Integrand size = 26, antiderivative size = 132 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx=\frac {a^2 c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 c \cos ^5(e+f x)}{33 f (c-c \sin (e+f x))^7}+\frac {2 a^2 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^6}+\frac {2 a^2 \cos ^5(e+f x)}{1155 c f (c-c \sin (e+f x))^5} \] Output:
1/11*a^2*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^8+1/33*a^2*c*cos(f*x+e)^5/f/( c-c*sin(f*x+e))^7+2/231*a^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^6+2/1155*a^2*c os(f*x+e)^5/c/f/(c-c*sin(f*x+e))^5
Time = 4.83 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.01 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx=\frac {a^2 \left (2079 \cos \left (\frac {1}{2} (e+f x)\right )-825 \cos \left (\frac {3}{2} (e+f x)\right )-55 \cos \left (\frac {7}{2} (e+f x)\right )+\cos \left (\frac {11}{2} (e+f x)\right )+2541 \sin \left (\frac {1}{2} (e+f x)\right )+1155 \sin \left (\frac {3}{2} (e+f x)\right )-165 \sin \left (\frac {5}{2} (e+f x)\right )+11 \sin \left (\frac {9}{2} (e+f x)\right )\right )}{9240 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}} \] Input:
Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^6,x]
Output:
(a^2*(2079*Cos[(e + f*x)/2] - 825*Cos[(3*(e + f*x))/2] - 55*Cos[(7*(e + f* x))/2] + Cos[(11*(e + f*x))/2] + 2541*Sin[(e + f*x)/2] + 1155*Sin[(3*(e + f*x))/2] - 165*Sin[(5*(e + f*x))/2] + 11*Sin[(9*(e + f*x))/2]))/(9240*c^6* f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11)
Time = 0.70 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3215, 3042, 3151, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2}{(c-c \sin (e+f x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2}{(c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^2 c^2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^8}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^8}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {3 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^7}dx}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {3 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^7}dx}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {3 \left (\frac {2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {3 \left (\frac {2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {3 \left (\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 \left (\frac {\cos ^5(e+f x)}{35 c f (c-c \sin (e+f x))^5}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}\right )}{11 c}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^6,x]
Output:
a^2*c^2*(Cos[e + f*x]^5/(11*f*(c - c*Sin[e + f*x])^8) + (3*(Cos[e + f*x]^5 /(9*f*(c - c*Sin[e + f*x])^7) + (2*(Cos[e + f*x]^5/(7*f*(c - c*Sin[e + f*x ])^6) + Cos[e + f*x]^5/(35*c*f*(c - c*Sin[e + f*x])^5)))/(9*c)))/(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 = 1.48 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(\frac {4 i a^{2} \left (2079 i {\mathrm e}^{6 i \left (f x +e \right )}+1155 \,{\mathrm e}^{7 i \left (f x +e \right )}-825 i {\mathrm e}^{4 i \left (f x +e \right )}-2541 \,{\mathrm e}^{5 i \left (f x +e \right )}-55 i {\mathrm e}^{2 i \left (f x +e \right )}+165 \,{\mathrm e}^{3 i \left (f x +e \right )}+i-11 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{1155 f \,c^{6} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11}}\) | \(110\) |
| parallelrisch | \(-\frac {2 a^{2} \left (\frac {152}{1155}+\frac {164 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{7}+\frac {89 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{21}+12 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-20 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-\frac {68 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{7}-\frac {47 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{105}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+\frac {162 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{5}-\frac {142 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}\right )}{f \,c^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) | \(153\) |
| derivativedivides | \(\frac {2 a^{2} \left (-\frac {128}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {292}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {932}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {7}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {30}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {88}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {512}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {288}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {2376}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\right )}{f \,c^{6}}\) | \(178\) |
| default | \(\frac {2 a^{2} \left (-\frac {128}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {292}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {932}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {7}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {30}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {88}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {512}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {288}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {2376}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\right )}{f \,c^{6}}\) | \(178\) |
| norman | \(\frac {\frac {52 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{f c}-\frac {304 a^{2}}{1155 c f}-\frac {2 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{f c}+\frac {6 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{f c}-\frac {28 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{f c}+\frac {94 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{105 c f}-\frac {574 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{5 f c}+\frac {714 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{5 f c}-\frac {17534 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{105 f c}+\frac {2228 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{105 c f}-\frac {3466 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{385 c f}-\frac {3524 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{55 c f}+\frac {6056 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{35 f c}-\frac {7016 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{35 f c}+\frac {10138 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{105 c f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{2} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) | \(351\) |
Input:
int((a+sin(f*x+e)*a)^2/(c-c*sin(f*x+e))^6,x,method=_RETURNVERBOSE)
Output:
4/1155*I*a^2*(2079*I*exp(6*I*(f*x+e))+1155*exp(7*I*(f*x+e))-825*I*exp(4*I* (f*x+e))-2541*exp(5*I*(f*x+e))-55*I*exp(2*I*(f*x+e))+165*exp(3*I*(f*x+e))+ I-11*exp(I*(f*x+e)))/f/c^6/(exp(I*(f*x+e))-I)^11
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (128) = 256\).
Time = 0.09 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.52 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx=-\frac {2 \, a^{2} \cos \left (f x + e\right )^{6} + 12 \, a^{2} \cos \left (f x + e\right )^{5} - 25 \, a^{2} \cos \left (f x + e\right )^{4} - 70 \, a^{2} \cos \left (f x + e\right )^{3} - 245 \, a^{2} \cos \left (f x + e\right )^{2} + 210 \, a^{2} \cos \left (f x + e\right ) + 420 \, a^{2} - {\left (2 \, a^{2} \cos \left (f x + e\right )^{5} - 10 \, a^{2} \cos \left (f x + e\right )^{4} - 35 \, a^{2} \cos \left (f x + e\right )^{3} + 35 \, a^{2} \cos \left (f x + e\right )^{2} - 210 \, a^{2} \cos \left (f x + e\right ) - 420 \, a^{2}\right )} \sin \left (f x + e\right )}{1155 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^6,x, algorithm="fricas")
Output:
-1/1155*(2*a^2*cos(f*x + e)^6 + 12*a^2*cos(f*x + e)^5 - 25*a^2*cos(f*x + e )^4 - 70*a^2*cos(f*x + e)^3 - 245*a^2*cos(f*x + e)^2 + 210*a^2*cos(f*x + e ) + 420*a^2 - (2*a^2*cos(f*x + e)^5 - 10*a^2*cos(f*x + e)^4 - 35*a^2*cos(f *x + e)^3 + 35*a^2*cos(f*x + e)^2 - 210*a^2*cos(f*x + e) - 420*a^2)*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. \(2 (117) = 234\).
Time = 29.20 (sec) , antiderivative size = 2509, normalized size of antiderivative = 19.01 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**6,x)
Output:
Piecewise((-2310*a**2*tan(e/2 + f*x/2)**10/(1155*c**6*f*tan(e/2 + f*x/2)** 11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*tan(e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 38 1150*c**6*f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan(e/2 + f*x/2)**3 - 6352 5*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f ) + 6930*a**2*tan(e/2 + f*x/2)**9/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 1270 5*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2 + f*x/2)**9 - 190575* c**6*f*tan(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c* *6*f*tan(e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6 *f*tan(e/2 + f*x/2)**4 + 190575*c**6*f*tan(e/2 + f*x/2)**3 - 63525*c**6*f* tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) - 27720 *a**2*tan(e/2 + f*x/2)**8/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f *tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2 + f*x/2)**9 - 190575*c**6*f*t an(e/2 + f*x/2)**8 + 381150*c**6*f*tan(e/2 + f*x/2)**7 - 533610*c**6*f*tan (e/2 + f*x/2)**6 + 533610*c**6*f*tan(e/2 + f*x/2)**5 - 381150*c**6*f*tan(e /2 + f*x/2)**4 + 190575*c**6*f*tan(e/2 + f*x/2)**3 - 63525*c**6*f*tan(e/2 + f*x/2)**2 + 12705*c**6*f*tan(e/2 + f*x/2) - 1155*c**6*f) + 46200*a**2*ta n(e/2 + f*x/2)**7/(1155*c**6*f*tan(e/2 + f*x/2)**11 - 12705*c**6*f*tan(e/2 + f*x/2)**10 + 63525*c**6*f*tan(e/2 + f*x/2)**9 - 190575*c**6*f*tan(e/...
Leaf count of result is larger than twice the leaf count of optimal. 1332 vs. \(2 (128) = 256\).
Time = 0.07 (sec) , antiderivative size = 1332, normalized size of antiderivative = 10.09 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^6,x, algorithm="maxima")
Output:
-2/3465*(5*a^2*(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) - 6*a ^2*(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 = 0.16 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.40 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (1155 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 3465 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 13860 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 23100 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 37422 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 32802 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 27060 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 11220 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4895 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 517 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 152 \, a^{2}\right )}}{1155 \, c^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}} \] Input:
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^6,x, algorithm="giac")
Output:
-2/1155*(1155*a^2*tan(1/2*f*x + 1/2*e)^10 - 3465*a^2*tan(1/2*f*x + 1/2*e)^ 9 + 13860*a^2*tan(1/2*f*x + 1/2*e)^8 - 23100*a^2*tan(1/2*f*x + 1/2*e)^7 + 37422*a^2*tan(1/2*f*x + 1/2*e)^6 - 32802*a^2*tan(1/2*f*x + 1/2*e)^5 + 2706 0*a^2*tan(1/2*f*x + 1/2*e)^4 - 11220*a^2*tan(1/2*f*x + 1/2*e)^3 + 4895*a^2 *tan(1/2*f*x + 1/2*e)^2 - 517*a^2*tan(1/2*f*x + 1/2*e) + 152*a^2)/(c^6*f*( tan(1/2*f*x + 1/2*e) - 1)^11)
Time = 18.88 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx=-\frac {\sqrt {2}\,a^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (697\,\cos \left (e+f\,x\right )+\frac {7623\,\sin \left (e+f\,x\right )}{4}+\frac {3977\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {3203\,\cos \left (3\,e+3\,f\,x\right )}{16}-\frac {461\,\cos \left (4\,e+4\,f\,x\right )}{8}+\frac {75\,\cos \left (5\,e+5\,f\,x\right )}{16}-462\,\sin \left (2\,e+2\,f\,x\right )-\frac {4983\,\sin \left (3\,e+3\,f\,x\right )}{16}+\frac {187\,\sin \left (4\,e+4\,f\,x\right )}{4}+\frac {77\,\sin \left (5\,e+5\,f\,x\right )}{16}-\frac {12721}{8}\right )}{36960\,c^6\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^{11}} \] Input:
int((a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^6,x)
Output:
-(2^(1/2)*a^2*cos(e/2 + (f*x)/2)*(697*cos(e + f*x) + (7623*sin(e + f*x))/4 + (3977*cos(2*e + 2*f*x))/4 - (3203*cos(3*e + 3*f*x))/16 - (461*cos(4*e + 4*f*x))/8 + (75*cos(5*e + 5*f*x))/16 - 462*sin(2*e + 2*f*x) - (4983*sin(3 *e + 3*f*x))/16 + (187*sin(4*e + 4*f*x))/4 + (77*sin(5*e + 5*f*x))/16 - 12 721/8))/(36960*c^6*f*cos(e/2 + pi/4 + (f*x)/2)^11)
Time = 0.19 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.29 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^6} \, dx=\frac {a^{2} \left (-60 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}+302 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}-611 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+626 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-47 \cos \left (f x +e \right ) \sin \left (f x +e \right )+210 \cos \left (f x +e \right )-56 \sin \left (f x +e \right )^{6}+338 \sin \left (f x +e \right )^{5}-851 \sin \left (f x +e \right )^{4}+1145 \sin \left (f x +e \right )^{3}-1159 \sin \left (f x +e \right )^{2}-47 \sin \left (f x +e \right )-210\right )}{1155 c^{6} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}-5 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}+10 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-10 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+5 \cos \left (f x +e \right ) \sin \left (f x +e \right )-\cos \left (f x +e \right )+\sin \left (f x +e \right )^{6}-6 \sin \left (f x +e \right )^{5}+15 \sin \left (f x +e \right )^{4}-20 \sin \left (f x +e \right )^{3}+15 \sin \left (f x +e \right )^{2}-6 \sin \left (f x +e \right )+1\right )} \] Input:
int((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^6,x)
Output:
(a**2*( - 60*cos(e + f*x)*sin(e + f*x)**5 + 302*cos(e + f*x)*sin(e + f*x)* *4 - 611*cos(e + f*x)*sin(e + f*x)**3 + 626*cos(e + f*x)*sin(e + f*x)**2 - 47*cos(e + f*x)*sin(e + f*x) + 210*cos(e + f*x) - 56*sin(e + f*x)**6 + 33 8*sin(e + f*x)**5 - 851*sin(e + f*x)**4 + 1145*sin(e + f*x)**3 - 1159*sin( e + f*x)**2 - 47*sin(e + f*x) - 210))/(1155*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))