Integrand size = 26, antiderivative size = 101 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7} \] Output:
1/11*a^3*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^9+2/99*a^3*c^2*cos(f*x+e)^7/f /(c-c*sin(f*x+e))^8+2/693*a^3*c*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^7
Time = 6.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.44 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx=-\frac {a^3 \left (-2541 \cos \left (\frac {1}{2} (e+f x)\right )+1485 \cos \left (\frac {3}{2} (e+f x)\right )+462 \cos \left (\frac {5}{2} (e+f x)\right )-55 \cos \left (\frac {7}{2} (e+f x)\right )+\cos \left (\frac {11}{2} (e+f x)\right )-2079 \sin \left (\frac {1}{2} (e+f x)\right )-1155 \sin \left (\frac {3}{2} (e+f x)\right )+297 \sin \left (\frac {5}{2} (e+f x)\right )+11 \sin \left (\frac {9}{2} (e+f x)\right )\right )}{5544 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}} \] Input:
Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6,x]
Output:
-1/5544*(a^3*(-2541*Cos[(e + f*x)/2] + 1485*Cos[(3*(e + f*x))/2] + 462*Cos [(5*(e + f*x))/2] - 55*Cos[(7*(e + f*x))/2] + Cos[(11*(e + f*x))/2] - 2079 *Sin[(e + f*x)/2] - 1155*Sin[(3*(e + f*x))/2] + 297*Sin[(5*(e + f*x))/2] + 11*Sin[(9*(e + f*x))/2]))/(c^6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11 )
Time = 0.55 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 3215, 3042, 3151, 3042, 3151, 3042, 3150}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^9}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^9}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^8}dx}{11 c}+\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^8}dx}{11 c}+\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \left (\frac {\int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7}dx}{9 c}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )}{11 c}+\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {2 \left (\frac {\int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^7}dx}{9 c}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )}{11 c}+\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {2 \left (\frac {\cos ^7(e+f x)}{63 c f (c-c \sin (e+f x))^7}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )}{11 c}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6,x]
Output:
a^3*c^3*(Cos[e + f*x]^7/(11*f*(c - c*Sin[e + f*x])^9) + (2*(Cos[e + f*x]^7 /(9*f*(c - c*Sin[e + f*x])^8) + Cos[e + f*x]^7/(63*c*f*(c - c*Sin[e + f*x] )^7)))/(11*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/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
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/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) 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] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 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]))
Result contains complex when optimal does not.
Time = 2.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(\frac {\frac {60 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}}{7}-12 i a^{3} {\mathrm e}^{5 i \left (f x +e \right )}-\frac {20 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{63}+\frac {12 i a^{3} {\mathrm e}^{3 i \left (f x +e \right )}}{7}+\frac {4 i a^{3} {\mathrm e}^{i \left (f x +e \right )}}{63}-\frac {44 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}}{3}+\frac {4 a^{3}}{693}+\frac {20 i a^{3} {\mathrm e}^{7 i \left (f x +e \right )}}{3}+\frac {8 a^{3} {\mathrm e}^{8 i \left (f x +e \right )}}{3}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11} f \,c^{6}}\) | \(143\) |
| parallelrisch | \(-\frac {2 a^{3} \left (693 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-1386 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+8085 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-10626 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+21252 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-15246 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+15444 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-4950 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+2959 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-176 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+79\right )}{693 f \,c^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) | \(155\) |
| derivativedivides | \(\frac {2 a^{3} \left (-\frac {1480}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {116}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {126}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {256}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4272}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {544}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {292}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {3008}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{6}}\) | \(178\) |
| default | \(\frac {2 a^{3} \left (-\frac {1480}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {116}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {126}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {256}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4272}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {544}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {292}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {3008}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{6}}\) | \(178\) |
| norman | \(\frac {-\frac {158 a^{3}}{693 c f}-\frac {2 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{16}}{f c}+\frac {4 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{f c}-\frac {88 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{3 f c}+\frac {32 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{63 c f}+\frac {128 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{3 f c}+\frac {148 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{f c}+\frac {332 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{21 c f}-\frac {412 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{3 f c}+\frac {1696 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{7 f c}+\frac {12980 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{63 c f}+\frac {1856 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{21 c f}-\frac {2104 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{7 f c}-\frac {6392 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{693 c f}-\frac {22024 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{63 c f}-\frac {16372 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{231 c f}-\frac {153080 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{693 c f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) | \(395\) |
Input:
int((a+sin(f*x+e)*a)^3/(c-c*sin(f*x+e))^6,x,method=_RETURNVERBOSE)
Output:
4/693*(1485*a^3*exp(4*I*(f*x+e))-2079*I*a^3*exp(5*I*(f*x+e))-55*a^3*exp(2* I*(f*x+e))+297*I*a^3*exp(3*I*(f*x+e))+11*I*a^3*exp(I*(f*x+e))-2541*a^3*exp (6*I*(f*x+e))+a^3+1155*I*a^3*exp(7*I*(f*x+e))+462*a^3*exp(8*I*(f*x+e)))/(e xp(I*(f*x+e))-I)^11/f/c^6
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (98) = 196\).
