Integrand size = 26, antiderivative size = 166 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7} \] Output:
1/15*a^3*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^11+4/195*a^3*c^2*cos(f*x+e)^7 /f/(c-c*sin(f*x+e))^10+4/715*a^3*c*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^9+8/643 5*a^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^8+8/45045*a^3*cos(f*x+e)^7/c/f/(c-c* sin(f*x+e))^7
Time = 9.40 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.26 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a+a \sin (e+f x))^3 \left (115830 \cos \left (\frac {1}{2} (e+f x)\right )-65065 \cos \left (\frac {3}{2} (e+f x)\right )-18018 \cos \left (\frac {5}{2} (e+f x)\right )+1365 \cos \left (\frac {7}{2} (e+f x)\right )-105 \cos \left (\frac {11}{2} (e+f x)\right )+\cos \left (\frac {15}{2} (e+f x)\right )+109395 \sin \left (\frac {1}{2} (e+f x)\right )+60060 \sin \left (\frac {3}{2} (e+f x)\right )-15015 \sin \left (\frac {5}{2} (e+f x)\right )-455 \sin \left (\frac {9}{2} (e+f x)\right )+15 \sin \left (\frac {13}{2} (e+f x)\right )\right )}{360360 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^8} \] Input:
Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^8,x]
Output:
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a + a*Sin[e + f*x])^3*(115830*Cos[ (e + f*x)/2] - 65065*Cos[(3*(e + f*x))/2] - 18018*Cos[(5*(e + f*x))/2] + 1 365*Cos[(7*(e + f*x))/2] - 105*Cos[(11*(e + f*x))/2] + Cos[(15*(e + f*x))/ 2] + 109395*Sin[(e + f*x)/2] + 60060*Sin[(3*(e + f*x))/2] - 15015*Sin[(5*( e + f*x))/2] - 455*Sin[(9*(e + f*x))/2] + 15*Sin[(13*(e + f*x))/2]))/(3603 60*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^8)
Time = 0.86 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 3215, 3042, 3151, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^8}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{11}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{11}}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^3 c^3 \left (\frac {4 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{10}}dx}{15 c}+\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {4 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{10}}dx}{15 c}+\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^3 c^3 \left (\frac {4 \left (\frac {3 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^9}dx}{13 c}+\frac {\cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}\right )}{15 c}+\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {4 \left (\frac {3 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^9}dx}{13 c}+\frac {\cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}\right )}{15 c}+\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^3 c^3 \left (\frac {4 \left (\frac {3 \left (\frac {2 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^8}dx}{11 c}+\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}\right )}{13 c}+\frac {\cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}\right )}{15 c}+\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {4 \left (\frac {3 \left (\frac {2 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^8}dx}{11 c}+\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}\right )}{13 c}+\frac {\cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}\right )}{15 c}+\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^3 c^3 \left (\frac {4 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7}dx}{9 c}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )}{11 c}+\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}\right )}{13 c}+\frac {\cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}\right )}{15 c}+\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {4 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^7}dx}{9 c}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )}{11 c}+\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}\right )}{13 c}+\frac {\cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}\right )}{15 c}+\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {4 \left (\frac {\cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {3 \left (\frac {\cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {2 \left (\frac {\cos ^7(e+f x)}{63 c f (c-c \sin (e+f x))^7}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )}{11 c}\right )}{13 c}\right )}{15 c}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^8,x]
Output:
a^3*c^3*(Cos[e + f*x]^7/(15*f*(c - c*Sin[e + f*x])^11) + (4*(Cos[e + f*x]^ 7/(13*f*(c - c*Sin[e + f*x])^10) + (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)))/(13*c)))/(15*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 = 2.95 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(-\frac {16 \left (-a^{3}-115830 a^{3} {\mathrm e}^{8 i \left (f x +e \right )}+65065 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-1365 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+105 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}+455 i a^{3} {\mathrm e}^{3 i \left (f x +e \right )}-15 i a^{3} {\mathrm e}^{i \left (f x +e \right )}-109395 i a^{3} {\mathrm e}^{7 i \left (f x +e \right )}+15015 i a^{3} {\mathrm e}^{5 i \left (f x +e \right )}+60060 i a^{3} {\mathrm e}^{9 i \left (f x +e \right )}+18018 a^{3} {\mathrm e}^{10 i \left (f x +e \right )}\right )}{45045 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{15} f \,c^{8}}\) | \(174\) |
| parallelrisch | \(-\frac {2 a^{3} \left (\frac {4243}{45045}+\frac {741 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{5}-\frac {680 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{3}+\frac {2195 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{7}-\frac {2048 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{7}-\frac {404 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3}-\frac {1900 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{99}-\frac {1240 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3003}+\frac {2263 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{33}+\frac {2527 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{429}+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}-4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}+\frac {71 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{3}-60 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}+\frac {2203 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{9}\right )}{f \,c^{8} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{15}}\) | \(205\) |
| derivativedivides | \(\frac {2 a^{3} \left (-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {276}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {512}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{14}}-\frac {2304}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {13184}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {32288}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1024}{15 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{15}}-\frac {10}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {81344}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4536}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {24320}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {188}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {7352}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {47072}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {84112}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{8}}\) | \(238\) |
| default | \(\frac {2 a^{3} \left (-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {276}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {512}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{14}}-\frac {2304}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {13184}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {32288}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1024}{15 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{15}}-\frac {10}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {81344}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4536}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {24320}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {188}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {7352}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {47072}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {84112}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{8}}\) | \(238\) |
Input:
int((a+sin(f*x+e)*a)^3/(c-c*sin(f*x+e))^8,x,method=_RETURNVERBOSE)
Output:
-16/45045*(-a^3-115830*a^3*exp(8*I*(f*x+e))+65065*a^3*exp(6*I*(f*x+e))-136 5*a^3*exp(4*I*(f*x+e))+105*a^3*exp(2*I*(f*x+e))+455*I*a^3*exp(3*I*(f*x+e)) -15*I*a^3*exp(I*(f*x+e))-109395*I*a^3*exp(7*I*(f*x+e))+15015*I*a^3*exp(5*I *(f*x+e))+60060*I*a^3*exp(9*I*(f*x+e))+18018*a^3*exp(10*I*(f*x+e)))/(exp(I *(f*x+e))-I)^15/f/c^8
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (161) = 322\).
Time = 0.09 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.65 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx=\frac {8 \, a^{3} \cos \left (f x + e\right )^{8} + 64 \, a^{3} \cos \left (f x + e\right )^{7} - 196 \, a^{3} \cos \left (f x + e\right )^{6} - 672 \, a^{3} \cos \left (f x + e\right )^{5} + 735 \, a^{3} \cos \left (f x + e\right )^{4} - 7161 \, a^{3} \cos \left (f x + e\right )^{3} - 20328 \, a^{3} \cos \left (f x + e\right )^{2} + 12012 \, a^{3} \cos \left (f x + e\right ) + 24024 \, a^{3} - {\left (8 \, a^{3} \cos \left (f x + e\right )^{7} - 56 \, a^{3} \cos \left (f x + e\right )^{6} - 252 \, a^{3} \cos \left (f x + e\right )^{5} + 420 \, a^{3} \cos \left (f x + e\right )^{4} + 1155 \, a^{3} \cos \left (f x + e\right )^{3} + 8316 \, a^{3} \cos \left (f x + e\right )^{2} - 12012 \, a^{3} \cos \left (f x + e\right ) - 24024 \, a^{3}\right )} \sin \left (f x + e\right )}{45045 \, {\left (c^{8} f \cos \left (f x + e\right )^{8} - 7 \, c^{8} f \cos \left (f x + e\right )^{7} - 32 \, c^{8} f \cos \left (f x + e\right )^{6} + 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 160 \, c^{8} f \cos \left (f x + e\right )^{4} - 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 256 \, c^{8} f \cos \left (f x + e\right )^{2} + 64 \, c^{8} f \cos \left (f x + e\right ) + 128 \, c^{8} f + {\left (c^{8} f \cos \left (f x + e\right )^{7} + 8 \, c^{8} f \cos \left (f x + e\right )^{6} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} - 80 \, c^{8} f \cos \left (f x + e\right )^{4} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} + 192 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} f \cos \left (f x + e\right ) - 128 \, c^{8} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x, algorithm="fricas")
Output:
1/45045*(8*a^3*cos(f*x + e)^8 + 64*a^3*cos(f*x + e)^7 - 196*a^3*cos(f*x + e)^6 - 672*a^3*cos(f*x + e)^5 + 735*a^3*cos(f*x + e)^4 - 7161*a^3*cos(f*x + e)^3 - 20328*a^3*cos(f*x + e)^2 + 12012*a^3*cos(f*x + e) + 24024*a^3 - ( 8*a^3*cos(f*x + e)^7 - 56*a^3*cos(f*x + e)^6 - 252*a^3*cos(f*x + e)^5 + 42 0*a^3*cos(f*x + e)^4 + 1155*a^3*cos(f*x + e)^3 + 8316*a^3*cos(f*x + e)^2 - 12012*a^3*cos(f*x + e) - 24024*a^3)*sin(f*x + e))/(c^8*f*cos(f*x + e)^8 - 7*c^8*f*cos(f*x + e)^7 - 32*c^8*f*cos(f*x + e)^6 + 56*c^8*f*cos(f*x + e)^ 5 + 160*c^8*f*cos(f*x + e)^4 - 112*c^8*f*cos(f*x + e)^3 - 256*c^8*f*cos(f* x + e)^2 + 64*c^8*f*cos(f*x + e) + 128*c^8*f + (c^8*f*cos(f*x + e)^7 + 8*c ^8*f*cos(f*x + e)^6 - 24*c^8*f*cos(f*x + e)^5 - 80*c^8*f*cos(f*x + e)^4 + 80*c^8*f*cos(f*x + e)^3 + 192*c^8*f*cos(f*x + e)^2 - 64*c^8*f*cos(f*x + e) - 128*c^8*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 4542 vs. \(2 (151) = 302\).
Time = 117.21 (sec) , antiderivative size = 4542, normalized size of antiderivative = 27.36 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**8,x)
Output:
Piecewise((-90090*a**3*tan(e/2 + f*x/2)**14/(45045*c**8*f*tan(e/2 + f*x/2) **15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2 )**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f *x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e /2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*t an(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8 *f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c* *8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c** 8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 360360*a**3*tan(e/2 + f*x/2)**13/(4 5045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 47 29725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)* *10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f* x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e /2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan( e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 2132130 *a**3*tan(e/2 + f*x/2)**12/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c** 8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475* c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 13...
Leaf count of result is larger than twice the leaf count of optimal. 2422 vs. \(2 (161) = 322\).
Time = 0.13 (sec) , antiderivative size = 2422, normalized size of antiderivative = 14.59 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x, algorithm="maxima")
Output:
2/45045*(3*a^3*(17715*sin(f*x + e)/(cos(f*x + e) + 1) - 78960*sin(f*x + e) ^2/(cos(f*x + e) + 1)^2 + 342160*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 891 345*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1960959*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 3043040*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3912480*sin(f* x + e)^7/(cos(f*x + e) + 1)^7 - 3687255*sin(f*x + e)^8/(cos(f*x + e) + 1)^ 8 + 2867865*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1585584*sin(f*x + e)^10/ (cos(f*x + e) + 1)^10 + 720720*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 195 195*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 1181)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105* c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin( f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e )^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/ (cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 10 5*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos( f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15) - 7*a^3*(784 5*sin(f*x + e)/(cos(f*x + e) + 1) - 54915*sin(f*x + e)^2/(cos(f*x + e) + 1 )^2 + 222950*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 668850*sin(f*x + e)^...
Time = 0.20 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.50 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx=-\frac {2 \, {\left (45045 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 180180 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} + 1066065 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 2702700 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 6675669 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 10210200 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 14124825 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 13178880 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 11026015 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 6066060 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3088995 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 864500 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 265335 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 18600 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4243 \, a^{3}\right )}}{45045 \, c^{8} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{15}} \] Input:
integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x, algorithm="giac")
Output:
