Integrand size = 30, antiderivative size = 178 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {3 \cos (e+f x) (3+3 \sin (e+f x))^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {27 \cos (e+f x) (3+3 \sin (e+f x))^{3/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {27 \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{56 c^2 f (c-c \sin (e+f x))^{13/2}}-\frac {81 \cos (e+f x)}{280 c^3 f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{11/2}} \]
1/8*a*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/f/(c-c*sin(f*x+e))^(17/2)-3/56*a^2 *cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/c/f/(c-c*sin(f*x+e))^(15/2)-1/280*a^4*c os(f*x+e)/c^3/f/(c-c*sin(f*x+e))^(11/2)/(a+a*sin(f*x+e))^(1/2)+1/56*a^3*co s(f*x+e)*(a+a*sin(f*x+e))^(1/2)/c^2/f/(c-c*sin(f*x+e))^(13/2)
Time = 11.15 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.85 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (3+3 \sin (e+f x))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}-\frac {12 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (3+3 \sin (e+f x))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}+\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (3+3 \sin (e+f x))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (3+3 \sin (e+f x))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}} \]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(3 + 3*Sin[e + f*x])^(7/2))/(f*(Cos [(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - (12*(Co s[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(3 + 3*Sin[e + f*x])^(7/2))/(7*f*(Cos [(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + ((Cos[( e + f*x)/2] - Sin[(e + f*x)/2])^5*(3 + 3*Sin[e + f*x])^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - ((Cos[(e + f *x)/2] - Sin[(e + f*x)/2])^7*(3 + 3*Sin[e + f*x])^(7/2))/(5*f*(Cos[(e + f* x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2))
Time = 0.86 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3218, 3042, 3218, 3042, 3218, 3042, 3217}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{17/2}}dx\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{15/2}}dx}{8 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{15/2}}dx}{8 c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{7 f (c-c \sin (e+f x))^{15/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{13/2}}dx}{7 c}\right )}{8 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{7 f (c-c \sin (e+f x))^{15/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{13/2}}dx}{7 c}\right )}{8 c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{7 f (c-c \sin (e+f x))^{15/2}}-\frac {2 a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{11/2}}dx}{6 c}\right )}{7 c}\right )}{8 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{7 f (c-c \sin (e+f x))^{15/2}}-\frac {2 a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{11/2}}dx}{6 c}\right )}{7 c}\right )}{8 c}\) |
| \(\Big \downarrow \) 3217 |
| \(\displaystyle \frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{17/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{7 f (c-c \sin (e+f x))^{15/2}}-\frac {2 a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{6 f (c-c \sin (e+f x))^{13/2}}-\frac {a^2 \cos (e+f x)}{30 c f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}\right )}{7 c}\right )}{8 c}\) |
(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*f*(c - c*Sin[e + f*x])^(17/ 2)) - (3*a*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(7*f*(c - c*Sin[e + f*x])^(15/2)) - (2*a*((a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(6*f*(c - c*Sin[e + f*x])^(13/2)) - (a^2*Cos[e + f*x])/(30*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(11/2))))/(7*c)))/(8*c)
3.4.82.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f _.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(2*n + 1))), x] - Simp[b*((2*m - 1)/(d*( 2*n + 1))) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b ^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Time = 3.57 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(-\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{3} \left (3 \left (\cos ^{7}\left (f x +e \right )\right )+24 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-96 \left (\cos ^{5}\left (f x +e \right )\right )-240 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+480 \left (\cos ^{3}\left (f x +e \right )\right )+583 \sin \left (f x +e \right ) \cos \left (f x +e \right )-754 \cos \left (f x +e \right )-402 \tan \left (f x +e \right )+367 \sec \left (f x +e \right )\right )}{35 f \left (\left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )-7 \left (\cos ^{6}\left (f x +e \right )\right )-24 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+56 \left (\cos ^{4}\left (f x +e \right )\right )+80 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-112 \left (\cos ^{2}\left (f x +e \right )\right )-64 \sin \left (f x +e \right )+64\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{8}}\) | \(227\) |
