Integrand size = 22, antiderivative size = 98 \[ \int \cos ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx=\frac {5 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )}{42 b}-\frac {5 \cos (2 a+2 b x) \sqrt {\sin (2 a+2 b x)}}{42 b}-\frac {\cos (2 a+2 b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{14 b}+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x)}{18 b} \]
-5/42*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi +b*x),2^(1/2))/b-1/14*cos(2*b*x+2*a)*sin(2*b*x+2*a)^(5/2)/b+1/18*sin(2*b*x +2*a)^(9/2)/b-5/42*cos(2*b*x+2*a)*sin(2*b*x+2*a)^(1/2)/b
Time = 0.72 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.98 \[ \int \cos ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx=\frac {240 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 (a+b x))}+70 \sin (2 (a+b x))-156 \sin (4 (a+b x))-35 \sin (6 (a+b x))+18 \sin (8 (a+b x))+7 \sin (10 (a+b x))}{2016 b \sqrt {\sin (2 (a+b x))}} \]
(240*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*(a + b*x)]] + 70*Sin[2*(a + b *x)] - 156*Sin[4*(a + b*x)] - 35*Sin[6*(a + b*x)] + 18*Sin[8*(a + b*x)] + 7*Sin[10*(a + b*x)])/(2016*b*Sqrt[Sin[2*(a + b*x)]])
Time = 0.41 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4785, 3042, 3115, 3042, 3115, 3042, 3120}
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.
\(\displaystyle \int \sin ^{\frac {7}{2}}(2 a+2 b x) \cos ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (2 a+2 b x)^{7/2} \cos (a+b x)^2dx\) |
\(\Big \downarrow \) 4785 |
\(\displaystyle \frac {1}{2} \int \sin ^{\frac {7}{2}}(2 a+2 b x)dx+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x)}{18 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \sin (2 a+2 b x)^{7/2}dx+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x)}{18 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{7} \int \sin ^{\frac {3}{2}}(2 a+2 b x)dx-\frac {\sin ^{\frac {5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}\right )+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x)}{18 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{7} \int \sin (2 a+2 b x)^{3/2}dx-\frac {\sin ^{\frac {5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}\right )+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x)}{18 b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx-\frac {\sqrt {\sin (2 a+2 b x)} \cos (2 a+2 b x)}{3 b}\right )-\frac {\sin ^{\frac {5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}\right )+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x)}{18 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}}dx-\frac {\sqrt {\sin (2 a+2 b x)} \cos (2 a+2 b x)}{3 b}\right )-\frac {\sin ^{\frac {5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}\right )+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x)}{18 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\sin ^{\frac {9}{2}}(2 a+2 b x)}{18 b}+\frac {1}{2} \left (\frac {5}{7} \left (\frac {\operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{3 b}-\frac {\sqrt {\sin (2 a+2 b x)} \cos (2 a+2 b x)}{3 b}\right )-\frac {\sin ^{\frac {5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}\right )\) |
Sin[2*a + 2*b*x]^(9/2)/(18*b) + ((5*(EllipticF[a - Pi/4 + b*x, 2]/(3*b) - (Cos[2*a + 2*b*x]*Sqrt[Sin[2*a + 2*b*x]])/(3*b)))/7 - (Cos[2*a + 2*b*x]*Si n[2*a + 2*b*x]^(5/2))/(7*b))/2
3.2.69.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
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*(m + 2*p))), x] + Simp[e^2*((m + p - 1)/(m + 2*p)) Int[(e*Cos[a + b*x])^(m - 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && !IntegerQ[p] && GtQ[m, 1] && NeQ [m + 2*p, 0] && IntegersQ[2*m, 2*p]
Timed out.
\[\int \cos \left (x b +a \right )^{2} \sin \left (2 x b +2 a \right )^{\frac {7}{2}}d x\]
\[ \int \cos ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx=\int { \cos \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \cos ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx=\text {Timed out} \]
\[ \int \cos ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx=\int { \cos \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {7}{2}} \,d x } \]
\[ \int \cos ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx=\int { \cos \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \cos ^2(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{7/2} \,d x \]