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]

(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

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Rubi [A]  time = 0.0658991, antiderivative size = 71, normalized size of antiderivative = 1., 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 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]

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]

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 = 0.15787, 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)

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Maple [A]  time = 1.081, size = 49, normalized size = 0.7 \begin{align*}{\frac{32}{984375}\sqrt{10\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}-4} \left ( 21875\, \left ( \sin \left ( x/2 \right ) \right ) ^{8}-46250\, \left ( \sin \left ( x/2 \right ) \right ) ^{6}+17175\, \left ( \sin \left ( x/2 \right ) \right ) ^{4}+9160\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+7328 \right ) } \end{align*}

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)

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Maxima [A]  time = 0.985314, size = 69, normalized size = 0.97 \begin{align*} \frac{2}{28125} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{9}{2}} - \frac{8}{21875} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{7}{2}} - \frac{88}{15625} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{5}{2}} + \frac{64}{3125} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{3}{2}} + \frac{1152}{3125} \, \sqrt{-5 \, \cos \left (x\right ) + 1} \end{align*}

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)

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Fricas [A]  time = 2.1865, size = 140, normalized size = 1.97 \begin{align*} \frac{2}{984375} \,{\left (21875 \, \cos \left (x\right )^{4} + 5000 \, \cos \left (x\right )^{3} - 77550 \, \cos \left (x\right )^{2} - 20680 \, \cos \left (x\right ) + 188603\right )} \sqrt{-5 \, \cos \left (x\right ) + 1} \end{align*}

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)

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

[Out]

Timed out

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Giac [A]  time = 1.11028, size = 101, normalized size = 1.42 \begin{align*} \frac{2}{28125} \,{\left (5 \, \cos \left (x\right ) - 1\right )}^{4} \sqrt{-5 \, \cos \left (x\right ) + 1} + \frac{8}{21875} \,{\left (5 \, \cos \left (x\right ) - 1\right )}^{3} \sqrt{-5 \, \cos \left (x\right ) + 1} - \frac{88}{15625} \,{\left (5 \, \cos \left (x\right ) - 1\right )}^{2} \sqrt{-5 \, \cos \left (x\right ) + 1} + \frac{64}{3125} \,{\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac{3}{2}} + \frac{1152}{3125} \, \sqrt{-5 \, \cos \left (x\right ) + 1} \end{align*}

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)