-1/35/f*(a*(sin(f*x+e)+1))^(1/2)*a^3/(cos(f*x+e)^6*sin(f*x+e)-7*cos(f*x+e) ^6-24*cos(f*x+e)^4*sin(f*x+e)+56*cos(f*x+e)^4+80*sin(f*x+e)*cos(f*x+e)^2-1 12*cos(f*x+e)^2-64*sin(f*x+e)+64)/(-c*(sin(f*x+e)-1))^(1/2)/c^8*(3*cos(f*x +e)^7+24*cos(f*x+e)^5*sin(f*x+e)-96*cos(f*x+e)^5-240*cos(f*x+e)^3*sin(f*x+ e)+480*cos(f*x+e)^3+583*sin(f*x+e)*cos(f*x+e)-754*cos(f*x+e)-402*tan(f*x+e )+367*sec(f*x+e))
Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.15 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=-\frac {{\left (14 \, a^{3} \cos \left (f x + e\right )^{2} - 17 \, a^{3} + {\left (7 \, a^{3} \cos \left (f x + e\right )^{2} - 18 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{35 \, {\left (c^{9} f \cos \left (f x + e\right )^{9} - 32 \, c^{9} f \cos \left (f x + e\right )^{7} + 160 \, c^{9} f \cos \left (f x + e\right )^{5} - 256 \, c^{9} f \cos \left (f x + e\right )^{3} + 128 \, c^{9} f \cos \left (f x + e\right ) + 8 \, {\left (c^{9} f \cos \left (f x + e\right )^{7} - 10 \, c^{9} f \cos \left (f x + e\right )^{5} + 24 \, c^{9} f \cos \left (f x + e\right )^{3} - 16 \, c^{9} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
-1/35*(14*a^3*cos(f*x + e)^2 - 17*a^3 + (7*a^3*cos(f*x + e)^2 - 18*a^3)*si n(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^9*f*cos( f*x + e)^9 - 32*c^9*f*cos(f*x + e)^7 + 160*c^9*f*cos(f*x + e)^5 - 256*c^9* f*cos(f*x + e)^3 + 128*c^9*f*cos(f*x + e) + 8*(c^9*f*cos(f*x + e)^7 - 10*c ^9*f*cos(f*x + e)^5 + 24*c^9*f*cos(f*x + e)^3 - 16*c^9*f*cos(f*x + e))*sin (f*x + e))
Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {17}{2}}} \,d x } \]
Time = 0.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.87 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {{\left (56 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 140 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 120 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 35 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{8960 \, c^{\frac {17}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{16}} \]
1/8960*(56*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^6 - 140*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2* f*x + 1/2*e)^4 + 120*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 35*a^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/ (c^(17/2)*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/ 2*e)^16)
Time = 15.33 (sec) , antiderivative size = 673, normalized size of antiderivative = 3.78 \[ \int \frac {(3+3 \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Too large to display} \]
-((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)* ((a^3*exp(e*6i + f*x*6i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*64i)/(5*c^9*f) + (256*a^3*exp(e*7i + f*x*7i)*(a + a *((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(5*c^9* f) - (a^3*exp(e*8i + f*x*8i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e* 1i + f*x*1i)*1i)/2))^(1/2)*832i)/(7*c^9*f) - (1024*a^3*exp(e*9i + f*x*9i)* (a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/( 7*c^9*f) + (a^3*exp(e*10i + f*x*10i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*832i)/(7*c^9*f) + (256*a^3*exp(e*11i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)) ^(1/2))/(5*c^9*f) - (a^3*exp(e*12i + f*x*12i)*(a + a*((exp(- e*1i - f*x*1i )*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*64i)/(5*c^9*f)))/(exp(e*1i + f *x*1i)*16i - 119*exp(e*2i + f*x*2i) - exp(e*3i + f*x*3i)*544i + 1700*exp(e *4i + f*x*4i) + exp(e*5i + f*x*5i)*3808i - 6188*exp(e*6i + f*x*6i) - exp(e *7i + f*x*7i)*7072i + 4862*exp(e*8i + f*x*8i) + 4862*exp(e*10i + f*x*10i) + exp(e*11i + f*x*11i)*7072i - 6188*exp(e*12i + f*x*12i) - exp(e*13i + f*x *13i)*3808i + 1700*exp(e*14i + f*x*14i) + exp(e*15i + f*x*15i)*544i - 119* exp(e*16i + f*x*16i) - exp(e*17i + f*x*17i)*16i + exp(e*18i + f*x*18i) + 1 )