Integrand size = 30, antiderivative size = 178 \[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}} \] Output:
1/14*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(15/2)+1/56*cos( f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c-c*sin(f*x+e))^(13/2)+1/280*cos(f*x+e) *(a+a*sin(f*x+e))^(7/2)/c^2/f/(c-c*sin(f*x+e))^(11/2)+1/2240*cos(f*x+e)*(a +a*sin(f*x+e))^(7/2)/c^3/f/(c-c*sin(f*x+e))^(9/2)
Time = 13.76 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.87 \[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {8 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {6 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (a (1+\sin (e+f x)))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}} \] Input:
Integrate[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(15/2),x]
Output:
(8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + 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])^(15/2)) - ( 2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(f *(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + (6 *(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + 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])^(15/2)) - ( (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(4*f *(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2))
Time = 0.87 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3222, 3042, 3222, 3042, 3222, 3042, 3221}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{15/2}}dx\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {3 \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{13/2}}dx}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{13/2}}dx}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{11/2}}dx}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{11/2}}dx}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx}{10 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx}{10 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3221 |
| \(\displaystyle \frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {3 \left (\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}\right )}{14 c}\) |
Input:
Int[(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(15/2),x]
Output:
(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x])^(15/2 )) + (3*((Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(12*f*(c - c*Sin[e + f* x])^(13/2)) + ((Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*f*(c - c*Sin[ e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(80*c*f*(c - c*Sin[e + f*x])^(9/2)))/(6*c)))/(14*c)
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*( (c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && Ne Q[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*( (c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp[(m + n + 1)/(a*(2*m + 1) ) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; Free Q[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
Time = 1.95 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(\frac {\tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{7} \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6} \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}+4 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+20\right ) a^{3} \sqrt {4}}{4480 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{7}}\) | \(127\) |
Input:
int((a+sin(f*x+e)*a)^(7/2)/(c-c*sin(f*x+e))^(15/2),x,method=_RETURNVERBOSE )
Output:
1/4480/f*tan(1/4*Pi+1/2*f*x+1/2*e)^7*sec(1/4*Pi+1/2*f*x+1/2*e)^6*(a*sin(1/ 4*Pi+1/2*f*x+1/2*e)^2)^(1/2)*(cos(1/4*Pi+1/2*f*x+1/2*e)^6+4*cos(1/4*Pi+1/2 *f*x+1/2*e)^4+10*cos(1/4*Pi+1/2*f*x+1/2*e)^2+20)*a^3/(c*cos(1/4*Pi+1/2*f*x +1/2*e)^2)^(1/2)/c^7*4^(1/2)
Time = 0.11 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08 \[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {{\left (63 \, a^{3} \cos \left (f x + e\right )^{2} - 76 \, a^{3} + 7 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 12 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{140 \, {\left (7 \, c^{8} f \cos \left (f x + e\right )^{7} - 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right ) - {\left (c^{8} f \cos \left (f x + e\right )^{7} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, algorithm="fri cas")
Output:
1/140*(63*a^3*cos(f*x + e)^2 - 76*a^3 + 7*(5*a^3*cos(f*x + e)^2 - 12*a^3)* sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(7*c^8*f* cos(f*x + e)^7 - 56*c^8*f*cos(f*x + e)^5 + 112*c^8*f*cos(f*x + e)^3 - 64*c ^8*f*cos(f*x + e) - (c^8*f*cos(f*x + e)^7 - 24*c^8*f*cos(f*x + e)^5 + 80*c ^8*f*cos(f*x + e)^3 - 64*c^8*f*cos(f*x + e))*sin(f*x + e))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(15/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, algorithm="max ima")
Output:
integrate((a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(15/2), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, algorithm="gia c")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 22.33 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.63 \[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Too large to display} \] Input:
int((a + a*sin(e + f*x))^(7/2)/(c - c*sin(e + f*x))^(15/2),x)
Output:
((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)*144i)/(5*c^8*f) - (8*a^3*exp(e*5i + f*x*5i)*(a + a*( (exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(c^8*f) + (344*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^8*f) - (a^3*exp(e*8i + f*x*8i)*(a + a*((e xp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*2848i)/(35*c ^8*f) - (344*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))/(5*c^8*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)*144 i)/(5*c^8*f) + (8*a^3*exp(e*11i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1 i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(c^8*f)))/(exp(e*1i + f*x*1i)*14 i - 90*exp(e*2i + f*x*2i) - exp(e*3i + f*x*3i)*350i + 910*exp(e*4i + f*x*4 i) + exp(e*5i + f*x*5i)*1638i - 2002*exp(e*6i + f*x*6i) - exp(e*7i + f*x*7 i)*1430i - exp(e*9i + f*x*9i)*1430i + 2002*exp(e*10i + f*x*10i) + exp(e*11 i + f*x*11i)*1638i - 910*exp(e*12i + f*x*12i) - exp(e*13i + f*x*13i)*350i + 90*exp(e*14i + f*x*14i) + exp(e*15i + f*x*15i)*14i - exp(e*16i + f*x*16i ) + 1)
\[ \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1}d x +3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1}d x \right )+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1}d x \right )+\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1}d x \right )}{c^{8}} \] Input:
int((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x)
Output:
(sqrt(c)*sqrt(a)*a**3*(int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**8 - 8*sin(e + f*x)**7 + 28*sin(e + f*x) **6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x)**3 + 28*si n(e + f*x)**2 - 8*sin(e + f*x) + 1),x) + 3*int((sqrt(sin(e + f*x) + 1)*sqr t( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**8 - 8*sin(e + f*x)* *7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin (e + f*x)**3 + 28*sin(e + f*x)**2 - 8*sin(e + f*x) + 1),x) + 3*int((sqrt(s in(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**8 - 8*sin(e + f*x)**7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x)**3 + 28*sin(e + f*x)**2 - 8*sin(e + f*x) + 1),x ) + int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1))/(sin(e + f*x)** 8 - 8*sin(e + f*x)**7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x)**3 + 28*sin(e + f*x)**2 - 8*sin(e + f*x) + 1) ,x)))/c**8