Integrand size = 15, antiderivative size = 71 \[ \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx=\frac {1152 \sqrt {1-5 \cos (x)}}{3125}+\frac {64 (1-5 \cos (x))^{3/2}}{3125}-\frac {88 (1-5 \cos (x))^{5/2}}{15625}-\frac {8 (1-5 \cos (x))^{7/2}}{21875}+\frac {2 (1-5 \cos (x))^{9/2}}{28125} \] Output:
1152/3125*(1-5*cos(x))^(1/2)+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)
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx=\frac {180607 \left (-1+\sqrt {1-5 \cos (x)}\right )}{562500}+\sqrt {1-5 \cos (x)} \left (-\frac {6772 \cos (x)}{196875}-\frac {2227 \cos (2 x)}{39375}+\frac {4 \cos (3 x)}{1575}+\frac {1}{180} \cos (4 x)\right ) \] Input:
Integrate[Sin[x]^5/Sqrt[1 - 5*Cos[x]],x]
Output:
(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)
Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3147, 476, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (x-\frac {\pi }{2}\right )^5}{\sqrt {5 \sin \left (x-\frac {\pi }{2}\right )+1}}dx\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle \frac {\int \frac {\left (25-25 \cos ^2(x)\right )^2}{\sqrt {1-5 \cos (x)}}d(-5 \cos (x))}{3125}\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \frac {\int \left ((1-5 \cos (x))^{7/2}-4 (1-5 \cos (x))^{5/2}-44 (1-5 \cos (x))^{3/2}+96 \sqrt {1-5 \cos (x)}+\frac {576}{\sqrt {1-5 \cos (x)}}\right )d(-5 \cos (x))}{3125}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {2}{9} (1-5 \cos (x))^{9/2}-\frac {8}{7} (1-5 \cos (x))^{7/2}-\frac {88}{5} (1-5 \cos (x))^{5/2}+64 (1-5 \cos (x))^{3/2}+1152 \sqrt {1-5 \cos (x)}}{3125}\) |
Input:
Int[Sin[x]^5/Sqrt[1 - 5*Cos[x]],x]
Output:
(1152*Sqrt[1 - 5*Cos[x]] + 64*(1 - 5*Cos[x])^(3/2) - (88*(1 - 5*Cos[x])^(5 /2))/5 - (8*(1 - 5*Cos[x])^(7/2))/7 + (2*(1 - 5*Cos[x])^(9/2))/9)/3125
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[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] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 0.35 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {1152 \sqrt {1-5 \cos \left (x \right )}}{3125}+\frac {64 \left (1-5 \cos \left (x \right )\right )^{\frac {3}{2}}}{3125}-\frac {88 \left (1-5 \cos \left (x \right )\right )^{\frac {5}{2}}}{15625}-\frac {8 \left (1-5 \cos \left (x \right )\right )^{\frac {7}{2}}}{21875}+\frac {2 \left (1-5 \cos \left (x \right )\right )^{\frac {9}{2}}}{28125}\) | \(52\) |
default | \(\frac {1152 \sqrt {1-5 \cos \left (x \right )}}{3125}+\frac {64 \left (1-5 \cos \left (x \right )\right )^{\frac {3}{2}}}{3125}-\frac {88 \left (1-5 \cos \left (x \right )\right )^{\frac {5}{2}}}{15625}-\frac {8 \left (1-5 \cos \left (x \right )\right )^{\frac {7}{2}}}{21875}+\frac {2 \left (1-5 \cos \left (x \right )\right )^{\frac {9}{2}}}{28125}\) | \(52\) |
Input:
int(sin(x)^5/(1-5*cos(x))^(1/2),x,method=_RETURNVERBOSE)
Output:
1152/3125*(1-5*cos(x))^(1/2)+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)
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.48 \[ \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx=\frac {2}{984375} \, {\left (21875 \, \cos \left (x\right )^{4} + 5000 \, \cos \left (x\right )^{3} - 77550 \, \cos \left (x\right )^{2} - 20680 \, \cos \left (x\right ) + 188603\right )} \sqrt {-5 \, \cos \left (x\right ) + 1} \] Input:
integrate(sin(x)^5/(1-5*cos(x))^(1/2),x, algorithm="fricas")
Output:
2/984375*(21875*cos(x)^4 + 5000*cos(x)^3 - 77550*cos(x)^2 - 20680*cos(x) + 188603)*sqrt(-5*cos(x) + 1)
Timed out. \[ \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx=\text {Timed out} \] Input:
integrate(sin(x)**5/(1-5*cos(x))**(1/2),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx=\frac {2}{28125} \, {\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac {9}{2}} - \frac {8}{21875} \, {\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac {7}{2}} - \frac {88}{15625} \, {\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac {5}{2}} + \frac {64}{3125} \, {\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac {3}{2}} + \frac {1152}{3125} \, \sqrt {-5 \, \cos \left (x\right ) + 1} \] Input:
integrate(sin(x)^5/(1-5*cos(x))^(1/2),x, algorithm="maxima")
Output:
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*cos(x) + 1)^(3/2) + 1152/3125*sqrt(-5*c os(x) + 1)
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06 \[ \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx=\frac {2}{28125} \, {\left (5 \, \cos \left (x\right ) - 1\right )}^{4} \sqrt {-5 \, \cos \left (x\right ) + 1} + \frac {8}{21875} \, {\left (5 \, \cos \left (x\right ) - 1\right )}^{3} \sqrt {-5 \, \cos \left (x\right ) + 1} - \frac {88}{15625} \, {\left (5 \, \cos \left (x\right ) - 1\right )}^{2} \sqrt {-5 \, \cos \left (x\right ) + 1} + \frac {64}{3125} \, {\left (-5 \, \cos \left (x\right ) + 1\right )}^{\frac {3}{2}} + \frac {1152}{3125} \, \sqrt {-5 \, \cos \left (x\right ) + 1} \] Input:
integrate(sin(x)^5/(1-5*cos(x))^(1/2),x, algorithm="giac")
Output:
2/28125*(5*cos(x) - 1)^4*sqrt(-5*cos(x) + 1) + 8/21875*(5*cos(x) - 1)^3*sq rt(-5*cos(x) + 1) - 88/15625*(5*cos(x) - 1)^2*sqrt(-5*cos(x) + 1) + 64/312 5*(-5*cos(x) + 1)^(3/2) + 1152/3125*sqrt(-5*cos(x) + 1)
Timed out. \[ \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx=\int \frac {{\sin \left (x\right )}^5}{\sqrt {1-5\,\cos \left (x\right )}} \,d x \] Input:
int(sin(x)^5/(1 - 5*cos(x))^(1/2),x)
Output:
int(sin(x)^5/(1 - 5*cos(x))^(1/2), x)
\[ \int \frac {\sin ^5(x)}{\sqrt {1-5 \cos (x)}} \, dx=-\left (\int \frac {\sqrt {-5 \cos \left (x \right )+1}\, \sin \left (x \right )^{5}}{5 \cos \left (x \right )-1}d x \right ) \] Input:
int(sin(x)^5/(1-5*cos(x))^(1/2),x)
Output:
- int((sqrt( - 5*cos(x) + 1)*sin(x)**5)/(5*cos(x) - 1),x)