-2/45045*(45045*a^3*tan(1/2*f*x + 1/2*e)^14 - 180180*a^3*tan(1/2*f*x + 1/2 *e)^13 + 1066065*a^3*tan(1/2*f*x + 1/2*e)^12 - 2702700*a^3*tan(1/2*f*x + 1 /2*e)^11 + 6675669*a^3*tan(1/2*f*x + 1/2*e)^10 - 10210200*a^3*tan(1/2*f*x + 1/2*e)^9 + 14124825*a^3*tan(1/2*f*x + 1/2*e)^8 - 13178880*a^3*tan(1/2*f* x + 1/2*e)^7 + 11026015*a^3*tan(1/2*f*x + 1/2*e)^6 - 6066060*a^3*tan(1/2*f *x + 1/2*e)^5 + 3088995*a^3*tan(1/2*f*x + 1/2*e)^4 - 864500*a^3*tan(1/2*f* x + 1/2*e)^3 + 265335*a^3*tan(1/2*f*x + 1/2*e)^2 - 18600*a^3*tan(1/2*f*x + 1/2*e) + 4243*a^3)/(c^8*f*(tan(1/2*f*x + 1/2*e) - 1)^15)
Time = 21.83 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx=\frac {\sqrt {2}\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3497111\,\cos \left (3\,e+3\,f\,x\right )}{128}-\frac {25501905\,\sin \left (e+f\,x\right )}{128}-\frac {257861\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {5734111\,\cos \left (e+f\,x\right )}{128}+\frac {72047\,\cos \left (4\,e+4\,f\,x\right )}{4}-\frac {378579\,\cos \left (5\,e+5\,f\,x\right )}{128}-\frac {1059\,\cos \left (6\,e+6\,f\,x\right )}{2}+\frac {4251\,\cos \left (7\,e+7\,f\,x\right )}{128}+\frac {2633345\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {7210775\,\sin \left (3\,e+3\,f\,x\right )}{128}-\frac {89375\,\sin \left (4\,e+4\,f\,x\right )}{8}-\frac {504205\,\sin \left (5\,e+5\,f\,x\right )}{128}+\frac {29765\,\sin \left (6\,e+6\,f\,x\right )}{64}+\frac {4235\,\sin \left (7\,e+7\,f\,x\right )}{128}+\frac {544369}{4}\right )}{5765760\,c^8\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^{15}} \] Input:
int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^8,x)
Output:
(2^(1/2)*a^3*cos(e/2 + (f*x)/2)*((3497111*cos(3*e + 3*f*x))/128 - (2550190 5*sin(e + f*x))/128 - (257861*cos(2*e + 2*f*x))/2 - (5734111*cos(e + f*x)) /128 + (72047*cos(4*e + 4*f*x))/4 - (378579*cos(5*e + 5*f*x))/128 - (1059* cos(6*e + 6*f*x))/2 + (4251*cos(7*e + 7*f*x))/128 + (2633345*sin(2*e + 2*f *x))/64 + (7210775*sin(3*e + 3*f*x))/128 - (89375*sin(4*e + 4*f*x))/8 - (5 04205*sin(5*e + 5*f*x))/128 + (29765*sin(6*e + 6*f*x))/64 + (4235*sin(7*e + 7*f*x))/128 + 544369/4))/(5765760*c^8*f*cos(e/2 + pi/4 + (f*x)/2)^15)
Time = 0.17 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.45 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx=\frac {a^{3} \left (-1755 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{7}+12277 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6}-36795 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}+61225 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}-61030 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+45336 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-1240 \cos \left (f x +e \right ) \sin \left (f x +e \right )+6006 \cos \left (f x +e \right )-1771 \sin \left (f x +e \right )^{8}+14160 \sin \left (f x +e \right )^{7}-49528 \sin \left (f x +e \right )^{6}+98980 \sin \left (f x +e \right )^{5}-123605 \sin \left (f x +e \right )^{4}+89740 \sin \left (f x +e \right )^{3}-68778 \sin \left (f x +e \right )^{2}-1240 \sin \left (f x +e \right )-6006\right )}{45045 c^{8} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{7}-7 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6}+21 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}-35 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}+35 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-21 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+7 \cos \left (f x +e \right ) \sin \left (f x +e \right )-\cos \left (f x +e \right )+\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1\right )} \] Input:
int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x)
Output:
(a**3*( - 1755*cos(e + f*x)*sin(e + f*x)**7 + 12277*cos(e + f*x)*sin(e + f *x)**6 - 36795*cos(e + f*x)*sin(e + f*x)**5 + 61225*cos(e + f*x)*sin(e + f *x)**4 - 61030*cos(e + f*x)*sin(e + f*x)**3 + 45336*cos(e + f*x)*sin(e + f *x)**2 - 1240*cos(e + f*x)*sin(e + f*x) + 6006*cos(e + f*x) - 1771*sin(e + f*x)**8 + 14160*sin(e + f*x)**7 - 49528*sin(e + f*x)**6 + 98980*sin(e + f *x)**5 - 123605*sin(e + f*x)**4 + 89740*sin(e + f*x)**3 - 68778*sin(e + f* x)**2 - 1240*sin(e + f*x) - 6006))/(45045*c**8*f*(cos(e + f*x)*sin(e + f*x )**7 - 7*cos(e + f*x)*sin(e + f*x)**6 + 21*cos(e + f*x)*sin(e + f*x)**5 - 35*cos(e + f*x)*sin(e + f*x)**4 + 35*cos(e + f*x)*sin(e + f*x)**3 - 21*cos (e + f*x)*sin(e + f*x)**2 + 7*cos(e + f*x)*sin(e + f*x) - cos(e + f*x) + s in(e + f*x)**8 - 8*sin(e + f*x)**7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)* *5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x)**3 + 28*sin(e + f*x)**2 - 8*sin( e + f*x) + 1))