Integrand size = 38, antiderivative size = 200 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {128 (7 A-9 B) c^4 \cos (e+f x)}{35 a f \sqrt {c-c \sin (e+f x)}}-\frac {32 (7 A-9 B) c^3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{35 a f}-\frac {12 (7 A-9 B) c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac {(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f} \]
-12/35*(7*A-9*B)*c^2*cos(f*x+e)*(c-c*sin(f*x+e))^(3/2)/a/f-1/7*(7*A-9*B)*c *cos(f*x+e)*(c-c*sin(f*x+e))^(5/2)/a/f-(A-B)*sec(f*x+e)*(c-c*sin(f*x+e))^( 9/2)/a/c/f-128/35*(7*A-9*B)*c^4*cos(f*x+e)/a/f/(c-c*sin(f*x+e))^(1/2)-32/3 5*(7*A-9*B)*c^3*cos(f*x+e)*(c-c*sin(f*x+e))^(1/2)/a/f
Time = 14.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.78 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (4900 A-6125 B+196 (A-2 B) \cos (2 (e+f x))+5 B \cos (4 (e+f x))+2450 A \sin (e+f x)-3430 B \sin (e+f x)-14 A \sin (3 (e+f x))+58 B \sin (3 (e+f x)))}{140 a f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))} \]
-1/140*(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]] *(4900*A - 6125*B + 196*(A - 2*B)*Cos[2*(e + f*x)] + 5*B*Cos[4*(e + f*x)] + 2450*A*Sin[e + f*x] - 3430*B*Sin[e + f*x] - 14*A*Sin[3*(e + f*x)] + 58*B *Sin[3*(e + f*x)]))/(a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))
Time = 0.94 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.91, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3042, 3446, 3042, 3334, 3042, 3126, 3042, 3126, 3042, 3126, 3042, 3125}
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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{a \sin (e+f x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{a \sin (e+f x)+a}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle \frac {\int \sec ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}dx}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2}}{\cos (e+f x)^2}dx}{a c}\) |
| \(\Big \downarrow \) 3334 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \int (c-c \sin (e+f x))^{7/2}dx-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \int (c-c \sin (e+f x))^{7/2}dx-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \left (\frac {12}{7} c \int (c-c \sin (e+f x))^{5/2}dx+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 f}\right )-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \left (\frac {12}{7} c \int (c-c \sin (e+f x))^{5/2}dx+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 f}\right )-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \left (\frac {12}{7} c \left (\frac {8}{5} c \int (c-c \sin (e+f x))^{3/2}dx+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 f}\right )-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \left (\frac {12}{7} c \left (\frac {8}{5} c \int (c-c \sin (e+f x))^{3/2}dx+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 f}\right )-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \left (\frac {12}{7} c \left (\frac {8}{5} c \left (\frac {4}{3} c \int \sqrt {c-c \sin (e+f x)}dx+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 f}\right )-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \left (\frac {12}{7} c \left (\frac {8}{5} c \left (\frac {4}{3} c \int \sqrt {c-c \sin (e+f x)}dx+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 f}\right )-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {-\frac {1}{2} c (7 A-9 B) \left (\frac {12}{7} c \left (\frac {8}{5} c \left (\frac {8 c^2 \cos (e+f x)}{3 f \sqrt {c-c \sin (e+f x)}}+\frac {2 c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{3 f}\right )+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{5 f}\right )+\frac {2 c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 f}\right )-\frac {(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{f}}{a c}\) |
(-(((A - B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/f) - ((7*A - 9*B)*c*( (2*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(7*f) + (12*c*((2*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(5*f) + (8*c*((8*c^2*Cos[e + f*x])/(3*f*S qrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3* f)))/5))/7))/2)/(a*c)
3.2.8.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b* c + a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(p + 1))) , x] + Simp[b*((a*d*m + b*c*(m + p + 1))/(a*g^2*(p + 1))) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 3.19 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.56
| method | result | size |
| default | \(\frac {2 c^{4} \left (\sin \left (f x +e \right )-1\right ) \left (5 B \left (\cos ^{4}\left (f x +e \right )\right )+\left (-7 A +29 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (49 A -103 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (308 A -436 B \right ) \sin \left (f x +e \right )+588 A -716 B \right )}{35 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(111\) |
