}\\ &=\frac {1152 \sqrt {1-5 \cos (x)}}{3125}+\frac {64 (1-5 \cos (x))^{3/2}}{3125}-\frac {88 (1-5 \cos (x))^{5/2}}{15625}-\frac {8 (1-5 \cos (x))^{7/2}}{21875}+\frac {2 (1-5 \cos (x))^{9/2}}{28125}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 59, normalized size = 0.83 \[ \frac {180607 \left (\sqrt {1-5 \cos (x)}-1\right )}{562500}+\sqrt {1-5 \cos (x)} \left (-\frac {6772 \cos (x)}{196875}-\frac {2227 \cos (2 x)}{39375}+\frac {4 \cos (3 x)}{1575}+\frac {1}{180} \cos (4 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^5/Sqrt[1 - 5*Cos[x]],x]

[Out]

(180607*(-1 + Sqrt[1 - 5*Cos[x]]))/562500 + Sqrt[1 - 5*Cos[x]]*((-6772*Cos[x])/196875 - (2227*Cos[2*x])/39375
+ (4*Cos[3*x])/1575 + Cos[4*x]/180)

________________________________________________________________________________________

fricas [A]  time = 1.68, size = 34, normalized size = 0.48 \[ \frac {2}{984375} \, {\left (21875 \, \cos \relax (x)^{4} + 5000 \, \cos \relax (x)^{3} - 77550 \, \cos \relax (x)^{2} - 20680 \, \cos \relax (x) + 188603\right )} \sqrt {-5 \, \cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^5/(1-5*cos(x))^(1/2),x, algorithm="fricas")

[Out]

2/984375*(21875*cos(x)^4 + 5000*cos(x)^3 - 77550*cos(x)^2 - 20680*cos(x) + 188603)*sqrt(-5*cos(x) + 1)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 75, normalized size = 1.06 \[ \frac {2}{28125} \, {\left (5 \, \cos \relax (x) - 1\right )}^{4} \sqrt {-5 \, \cos \relax (x) + 1} + \frac {8}{21875} \, {\left (5 \, \cos \relax (x) - 1\right )}^{3} \sqrt {-5 \, \cos \relax (x) + 1} - \frac {88}{15625} \, {\left (5 \, \cos \relax (x) - 1\right )}^{2} \sqrt {-5 \, \cos \relax (x) + 1} + \frac {64}{3125} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {3}{2}} + \frac {1152}{3125} \, \sqrt {-5 \, \cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^5/(1-5*cos(x))^(1/2),x, algorithm="giac")

[Out]

2/28125*(5*cos(x) - 1)^4*sqrt(-5*cos(x) + 1) + 8/21875*(5*cos(x) - 1)^3*sqrt(-5*cos(x) + 1) - 88/15625*(5*cos(
x) - 1)^2*sqrt(-5*cos(x) + 1) + 64/3125*(-5*cos(x) + 1)^(3/2) + 1152/3125*sqrt(-5*cos(x) + 1)

________________________________________________________________________________________

maple [A]  time = 0.17, size = 49, normalized size = 0.69 \[ \frac {32 \sqrt {10 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )-4}\, \left (21875 \left (\sin ^{8}\left (\frac {x}{2}\right )\right )-46250 \left (\sin ^{6}\left (\frac {x}{2}\right )\right )+17175 \left (\sin ^{4}\left (\frac {x}{2}\right )\right )+9160 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )+7328\right )}{984375} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5/(1-5*cos(x))^(1/2),x)

[Out]

32/984375*(10*sin(1/2*x)^2-4)^(1/2)*(21875*sin(1/2*x)^8-46250*sin(1/2*x)^6+17175*sin(1/2*x)^4+9160*sin(1/2*x)^
2+7328)

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 51, normalized size = 0.72 \[ \frac {2}{28125} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {9}{2}} - \frac {8}{21875} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {7}{2}} - \frac {88}{15625} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {5}{2}} + \frac {64}{3125} \, {\left (-5 \, \cos \relax (x) + 1\right )}^{\frac {3}{2}} + \frac {1152}{3125} \, \sqrt {-5 \, \cos \relax (x) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^5/(1-5*cos(x))^(1/2),x, algorithm="maxima")

[Out]

2/28125*(-5*cos(x) + 1)^(9/2) - 8/21875*(-5*cos(x) + 1)^(7/2) - 88/15625*(-5*cos(x) + 1)^(5/2) + 64/3125*(-5*c
os(x) + 1)^(3/2) + 1152/3125*sqrt(-5*cos(x) + 1)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \relax (x)}^5}{\sqrt {1-5\,\cos \relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5/(1 - 5*cos(x))^(1/2),x)

[Out]

int(sin(x)^5/(1 - 5*cos(x))^(1/2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**5/(1-5*cos(x))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________