Integrand size = 13, antiderivative size = 61 \[ \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=-\frac {1}{16} \arcsin (\cos (x)-\sin (x))-\frac {1}{16} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}+\frac {\cos (x)}{4 \sqrt {\sin (2 x)}} \]
-1/16*arcsin(cos(x)-sin(x))-1/16*ln(cos(x)+sin(x)+sin(2*x)^(1/2))-1/5*cos( x)^5/sin(2*x)^(5/2)+1/4*cos(x)/sin(2*x)^(1/2)
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\frac {1}{16} \left (-\arcsin (\cos (x)-\sin (x))-\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )\right )+\left (\frac {3 \csc (x)}{20}-\frac {\csc ^3(x)}{40}\right ) \sqrt {\sin (2 x)} \]
(-ArcSin[Cos[x] - Sin[x]] - Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]])/16 + (( 3*Csc[x])/20 - Csc[x]^3/40)*Sqrt[Sin[2*x]]
Time = 0.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 4781, 3042, 4781, 3042, 4795, 3042, 4794}
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.
\(\displaystyle \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (x)^7}{\sin (2 x)^{7/2}}dx\) |
\(\Big \downarrow \) 4781 |
\(\displaystyle -\frac {1}{4} \int \frac {\cos ^3(x)}{\sin ^{\frac {3}{2}}(2 x)}dx-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{4} \int \frac {\cos (x)^3}{\sin (2 x)^{3/2}}dx-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
\(\Big \downarrow \) 4781 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{4} \int \sec (x) \sqrt {\sin (2 x)}dx+\frac {\cos (x)}{\sqrt {\sin (2 x)}}\right )-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{4} \int \frac {\sqrt {\sin (2 x)}}{\cos (x)}dx+\frac {\cos (x)}{\sqrt {\sin (2 x)}}\right )-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
\(\Big \downarrow \) 4795 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {\sin (x)}{\sqrt {\sin (2 x)}}dx+\frac {\cos (x)}{\sqrt {\sin (2 x)}}\right )-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {\sin (x)}{\sqrt {\sin (2 x)}}dx+\frac {\cos (x)}{\sqrt {\sin (2 x)}}\right )-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
\(\Big \downarrow \) 4794 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {1}{2} \arcsin (\cos (x)-\sin (x))-\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )\right )+\frac {\cos (x)}{\sqrt {\sin (2 x)}}\right )-\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
((-1/2*ArcSin[Cos[x] - Sin[x]] - Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/2)/ 2 + Cos[x]/Sqrt[Sin[2*x]])/4 - Cos[x]^5/(5*Sin[2*x]^(5/2))
3.5.7.3.1 Defintions of rubi rules used
Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p _), x_Symbol] :> Simp[e^2*(e*Cos[a + b*x])^(m - 2)*((g*Sin[c + d*x])^(p + 1 )/(2*b*g*(p + 1))), x] + Simp[e^4*((m + p - 1)/(4*g^2*(p + 1))) Int[(e*Co s[a + b*x])^(m - 4)*(g*Sin[c + d*x])^(p + 2), x], x] /; FreeQ[{a, b, c, d, e, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 2] && LtQ[p, -1] && (GtQ[m, 3] || EqQ[p, -3/2]) && IntegersQ[2*m, 2*p]
Int[sin[(a_.) + (b_.)*(x_)]/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Sim p[-ArcSin[Cos[a + b*x] - Sin[a + b*x]]/d, x] - Simp[Log[Cos[a + b*x] + Sin[ a + b*x] + Sqrt[Sin[c + d*x]]]/d, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
Int[((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_)/cos[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[2*g Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p - 1), x], x] /; FreeQ[{ a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && IntegerQ[2*p]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.73 (sec) , antiderivative size = 1108, normalized size of antiderivative = 18.16
1/160*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)/tan(1/2*x)^3*(192*(tan(1/2*x)*( tan(1/2*x)^2-1))^(1/2)*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan( 1/2*x))^(1/2)*EllipticE((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*((1+tan(1/2*x))* (tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^6-96*(tan(1/2*x)*(tan(1/2*x)^2 -1))^(1/2)*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2 )*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*((1+tan(1/2*x))*(tan(1/2*x)- 1)*tan(1/2*x))^(1/2)*tan(1/2*x)^6-(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*((1+ tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^10+96*(tan(1/2*x)* (tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*tan(1/2*x)^8-384*( tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^ (1/2)*(-tan(1/2*x))^(1/2)*EllipticE((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*((1+ tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^4+192*(tan(1/2*x)* (tan(1/2*x)^2-1))^(1/2)*(1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan (1/2*x))^(1/2)*EllipticF((1+tan(1/2*x))^(1/2),1/2*2^(1/2))*((1+tan(1/2*x)) *(tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^4+3*(tan(1/2*x)*(tan(1/2*x)^2 -1))^(1/2)*((1+tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x))^(1/2)*tan(1/2*x)^8+4 8*(tan(1/2*x)^3-tan(1/2*x))^(1/2)*((1+tan(1/2*x))*(tan(1/2*x)-1)*tan(1/2*x ))^(1/2)*tan(1/2*x)^8-192*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^ 3-tan(1/2*x))^(1/2)*tan(1/2*x)^6+192*(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)*( 1+tan(1/2*x))^(1/2)*(-2*tan(1/2*x)+2)^(1/2)*(-tan(1/2*x))^(1/2)*Ellipti...
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (47) = 94\).
Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 3.36 \[ \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\frac {10 \, {\left (\cos \left (x\right )^{2} - 1\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) \sin \left (x\right ) - 10 \, {\left (\cos \left (x\right )^{2} - 1\right )} \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) \sin \left (x\right ) + 5 \, {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) + 8 \, \sqrt {2} {\left (6 \, \cos \left (x\right )^{2} - 5\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 48 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}{320 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
1/320*(10*(cos(x)^2 - 1)*arctan(-(sqrt(2)*sqrt(cos(x)*sin(x))*(cos(x) - si n(x)) + cos(x)*sin(x))/(cos(x)^2 + 2*cos(x)*sin(x) - 1))*sin(x) - 10*(cos( x)^2 - 1)*arctan(-(2*sqrt(2)*sqrt(cos(x)*sin(x)) - cos(x) - sin(x))/(cos(x ) - sin(x)))*sin(x) + 5*(cos(x)^2 - 1)*log(-32*cos(x)^4 + 4*sqrt(2)*(4*cos (x)^3 - (4*cos(x)^2 + 1)*sin(x) - 5*cos(x))*sqrt(cos(x)*sin(x)) + 32*cos(x )^2 + 16*cos(x)*sin(x) + 1)*sin(x) + 8*sqrt(2)*(6*cos(x)^2 - 5)*sqrt(cos(x )*sin(x)) + 48*(cos(x)^2 - 1)*sin(x))/((cos(x)^2 - 1)*sin(x))
Timed out. \[ \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int { \frac {\cos \left (x\right )^{7}}{\sin \left (2 \, x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int { \frac {\cos \left (x\right )^{7}}{\sin \left (2 \, x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int \frac {{\cos \left (x\right )}^7}{{\sin \left (2\,x\right )}^{7/2}} \,d x \]