Integrand size = 13, antiderivative size = 61 \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=-\frac {1}{16} \arcsin (\cos (x)-\sin (x))+\frac {1}{16} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}} \]
-1/16*arcsin(cos(x)-sin(x))+1/16*ln(cos(x)+sin(x)+sin(2*x)^(1/2))+1/5*sin( x)^5/sin(2*x)^(5/2)-1/4*sin(x)/sin(2*x)^(1/2)
Time = 0.13 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\frac {1}{80} \left (5 \left (-\arcsin (\cos (x)-\sin (x))+\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )\right )+2 \sec (x) \left (-6+\sec ^2(x)\right ) \sqrt {\sin (2 x)}\right ) \]
(5*(-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]) + 2* Sec[x]*(-6 + Sec[x]^2)*Sqrt[Sin[2*x]])/80
Time = 0.42 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 4782, 3042, 4782, 3042, 4796, 3042, 4793}
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.
| \(\displaystyle \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^7}{\sin (2 x)^{7/2}}dx\) |
| \(\Big \downarrow \) 4782 |
| \(\displaystyle \frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {1}{4} \int \frac {\sin ^3(x)}{\sin ^{\frac {3}{2}}(2 x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {1}{4} \int \frac {\sin (x)^3}{\sin (2 x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4782 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{4} \int \csc (x) \sqrt {\sin (2 x)}dx-\frac {\sin (x)}{\sqrt {\sin (2 x)}}\right )+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{4} \int \frac {\sqrt {\sin (2 x)}}{\sin (x)}dx-\frac {\sin (x)}{\sqrt {\sin (2 x)}}\right )+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
| \(\Big \downarrow \) 4796 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {\cos (x)}{\sqrt {\sin (2 x)}}dx-\frac {\sin (x)}{\sqrt {\sin (2 x)}}\right )+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {\cos (x)}{\sqrt {\sin (2 x)}}dx-\frac {\sin (x)}{\sqrt {\sin (2 x)}}\right )+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
| \(\Big \downarrow \) 4793 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right )-\frac {1}{2} \arcsin (\cos (x)-\sin (x))\right )-\frac {\sin (x)}{\sqrt {\sin (2 x)}}\right )+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}\) |
((-1/2*ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]/2)/ 2 - Sin[x]/Sqrt[Sin[2*x]])/4 + Sin[x]^5/(5*Sin[2*x]^(5/2))
3.5.6.3.1 Defintions of rubi rules used
Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p _), x_Symbol] :> Simp[(-e^2)*(e*Sin[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 *Sin[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[cos[(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_)/sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[2*g Int[Cos[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 = 6.09 (sec) , antiderivative size = 510, normalized size of antiderivative = 8.36
| method | result | size |
| default | \(\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (5 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{14}\left (\frac {x}{2}\right )\right )+35 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{12}\left (\frac {x}{2}\right )\right )+10 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )+105 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{10}\left (\frac {x}{2}\right )\right )+66 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )+175 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{8}\left (\frac {x}{2}\right )\right )-1014 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )+175 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+2002 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )+105 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-2002 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+35 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+1014 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+5 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )-66 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-10 \tan \left (\frac {x}{2}\right )\right )}{2688 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7} \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}}\) | \(510\) |
1/2688*(-tan(1/2*x)/(tan(1/2*x)^2-1))^(1/2)*(tan(1/2*x)^2-1)*(5*(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))*tan(1/2*x)^14+35*(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 ))*tan(1/2*x)^12+10*tan(1/2*x)^15+105*(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))*t an(1/2*x)^10+66*tan(1/2*x)^13+175*(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))*tan(1 /2*x)^8-1014*tan(1/2*x)^11+175*(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))*tan(1/2* x)^6+2002*tan(1/2*x)^9+105*(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))*tan(1/2*x)^4 -2002*tan(1/2*x)^7+35*(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))*tan(1/2*x)^2+1014 *tan(1/2*x)^5+5*(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))-66*tan(1/2*x)^3-10*tan( 1/2*x))/(tan(1/2*x)*(tan(1/2*x)^2-1))^(1/2)/(1+tan(1/2*x)^2)^7/(tan(1/2*x) ^3-tan(1/2*x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (47) = 94\).
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.97 \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\frac {10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) \cos \left (x\right )^{3} - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{3} \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - 48 \, \cos \left (x\right )^{3} - 8 \, \sqrt {2} {\left (6 \, \cos \left (x\right )^{2} - 1\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )}}{320 \, \cos \left (x\right )^{3}} \]
1/320*(10*arctan(-(sqrt(2)*sqrt(cos(x)*sin(x))*(cos(x) - sin(x)) + cos(x)* sin(x))/(cos(x)^2 + 2*cos(x)*sin(x) - 1))*cos(x)^3 - 10*arctan(-(2*sqrt(2) *sqrt(cos(x)*sin(x)) - cos(x) - sin(x))/(cos(x) - sin(x)))*cos(x)^3 - 5*co s(x)^3*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) - 48 *cos(x)^3 - 8*sqrt(2)*(6*cos(x)^2 - 1)*sqrt(cos(x)*sin(x)))/cos(x)^3
Timed out. \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int { \frac {\sin \left (x\right )^{7}}{\sin \left (2 \, x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int { \frac {\sin \left (x\right )^{7}}{\sin \left (2 \, x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int \frac {{\sin \left (x\right )}^7}{{\sin \left (2\,x\right )}^{7/2}} \,d x \]
\[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int \frac {\sqrt {\sin \left (2 x \right )}\, \sin \left (x \right )^{7}}{\sin \left (2 x \right )^{4}}d x \]