Integrand size = 28, antiderivative size = 134 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx=-\frac {256 c \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^3 f}+\frac {64 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{a^3 f}-\frac {24 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c f}+\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{a^3 c^2 f} \] Output:
-256/5*c*sec(f*x+e)^5*(c-c*sin(f*x+e))^(5/2)/a^3/f+64*sec(f*x+e)^5*(c-c*si n(f*x+e))^(7/2)/a^3/f-24*sec(f*x+e)^5*(c-c*sin(f*x+e))^(9/2)/a^3/c/f+2*sec (f*x+e)^5*(c-c*sin(f*x+e))^(11/2)/a^3/c^2/f
Time = 4.83 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.85 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx=\frac {c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (-182+90 \cos (2 (e+f x))-235 \sin (e+f x)+5 \sin (3 (e+f x)))}{10 a^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3} \] Input:
Integrate[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x])^3,x]
Output:
(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-182 + 90*Cos[2*(e + f*x)] - 235*Sin[e + f*x] + 5*Sin[3*(e + f*x)]))/(10*a^3*f* (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)
Time = 0.78 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 3215, 3042, 3153, 3042, 3153, 3042, 3153, 3042, 3152}
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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{7/2}}{(a \sin (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{7/2}}{(a \sin (e+f x)+a)^3}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \sec ^6(e+f x) (c-c \sin (e+f x))^{13/2}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c-c \sin (e+f x))^{13/2}}{\cos (e+f x)^6}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {12 c \int \sec ^6(e+f x) (c-c \sin (e+f x))^{11/2}dx+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {12 c \int \frac {(c-c \sin (e+f x))^{11/2}}{\cos (e+f x)^6}dx+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {12 c \left (-8 c \int \sec ^6(e+f x) (c-c \sin (e+f x))^{9/2}dx-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {12 c \left (-8 c \int \frac {(c-c \sin (e+f x))^{9/2}}{\cos (e+f x)^6}dx-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {12 c \left (-8 c \left (-\frac {4}{3} c \int \sec ^6(e+f x) (c-c \sin (e+f x))^{7/2}dx-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 f}\right )-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {12 c \left (-8 c \left (-\frac {4}{3} c \int \frac {(c-c \sin (e+f x))^{7/2}}{\cos (e+f x)^6}dx-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 f}\right )-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle \frac {12 c \left (-8 c \left (\frac {8 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 f}-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 f}\right )-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}}{a^3 c^3}\) |
Input:
Int[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x])^3,x]
Output:
((2*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(11/2))/f + 12*c*((-2*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/f - 8*c*((8*c^2*Sec[e + f*x]^5*(c - c*S in[e + f*x])^(5/2))/(15*f) - (2*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2 ))/(3*f))))/(a^3*c^3)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 133.70 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.60
| method | result | size |
| default | \(\frac {2 c^{4} \left (-1+\sin \left (f x +e \right )\right ) \left (5 \sin \left (f x +e \right )^{3}+45 \sin \left (f x +e \right )^{2}+55 \sin \left (f x +e \right )+23\right )}{5 a^{3} \left (1+\sin \left (f x +e \right )\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(81\) |
Input:
int((c-c*sin(f*x+e))^(7/2)/(a+sin(f*x+e)*a)^3,x,method=_RETURNVERBOSE)
Output:
2/5*c^4/a^3*(-1+sin(f*x+e))/(1+sin(f*x+e))^2*(5*sin(f*x+e)^3+45*sin(f*x+e) ^2+55*sin(f*x+e)+23)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.79 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (45 \, c^{3} \cos \left (f x + e\right )^{2} - 68 \, c^{3} + 5 \, {\left (c^{3} \cos \left (f x + e\right )^{2} - 12 \, c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{5 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} - 2 \, a^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} f \cos \left (f x + e\right )\right )}} \] Input:
integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-2/5*(45*c^3*cos(f*x + e)^2 - 68*c^3 + 5*(c^3*cos(f*x + e)^2 - 12*c^3)*sin (f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a^3*f*cos(f*x + e)^3 - 2*a^3*f*cos(f *x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))
Timed out. \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:
integrate((c-c*sin(f*x+e))**(7/2)/(a+a*sin(f*x+e))**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (124) = 248\).
Time = 0.13 (sec) , antiderivative size = 426, normalized size of antiderivative = 3.18 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx=\frac {2 \, {\left (23 \, c^{\frac {7}{2}} + \frac {110 \, c^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {318 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {590 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {1065 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {1220 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {1540 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {1220 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {1065 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {590 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {318 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {110 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} + \frac {23 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}}\right )}}{5 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} \] Input:
integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^3,x, algorithm="maxima")
Output:
2/5*(23*c^(7/2) + 110*c^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 318*c^(7/2 )*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 590*c^(7/2)*sin(f*x + e)^3/(cos(f* x + e) + 1)^3 + 1065*c^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1220*c^ (7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1540*c^(7/2)*sin(f*x + e)^6/(c os(f*x + e) + 1)^6 + 1220*c^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 10 65*c^(7/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 590*c^(7/2)*sin(f*x + e)^ 9/(cos(f*x + e) + 1)^9 + 318*c^(7/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 110*c^(7/2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 23*c^(7/2)*sin(f*x + e)^12/(cos(f*x + e) + 1)^12)/((a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(c os(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f *x + e)^5/(cos(f*x + e) + 1)^5)*f*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1 )^(7/2))
Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (124) = 248\).
Time = 0.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.43 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx=-\frac {4 \, \sqrt {2} \sqrt {c} {\left (\frac {5 \, c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{a^{3} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )}} - \frac {11 \, c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + \frac {50 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {80 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {30 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {5 \, c^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}}{a^{3} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}^{5}}\right )}}{5 \, f} \] Input:
integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^3,x, algorithm="giac")
Output:
-4/5*sqrt(2)*sqrt(c)*(5*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(a^3*((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) ) - (11*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 50*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) + 80*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 + 30*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 + 5*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)/(a^3*((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)^5))/f
Time = 24.13 (sec) , antiderivative size = 542, normalized size of antiderivative = 4.04 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:
int((c - c*sin(e + f*x))^(7/2)/(a + a*sin(e + f*x))^3,x)
Output:
(c^3*exp(e*1i + f*x*1i)*(c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*32i)/(a^3*f*(exp(e*1i + f*x*1i) - 1i)*(exp(e*1i + f* x*1i) + 1i)^2) - (24*c^3*exp(e*1i + f*x*1i)*(c - c*((exp(- e*1i - f*x*1i)* 1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(a^3*f*(exp(e*1i + f*x*1i) - 1i )*(exp(e*1i + f*x*1i) + 1i)) - ((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp (e*1i + f*x*1i)*1i)/2))^(1/2)*((2*c^3)/(a^3*f) - (c^3*exp(e*1i + f*x*1i)*2 i)/(a^3*f)))/(exp(e*1i + f*x*1i) - 1i) + (288*c^3*exp(e*1i + f*x*1i)*(c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(5*a^3 *f*(exp(e*1i + f*x*1i) - 1i)*(exp(e*1i + f*x*1i) + 1i)^3) - (c^3*exp(e*1i + f*x*1i)*(c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2) )^(1/2)*256i)/(5*a^3*f*(exp(e*1i + f*x*1i) - 1i)*(exp(e*1i + f*x*1i) + 1i) ^4) - (128*c^3*exp(e*1i + f*x*1i)*(c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (e xp(e*1i + f*x*1i)*1i)/2))^(1/2))/(5*a^3*f*(exp(e*1i + f*x*1i) - 1i)*(exp(e *1i + f*x*1i) + 1i)^5)
\[ \int \frac {(c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^3} \, dx=\frac {\sqrt {c}\, c^{3} \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x -\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right )+3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right )-3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3}+3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+1}d x \right )\right )}{a^{3}} \] Input:
int((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^3,x)
Output:
(sqrt(c)*c**3*(int(sqrt( - sin(e + f*x) + 1)/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x) - int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x) + 3 *int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**3 + 3*sin( e + f*x)**2 + 3*sin(e + f*x) + 1),x) - 3*int((sqrt( - sin(e + f*x) + 1)*si n(e + f*x))/(sin(e + f*x)**3 + 3*sin(e + f*x)**2 + 3*sin(e + f*x) + 1),x)) )/a**3