Integrand size = 36, antiderivative size = 121 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\frac {(A+B) \sec ^5(e+f x)}{7 a^3 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {(6 A-B) \tan (e+f x)}{7 a^3 c^4 f}+\frac {2 (6 A-B) \tan ^3(e+f x)}{21 a^3 c^4 f}+\frac {(6 A-B) \tan ^5(e+f x)}{35 a^3 c^4 f} \] Output:
1/7*(A+B)*sec(f*x+e)^5/a^3/f/(c^4-c^4*sin(f*x+e))+1/7*(6*A-B)*tan(f*x+e)/a ^3/c^4/f+2/21*(6*A-B)*tan(f*x+e)^3/a^3/c^4/f+1/35*(6*A-B)*tan(f*x+e)^5/a^3 /c^4/f
Leaf count is larger than twice the leaf count of optimal. \(325\) vs. \(2(121)=242\).
Time = 7.38 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.69 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-8960 B+1500 (A+B) \cos (e+f x)-640 (6 A-B) \cos (2 (e+f x))+750 A \cos (3 (e+f x))+750 B \cos (3 (e+f x))-3072 A \cos (4 (e+f x))+512 B \cos (4 (e+f x))+150 A \cos (5 (e+f x))+150 B \cos (5 (e+f x))-768 A \cos (6 (e+f x))+128 B \cos (6 (e+f x))-15360 A \sin (e+f x)+2560 B \sin (e+f x)-375 A \sin (2 (e+f x))-375 B \sin (2 (e+f x))-7680 A \sin (3 (e+f x))+1280 B \sin (3 (e+f x))-300 A \sin (4 (e+f x))-300 B \sin (4 (e+f x))-1536 A \sin (5 (e+f x))+256 B \sin (5 (e+f x))-75 A \sin (6 (e+f x))-75 B \sin (6 (e+f x)))}{53760 a^3 c^4 f (-1+\sin (e+f x))^4 (1+\sin (e+f x))^3} \] Input:
Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x] )^4),x]
Output:
-1/53760*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-8960*B + 1500*(A + B)*Cos[e + f*x] - 640*(6*A - B)*Cos[2*(e + f*x)] + 750*A*Cos[3*(e + f*x)] + 750*B*Cos[3*(e + f*x)] - 3072*A*Cos[4*( e + f*x)] + 512*B*Cos[4*(e + f*x)] + 150*A*Cos[5*(e + f*x)] + 150*B*Cos[5* (e + f*x)] - 768*A*Cos[6*(e + f*x)] + 128*B*Cos[6*(e + f*x)] - 15360*A*Sin [e + f*x] + 2560*B*Sin[e + f*x] - 375*A*Sin[2*(e + f*x)] - 375*B*Sin[2*(e + f*x)] - 7680*A*Sin[3*(e + f*x)] + 1280*B*Sin[3*(e + f*x)] - 300*A*Sin[4* (e + f*x)] - 300*B*Sin[4*(e + f*x)] - 1536*A*Sin[5*(e + f*x)] + 256*B*Sin[ 5*(e + f*x)] - 75*A*Sin[6*(e + f*x)] - 75*B*Sin[6*(e + f*x)]))/(a^3*c^4*f* (-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^3)
Time = 0.55 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3042, 3446, 3042, 3338, 3042, 4254, 2009}
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.
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^4}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle \frac {\int \frac {\sec ^6(e+f x) (A+B \sin (e+f x))}{c-c \sin (e+f x)}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {A+B \sin (e+f x)}{\cos (e+f x)^6 (c-c \sin (e+f x))}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle \frac {\frac {(6 A-B) \int \sec ^6(e+f x)dx}{7 c}+\frac {(A+B) \sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(6 A-B) \int \csc \left (e+f x+\frac {\pi }{2}\right )^6dx}{7 c}+\frac {(A+B) \sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}}{a^3 c^3}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {(A+B) \sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}-\frac {(6 A-B) \int \left (\tan ^4(e+f x)+2 \tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{7 c f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(A+B) \sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}-\frac {(6 A-B) \left (-\frac {1}{5} \tan ^5(e+f x)-\frac {2}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{7 c f}}{a^3 c^3}\) |
Input:
Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4),x ]
Output:
(((A + B)*Sec[e + f*x]^5)/(7*f*(c - c*Sin[e + f*x])) - ((6*A - B)*(-Tan[e + f*x] - (2*Tan[e + f*x]^3)/3 - Tan[e + f*x]^5/5))/(7*c*f))/(a^3*c^3)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) )), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 ]) && NeQ[2*m + p + 1, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains complex when optimal does not.
Time = 2.21 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.54
| method | result | size |
| risch | \(-\frac {16 i \left (120 i A \,{\mathrm e}^{5 i \left (f x +e \right )}-20 i B \,{\mathrm e}^{5 i \left (f x +e \right )}+70 B \,{\mathrm e}^{6 i \left (f x +e \right )}+60 i A \,{\mathrm e}^{3 i \left (f x +e \right )}+30 A \,{\mathrm e}^{4 i \left (f x +e \right )}-10 i B \,{\mathrm e}^{3 i \left (f x +e \right )}-5 B \,{\mathrm e}^{4 i \left (f x +e \right )}+12 i A \,{\mathrm e}^{i \left (f x +e \right )}+24 A \,{\mathrm e}^{2 i \left (f x +e \right )}-2 i B \,{\mathrm e}^{i \left (f x +e \right )}-4 B \,{\mathrm e}^{2 i \left (f x +e \right )}+6 A -B \right )}{105 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,a^{3} c^{4}}\) | \(186\) |
| parallelrisch | \(\frac {-210 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}+\left (210 A -210 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}+\left (210 A +140 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+\left (-630 A -70 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\left (-756 A -224 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\left (1092 A -532 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (-156 A +376 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (-780 A -220 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-90 A -160 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (330 A -90 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-150 A +60 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-30 A -30 B}{105 f \,a^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(247\) |
| derivativedivides | \(\frac {-\frac {-\frac {A}{2}+\frac {3 B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-\frac {A}{2}+\frac {B}{2}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (\frac {A}{4}-\frac {B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (\frac {3 A}{4}-\frac {5 B}{8}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {11 A}{32}-\frac {5 B}{32}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (A +B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {3 A +3 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {\frac {11 A}{2}+\frac {9 B}{2}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {\frac {15 A}{8}+B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {21 A}{32}+\frac {5 B}{32}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {21 A}{4}+\frac {19 B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 \left (\frac {33 A}{8}+\frac {11 B}{4}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}}{f \,a^{3} c^{4}}\) | \(271\) |
| default | \(\frac {-\frac {-\frac {A}{2}+\frac {3 B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-\frac {A}{2}+\frac {B}{2}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (\frac {A}{4}-\frac {B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (\frac {3 A}{4}-\frac {5 B}{8}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {11 A}{32}-\frac {5 B}{32}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (A +B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {3 A +3 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {\frac {11 A}{2}+\frac {9 B}{2}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {\frac {15 A}{8}+B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {21 A}{32}+\frac {5 B}{32}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {21 A}{4}+\frac {19 B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 \left (\frac {33 A}{8}+\frac {11 B}{4}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}}{f \,a^{3} c^{4}}\) | \(271\) |
| norman | \(\frac {\frac {\left (2 A -2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{f c a}-\frac {2 A +2 B}{7 f c a}-\frac {152 \left (6 A -B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{105 f c a}-\frac {2 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{a f c}+\frac {4 B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{3 f c a}+\frac {\left (312 A -752 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{105 f c a}-\frac {2 \left (41 A -36 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{35 f c a}-\frac {2 \left (5 A -2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7 f c a}-\frac {\left (12 A +8 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{3 f c a}+\frac {\left (20 A -8 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{7 f c a}-\frac {\left (90 A +62 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{21 f c a}-\frac {2 \left (13 A +2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{5 f c a}-\frac {4 \left (12 A +5 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{21 f c a}+\frac {\left (66 A -86 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{15 f c a}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(430\) |
Input:
int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x,method=_RETUR NVERBOSE)
Output:
-16/105*I*(120*I*A*exp(5*I*(f*x+e))-20*I*B*exp(5*I*(f*x+e))+70*B*exp(6*I*( f*x+e))+60*I*A*exp(3*I*(f*x+e))+30*A*exp(4*I*(f*x+e))-10*I*B*exp(3*I*(f*x+ e))-5*B*exp(4*I*(f*x+e))+12*I*A*exp(I*(f*x+e))+24*A*exp(2*I*(f*x+e))-2*I*B *exp(I*(f*x+e))-4*B*exp(2*I*(f*x+e))+6*A-B)/(exp(I*(f*x+e))+I)^5/(exp(I*(f *x+e))-I)^7/f/a^3/c^4
Time = 0.10 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=-\frac {8 \, {\left (6 \, A - B\right )} \cos \left (f x + e\right )^{6} - 4 \, {\left (6 \, A - B\right )} \cos \left (f x + e\right )^{4} - {\left (6 \, A - B\right )} \cos \left (f x + e\right )^{2} + {\left (8 \, {\left (6 \, A - B\right )} \cos \left (f x + e\right )^{4} + 4 \, {\left (6 \, A - B\right )} \cos \left (f x + e\right )^{2} + 18 \, A - 3 \, B\right )} \sin \left (f x + e\right ) - 3 \, A + 18 \, B}{105 \, {\left (a^{3} c^{4} f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) - a^{3} c^{4} f \cos \left (f x + e\right )^{5}\right )}} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algori thm="fricas")
Output:
-1/105*(8*(6*A - B)*cos(f*x + e)^6 - 4*(6*A - B)*cos(f*x + e)^4 - (6*A - B )*cos(f*x + e)^2 + (8*(6*A - B)*cos(f*x + e)^4 + 4*(6*A - B)*cos(f*x + e)^ 2 + 18*A - 3*B)*sin(f*x + e) - 3*A + 18*B)/(a^3*c^4*f*cos(f*x + e)^5*sin(f *x + e) - a^3*c^4*f*cos(f*x + e)^5)
Leaf count of result is larger than twice the leaf count of optimal. 6135 vs. \(2 (109) = 218\).