2/35*c^4/a*(sin(f*x+e)-1)*(5*B*cos(f*x+e)^4+(-7*A+29*B)*cos(f*x+e)^2*sin(f *x+e)+(49*A-103*B)*cos(f*x+e)^2+(308*A-436*B)*sin(f*x+e)+588*A-716*B)/cos( f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.58 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {2 \, {\left (5 \, B c^{3} \cos \left (f x + e\right )^{4} + {\left (49 \, A - 103 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (147 \, A - 179 \, B\right )} c^{3} - {\left ({\left (7 \, A - 29 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 4 \, {\left (77 \, A - 109 \, B\right )} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{35 \, a f \cos \left (f x + e\right )} \]
-2/35*(5*B*c^3*cos(f*x + e)^4 + (49*A - 103*B)*c^3*cos(f*x + e)^2 + 4*(147 *A - 179*B)*c^3 - ((7*A - 29*B)*c^3*cos(f*x + e)^2 - 4*(77*A - 109*B)*c^3) *sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(f*x + e))
Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (182) = 364\).
Time = 0.34 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.39 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (91 \, c^{\frac {7}{2}} + \frac {86 \, c^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {336 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {266 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {490 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {266 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {336 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {86 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {91 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} A}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} - \frac {2 \, {\left (407 \, c^{\frac {7}{2}} + \frac {407 \, c^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {1442 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {1337 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {2030 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {1337 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {1442 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {407 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {407 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} B}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}}\right )}}{35 \, f} \]
2/35*(7*(91*c^(7/2) + 86*c^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 336*c^( 7/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 266*c^(7/2)*sin(f*x + e)^3/(cos (f*x + e) + 1)^3 + 490*c^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 266*c ^(7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 336*c^(7/2)*sin(f*x + e)^6/(c os(f*x + e) + 1)^6 + 86*c^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 91*c ^(7/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8)*A/((a + a*sin(f*x + e)/(cos(f* x + e) + 1))*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(7/2)) - 2*(407*c^( 7/2) + 407*c^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 1442*c^(7/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1337*c^(7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 2030*c^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1337*c^(7/2)*sin (f*x + e)^5/(cos(f*x + e) + 1)^5 + 1442*c^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 407*c^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 407*c^(7/2)* sin(f*x + e)^8/(cos(f*x + e) + 1)^8)*B/((a + a*sin(f*x + e)/(cos(f*x + e) + 1))*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(7/2)))/f
Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (182) = 364\).
Time = 0.51 (sec) , antiderivative size = 777, normalized size of antiderivative = 3.88 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \]
16/35*sqrt(2)*sqrt(c)*(35*(A*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - B*c ^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))/(a*((cos(-1/4*pi + 1/2*f*x + 1/2*e ) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 1)) - (77*A*c^3*sgn(sin(-1/4 *pi + 1/2*f*x + 1/2*e)) - 109*B*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 504*A*c^3*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 728*B*c^3*(cos(-1/4*pi + 1 /2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/ 2*f*x + 1/2*e) + 1) + 1337*A*c^3*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2*sg n(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 2009*B*c^3*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2*sgn(sin(-1/4*pi + 1/2*f *x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 - 1680*A*c^3*(cos(-1/4 *pi + 1/2*f*x + 1/2*e) - 1)^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/ 4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 2800*B*c^3*(cos(-1/4*pi + 1/2*f*x + 1/2*e ) - 1)^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2* e) + 1)^3 + 1015*A*c^3*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4*sgn(sin(-1/4 *pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4 - 1015*B*c^ 3*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e ))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4 - 280*A*c^3*(cos(-1/4*pi + 1/2*f *x + 1/2*e) - 1)^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2* f*x + 1/2*e) + 1)^5 + 280*B*c^3*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^5*...
Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \]