Integrand size = 28, antiderivative size = 132 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {256 c^3 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{5 a f}+\frac {64 c^2 \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{5 a f}+\frac {8 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{5 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 a f} \] Output:
-256/5*c^3*sec(f*x+e)*(c-c*sin(f*x+e))^(1/2)/a/f+64/5*c^2*sec(f*x+e)*(c-c* sin(f*x+e))^(3/2)/a/f+8/5*c*sec(f*x+e)*(c-c*sin(f*x+e))^(5/2)/a/f+2/5*sec( f*x+e)*(c-c*sin(f*x+e))^(7/2)/a/f
Time = 9.93 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\frac {c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (-350-14 \cos (2 (e+f x))-175 \sin (e+f x)+\sin (3 (e+f x)))}{10 a f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))} \] Input:
Integrate[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x]),x]
Output:
(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-350 - 14*Cos[2*(e + f*x)] - 175*Sin[e + f*x] + Sin[3*(e + f*x)]))/(10*a*f*(Cos [(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))
Time = 0.82 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{7/2}}{a \sin (e+f x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{7/2}}{a \sin (e+f x)+a}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \sec ^2(e+f x) (c-c \sin (e+f x))^{9/2}dx}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c-c \sin (e+f x))^{9/2}}{\cos (e+f x)^2}dx}{a c}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {12}{5} c \int \sec ^2(e+f x) (c-c \sin (e+f x))^{7/2}dx+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 f}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {12}{5} c \int \frac {(c-c \sin (e+f x))^{7/2}}{\cos (e+f x)^2}dx+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 f}}{a c}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {12}{5} c \left (\frac {8}{3} c \int \sec ^2(e+f x) (c-c \sin (e+f x))^{5/2}dx+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f}\right )+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 f}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {12}{5} c \left (\frac {8}{3} c \int \frac {(c-c \sin (e+f x))^{5/2}}{\cos (e+f x)^2}dx+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f}\right )+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 f}}{a c}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {12}{5} c \left (\frac {8}{3} c \left (4 c \int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2}dx+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{f}\right )+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f}\right )+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 f}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {12}{5} c \left (\frac {8}{3} c \left (4 c \int \frac {(c-c \sin (e+f x))^{3/2}}{\cos (e+f x)^2}dx+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{f}\right )+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f}\right )+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 f}}{a c}\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle \frac {\frac {12}{5} c \left (\frac {8}{3} c \left (\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{f}-\frac {8 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{f}\right )+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f}\right )+\frac {2 c \sec (e+f x) (c-c \sin (e+f x))^{7/2}}{5 f}}{a c}\) |
Input:
Int[(c - c*Sin[e + f*x])^(7/2)/(a + a*Sin[e + f*x]),x]
Output:
((2*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(5*f) + (12*c*((2*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*f) + (8*c*((-8*c^2*Sec[e + f*x]*Sqrt[ c - c*Sin[e + f*x]])/f + (2*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/f)) /3))/5)/(a*c)
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 = 1.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.52
| method | result | size |
| default | \(\frac {2 c^{4} \left (-1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )^{3}-7 \sin \left (f x +e \right )^{2}+43 \sin \left (f x +e \right )+91\right )}{5 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(69\) |
Input:
int((c-c*sin(f*x+e))^(7/2)/(a+sin(f*x+e)*a),x,method=_RETURNVERBOSE)
Output:
2/5*c^4/a*(-1+sin(f*x+e))*(sin(f*x+e)^3-7*sin(f*x+e)^2+43*sin(f*x+e)+91)/c os(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=-\frac {2 \, {\left (7 \, c^{3} \cos \left (f x + e\right )^{2} + 84 \, c^{3} - {\left (c^{3} \cos \left (f x + e\right )^{2} - 44 \, c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{5 \, a f \cos \left (f x + e\right )} \] Input:
integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x, algorithm="fricas")
Output:
-2/5*(7*c^3*cos(f*x + e)^2 + 84*c^3 - (c^3*cos(f*x + e)^2 - 44*c^3)*sin(f* x + e))*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(f*x + e))
Timed out. \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\text {Timed out} \] Input:
integrate((c-c*sin(f*x+e))**(7/2)/(a+a*sin(f*x+e)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (116) = 232\).
Time = 0.12 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.80 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\frac {2 \, {\left (91 \, c^{\frac {7}{2}} + \frac {86 \, c^{\frac {7}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {336 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {266 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {490 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {266 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {336 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {86 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {91 \, c^{\frac {7}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )}}{5 \, {\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} \] Input:
integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x, algorithm="maxima")
Output:
2/5*(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/(cos(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*sin(f*x + e)/(cos(f*x + e) + 1))*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. \(2 (116) = 232\).
Time = 0.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.46 \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\frac {16 \, \sqrt {2} {\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 {\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 {\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 )} \sqrt {c}}{5 \, f} \] Input:
integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x, algorithm="giac")
Output:
16/5*sqrt(2)*(5*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)) - (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*((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))*sqrt(c)/f
Timed out. \[ \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \] Input:
int((c - c*sin(e + f*x))^(7/2)/(a + a*sin(e + f*x)),x)
Output:
int((c - c*sin(e + f*x))^(7/2)/(a + a*sin(e + f*x)), x)
\[ \int \frac {(c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx=\frac {\sqrt {c}\, c^{3} \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )+1}d x -\left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )+1}d x \right )+3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )+1}d x \right )-3 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )+1}d x \right )\right )}{a} \] Input:
int((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x)
Output:
(sqrt(c)*c**3*(int(sqrt( - sin(e + f*x) + 1)/(sin(e + f*x) + 1),x) - int(( sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x) + 1),x) + 3*int(( sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x) + 1),x) - 3*int(( sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x) + 1),x)))/a