Time = 0.08 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.29 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx=\frac {2 \, a^{3} \cos \left (f x + e\right )^{6} + 12 \, a^{3} \cos \left (f x + e\right )^{5} - 25 \, a^{3} \cos \left (f x + e\right )^{4} + 161 \, a^{3} \cos \left (f x + e\right )^{3} + 448 \, a^{3} \cos \left (f x + e\right )^{2} - 252 \, a^{3} \cos \left (f x + e\right ) - 504 \, a^{3} - {\left (2 \, a^{3} \cos \left (f x + e\right )^{5} - 10 \, a^{3} \cos \left (f x + e\right )^{4} - 35 \, a^{3} \cos \left (f x + e\right )^{3} - 196 \, a^{3} \cos \left (f x + e\right )^{2} + 252 \, a^{3} \cos \left (f x + e\right ) + 504 \, a^{3}\right )} \sin \left (f x + e\right )}{693 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="fricas")
Output:
1/693*(2*a^3*cos(f*x + e)^6 + 12*a^3*cos(f*x + e)^5 - 25*a^3*cos(f*x + e)^ 4 + 161*a^3*cos(f*x + e)^3 + 448*a^3*cos(f*x + e)^2 - 252*a^3*cos(f*x + e) - 504*a^3 - (2*a^3*cos(f*x + e)^5 - 10*a^3*cos(f*x + e)^4 - 35*a^3*cos(f* x + e)^3 - 196*a^3*cos(f*x + e)^2 + 252*a^3*cos(f*x + e) + 504*a^3)*sin(f* x + e))/(c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*cos( f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 - 12* c^6*f*cos(f*x + e)^3 - 32*c^6*f*cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 3 2*c^6*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 2509 vs. \(2 (92) = 184\).
Time = 46.28 (sec) , antiderivative size = 2509, normalized size of antiderivative = 24.84 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**6,x)
Output:
Piecewise((-1386*a**3*tan(e/2 + f*x/2)**10/(693*c**6*f*tan(e/2 + f*x/2)**1 1 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 32 0166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 2286 90*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115* c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 2772*a**3*tan(e/2 + f*x/2)**9/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6 *f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f *tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*t an(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan (e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/ 2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 16170*a**3*ta n(e/2 + f*x/2)**8/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f *x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x /2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2 )**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)** 2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 21252*a**3*tan(e/2 + f*x/ 2)**7/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 ...
Leaf count of result is larger than twice the leaf count of optimal. 1734 vs. \(2 (98) = 196\).
Time = 0.09 (sec) , antiderivative size = 1734, normalized size of antiderivative = 17.17 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="maxima")
Output:
-2/3465*(5*a^3*(913*sin(f*x + e)/(cos(f*x + e) + 1) - 4565*sin(f*x + e)^2/ (cos(f*x + e) + 1)^2 + 12540*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 25080*s in(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33726*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 33726*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 23100*sin(f*x + e)^7/(c os(f*x + e) + 1)^7 - 11550*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 3465*sin( f*x + e)^9/(cos(f*x + e) + 1)^9 - 693*sin(f*x + e)^10/(cos(f*x + e) + 1)^1 0 - 146)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos( f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*si n(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^1 0/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 9*a ^3*(671*sin(f*x + e)/(cos(f*x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e ) + 1)^2 + 6600*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 10890*sin(f*x + e)^4 /(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 12936* sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1155*sin(f*x + e)^9/(cos (f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55* c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f...
