Integrand size = 28, antiderivative size = 176 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {4096 c^3 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{15 a^2 f}-\frac {1024 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{5 a^2 f}+\frac {128 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{5 a^2 f}+\frac {32 \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{15 a^2 f}+\frac {2 \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 a^2 c f} \] Output:
4096/15*c^3*sec(f*x+e)^3*(c-c*sin(f*x+e))^(3/2)/a^2/f-1024/5*c^2*sec(f*x+e )^3*(c-c*sin(f*x+e))^(5/2)/a^2/f+128/5*c*sec(f*x+e)^3*(c-c*sin(f*x+e))^(7/ 2)/a^2/f+32/15*sec(f*x+e)^3*(c-c*sin(f*x+e))^(9/2)/a^2/f+2/5*sec(f*x+e)^3* (c-c*sin(f*x+e))^(11/2)/a^2/c/f
Time = 12.39 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.70 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {c^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (6825-1044 \cos (2 (e+f x))+3 \cos (4 (e+f x))+8568 \sin (e+f x)+56 \sin (3 (e+f x)))}{60 a^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2} \] Input:
Integrate[(c - c*Sin[e + f*x])^(9/2)/(a + a*Sin[e + f*x])^2,x]
Output:
(c^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(6825 - 1044*Cos[2*(e + f*x)] + 3*Cos[4*(e + f*x)] + 8568*Sin[e + f*x] + 56*Sin[ 3*(e + f*x)]))/(60*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2)
Time = 0.98 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3215, 3042, 3153, 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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{9/2}}{(a \sin (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{9/2}}{(a \sin (e+f x)+a)^2}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \sec ^4(e+f x) (c-c \sin (e+f x))^{13/2}dx}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c-c \sin (e+f x))^{13/2}}{\cos (e+f x)^4}dx}{a^2 c^2}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {16}{5} c \int \sec ^4(e+f x) (c-c \sin (e+f x))^{11/2}dx+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16}{5} c \int \frac {(c-c \sin (e+f x))^{11/2}}{\cos (e+f x)^4}dx+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {16}{5} c \left (4 c \int \sec ^4(e+f x) (c-c \sin (e+f x))^{9/2}dx+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16}{5} c \left (4 c \int \frac {(c-c \sin (e+f x))^{9/2}}{\cos (e+f x)^4}dx+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {16}{5} c \left (4 c \left (8 c \int \sec ^4(e+f x) (c-c \sin (e+f x))^{7/2}dx+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16}{5} c \left (4 c \left (8 c \int \frac {(c-c \sin (e+f x))^{7/2}}{\cos (e+f x)^4}dx+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {16}{5} c \left (4 c \left (8 c \left (-4 c \int \sec ^4(e+f x) (c-c \sin (e+f x))^{5/2}dx-\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16}{5} c \left (4 c \left (8 c \left (-4 c \int \frac {(c-c \sin (e+f x))^{5/2}}{\cos (e+f x)^4}dx-\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle \frac {\frac {16}{5} c \left (4 c \left (8 c \left (\frac {8 c^2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 f}-\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{5/2}}{f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{7/2}}{f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{9/2}}{3 f}\right )+\frac {2 c \sec ^3(e+f x) (c-c \sin (e+f x))^{11/2}}{5 f}}{a^2 c^2}\) |
Input:
Int[(c - c*Sin[e + f*x])^(9/2)/(a + a*Sin[e + f*x])^2,x]
Output:
((2*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(11/2))/(5*f) + (16*c*((2*c*Sec[ e + f*x]^3*(c - c*Sin[e + f*x])^(9/2))/(3*f) + 4*c*((2*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/f + 8*c*((8*c^2*Sec[e + f*x]^3*(c - c*Sin[e + f* x])^(3/2))/(3*f) - (2*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/f))))/5 )/(a^2*c^2)
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 = 20.75 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.52
| method | result | size |
| default | \(-\frac {2 c^{5} \left (-1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{4}-28 \sin \left (f x +e \right )^{3}+258 \sin \left (f x +e \right )^{2}+1092 \sin \left (f x +e \right )+723\right )}{15 a^{2} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(91\) |
Input:
int((c-c*sin(f*x+e))^(9/2)/(a+sin(f*x+e)*a)^2,x,method=_RETURNVERBOSE)
Output:
-2/15*c^5/a^2*(-1+sin(f*x+e))/(1+sin(f*x+e))*(3*sin(f*x+e)^4-28*sin(f*x+e) ^3+258*sin(f*x+e)^2+1092*sin(f*x+e)+723)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2) /f
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.59 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {2 \, {\left (3 \, c^{4} \cos \left (f x + e\right )^{4} - 264 \, c^{4} \cos \left (f x + e\right )^{2} + 984 \, c^{4} + 28 \, {\left (c^{4} \cos \left (f x + e\right )^{2} + 38 \, c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15 \, {\left (a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} f \cos \left (f x + e\right )\right )}} \] Input:
integrate((c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
Output:
2/15*(3*c^4*cos(f*x + e)^4 - 264*c^4*cos(f*x + e)^2 + 984*c^4 + 28*(c^4*co s(f*x + e)^2 + 38*c^4)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a^2*f*cos( f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))
Timed out. \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\text {Timed out} \] Input:
integrate((c-c*sin(f*x+e))**(9/2)/(a+a*sin(f*x+e))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (156) = 312\).
Time = 0.13 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.16 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (723 \, c^{\frac {9}{2}} + \frac {2184 \, c^{\frac {9}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5370 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10696 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15021 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {21168 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {20748 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {21168 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {15021 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {10696 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {5370 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {2184 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} + \frac {723 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}}\right )}}{15 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {9}{2}}} \] Input:
integrate((c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
Output:
-2/15*(723*c^(9/2) + 2184*c^(9/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 5370*c ^(9/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10696*c^(9/2)*sin(f*x + e)^3/ (cos(f*x + e) + 1)^3 + 15021*c^(9/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 21168*c^(9/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 20748*c^(9/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 21168*c^(9/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 15021*c^(9/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 10696*c^(9/2) *sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 5370*c^(9/2)*sin(f*x + e)^10/(cos(f *x + e) + 1)^10 + 2184*c^(9/2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 723 *c^(9/2)*sin(f*x + e)^12/(cos(f*x + e) + 1)^12)/((a^2 + 3*a^2*sin(f*x + e) /(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin( f*x + e)^3/(cos(f*x + e) + 1)^3)*f*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(9/2))
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (156) = 312\).
Time = 0.22 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.42 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx =\text {Too large to display} \] Input:
integrate((c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x, algorithm="giac")
Output:
-16/15*sqrt(2)*sqrt(c)*(5*(11*c^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 24 *c^4*(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) + 9*c^4*(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)/(a^2*((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)^3) - (73*c^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e) ) - 320*c^4*(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) + 490*c^4*(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 - 240*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3*s gn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 45*c^4*(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^2*((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
Timed out. \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \] Input:
int((c - c*sin(e + f*x))^(9/2)/(a + a*sin(e + f*x))^2,x)
Output:
int((c - c*sin(e + f*x))^(9/2)/(a + a*sin(e + f*x))^2, x)
\[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^2} \, dx=\frac {\sqrt {c}\, c^{4} \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x +\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x -4 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x \right )+6 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x \right )-4 \left (\int \frac {\sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x \right )\right )}{a^{2}} \] Input:
int((c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x)
Output:
(sqrt(c)*c**4*(int(sqrt( - sin(e + f*x) + 1)/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1),x) + int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**4)/(sin(e + f* x)**2 + 2*sin(e + f*x) + 1),x) - 4*int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1),x) + 6*int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1),x) - 4 *int((sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1),x)))/a**2