Time = 0.18 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.83 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (693 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 1386 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 8085 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 10626 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 21252 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 15246 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15444 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4950 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2959 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 176 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 79 \, a^{3}\right )}}{693 \, c^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="giac")
Output:
-2/693*(693*a^3*tan(1/2*f*x + 1/2*e)^10 - 1386*a^3*tan(1/2*f*x + 1/2*e)^9 + 8085*a^3*tan(1/2*f*x + 1/2*e)^8 - 10626*a^3*tan(1/2*f*x + 1/2*e)^7 + 212 52*a^3*tan(1/2*f*x + 1/2*e)^6 - 15246*a^3*tan(1/2*f*x + 1/2*e)^5 + 15444*a ^3*tan(1/2*f*x + 1/2*e)^4 - 4950*a^3*tan(1/2*f*x + 1/2*e)^3 + 2959*a^3*tan (1/2*f*x + 1/2*e)^2 - 176*a^3*tan(1/2*f*x + 1/2*e) + 79*a^3)/(c^6*f*(tan(1 /2*f*x + 1/2*e) - 1)^11)
Time = 19.38 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.42 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx=-\frac {\sqrt {2}\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {6635\,\cos \left (e+f\,x\right )}{16}+\frac {13629\,\sin \left (e+f\,x\right )}{16}+565\,\cos \left (2\,e+2\,f\,x\right )-\frac {3527\,\cos \left (3\,e+3\,f\,x\right )}{32}-29\,\cos \left (4\,e+4\,f\,x\right )+\frac {81\,\cos \left (5\,e+5\,f\,x\right )}{32}-\frac {1617\,\sin \left (2\,e+2\,f\,x\right )}{8}-\frac {5049\,\sin \left (3\,e+3\,f\,x\right )}{32}+\frac {407\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {77\,\sin \left (5\,e+5\,f\,x\right )}{32}-922\right )}{22176\,c^6\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^{11}} \] Input:
int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^6,x)
Output:
-(2^(1/2)*a^3*cos(e/2 + (f*x)/2)*((6635*cos(e + f*x))/16 + (13629*sin(e + f*x))/16 + 565*cos(2*e + 2*f*x) - (3527*cos(3*e + 3*f*x))/32 - 29*cos(4*e + 4*f*x) + (81*cos(5*e + 5*f*x))/32 - (1617*sin(2*e + 2*f*x))/8 - (5049*si n(3*e + 3*f*x))/32 + (407*sin(4*e + 4*f*x))/16 + (77*sin(5*e + 5*f*x))/32 - 922))/(22176*c^6*f*cos(e/2 + pi/4 + (f*x)/2)^11)
Time = 0.19 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.99 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx=\frac {a^{3} \left (-45 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}+223 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}-439 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+655 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-16 \cos \left (f x +e \right ) \sin \left (f x +e \right )+126 \cos \left (f x +e \right )-49 \sin \left (f x +e \right )^{6}+292 \sin \left (f x +e \right )^{5}-724 \sin \left (f x +e \right )^{4}+724 \sin \left (f x +e \right )^{3}-1109 \sin \left (f x +e \right )^{2}-16 \sin \left (f x +e \right )-126\right )}{693 c^{6} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}-5 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}+10 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-10 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+5 \cos \left (f x +e \right ) \sin \left (f x +e \right )-\cos \left (f x +e \right )+\sin \left (f x +e \right )^{6}-6 \sin \left (f x +e \right )^{5}+15 \sin \left (f x +e \right )^{4}-20 \sin \left (f x +e \right )^{3}+15 \sin \left (f x +e \right )^{2}-6 \sin \left (f x +e \right )+1\right )} \] Input:
int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x)
Output:
(a**3*( - 45*cos(e + f*x)*sin(e + f*x)**5 + 223*cos(e + f*x)*sin(e + f*x)* *4 - 439*cos(e + f*x)*sin(e + f*x)**3 + 655*cos(e + f*x)*sin(e + f*x)**2 - 16*cos(e + f*x)*sin(e + f*x) + 126*cos(e + f*x) - 49*sin(e + f*x)**6 + 29 2*sin(e + f*x)**5 - 724*sin(e + f*x)**4 + 724*sin(e + f*x)**3 - 1109*sin(e + f*x)**2 - 16*sin(e + f*x) - 126))/(693*c**6*f*(cos(e + f*x)*sin(e + f*x )**5 - 5*cos(e + f*x)*sin(e + f*x)**4 + 10*cos(e + f*x)*sin(e + f*x)**3 - 10*cos(e + f*x)*sin(e + f*x)**2 + 5*cos(e + f*x)*sin(e + f*x) - cos(e + f* x) + sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 1))