Time = 35.50 (sec) , antiderivative size = 6135, normalized size of antiderivative = 50.70 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**4,x)
Output:
Piecewise((-210*A*tan(e/2 + f*x/2)**11/(105*a**3*c**4*f*tan(e/2 + f*x/2)** 12 - 210*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 420*a**3*c**4*f*tan(e/2 + f*x/ 2)**10 + 1050*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 525*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 2100*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 2100*a**3*c**4*f*tan(e/ 2 + f*x/2)**5 - 525*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 1050*a**3*c**4*f*tan (e/2 + f*x/2)**3 + 420*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 210*a**3*c**4*f*t an(e/2 + f*x/2) - 105*a**3*c**4*f) + 210*A*tan(e/2 + f*x/2)**10/(105*a**3* c**4*f*tan(e/2 + f*x/2)**12 - 210*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 420*a **3*c**4*f*tan(e/2 + f*x/2)**10 + 1050*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 5 25*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 2100*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 2100*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 525*a**3*c**4*f*tan(e/2 + f*x/2)* *4 - 1050*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 420*a**3*c**4*f*tan(e/2 + f*x/ 2)**2 + 210*a**3*c**4*f*tan(e/2 + f*x/2) - 105*a**3*c**4*f) + 210*A*tan(e/ 2 + f*x/2)**9/(105*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 210*a**3*c**4*f*tan( e/2 + f*x/2)**11 - 420*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 1050*a**3*c**4*f *tan(e/2 + f*x/2)**9 + 525*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 2100*a**3*c** 4*f*tan(e/2 + f*x/2)**7 + 2100*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 525*a**3* c**4*f*tan(e/2 + f*x/2)**4 - 1050*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 420*a* *3*c**4*f*tan(e/2 + f*x/2)**2 + 210*a**3*c**4*f*tan(e/2 + f*x/2) - 105*a** 3*c**4*f) - 630*A*tan(e/2 + f*x/2)**8/(105*a**3*c**4*f*tan(e/2 + f*x/2)...
Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (114) = 228\).
Time = 0.08 (sec) , antiderivative size = 1019, normalized size of antiderivative = 8.42 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algori thm="maxima")
Output:
-2/105*(B*(30*sin(f*x + e)/(cos(f*x + e) + 1) - 45*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 80*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 110*sin(f*x + e)^4 /(cos(f*x + e) + 1)^4 + 188*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 266*sin( f*x + e)^6/(cos(f*x + e) + 1)^6 - 112*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 35*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 70*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 105*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 15)/(a^3*c^4 - 2*a^3 *c^4*sin(f*x + e)/(cos(f*x + e) + 1) - 4*a^3*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*c^4*si n(f*x + e)^4/(cos(f*x + e) + 1)^4 - 20*a^3*c^4*sin(f*x + e)^5/(cos(f*x + e ) + 1)^5 + 20*a^3*c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 5*a^3*c^4*sin( f*x + e)^8/(cos(f*x + e) + 1)^8 - 10*a^3*c^4*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 4*a^3*c^4*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 2*a^3*c^4*sin(f *x + e)^11/(cos(f*x + e) + 1)^11 - a^3*c^4*sin(f*x + e)^12/(cos(f*x + e) + 1)^12) - 3*A*(25*sin(f*x + e)/(cos(f*x + e) + 1) - 55*sin(f*x + e)^2/(cos (f*x + e) + 1)^2 + 15*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 130*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 182*s in(f*x + e)^6/(cos(f*x + e) + 1)^6 + 126*sin(f*x + e)^7/(cos(f*x + e) + 1) ^7 + 105*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 35*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 35*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 35*sin(f*x + e)^1 1/(cos(f*x + e) + 1)^11 + 5)/(a^3*c^4 - 2*a^3*c^4*sin(f*x + e)/(cos(f*x...
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (114) = 228\).
Time = 0.30 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.75 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=-\frac {\frac {7 \, {\left (165 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 75 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 540 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 210 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 750 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 280 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 480 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 170 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 129 \, A - 49 \, B\right )}}{a^{3} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {2205 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 525 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 10080 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 1470 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 21945 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2555 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 26460 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2240 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 18963 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1407 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7476 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 434 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1383 \, A + 137 \, B}{a^{3} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{1680 \, f} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algori thm="giac")
Output:
-1/1680*(7*(165*A*tan(1/2*f*x + 1/2*e)^4 - 75*B*tan(1/2*f*x + 1/2*e)^4 + 5 40*A*tan(1/2*f*x + 1/2*e)^3 - 210*B*tan(1/2*f*x + 1/2*e)^3 + 750*A*tan(1/2 *f*x + 1/2*e)^2 - 280*B*tan(1/2*f*x + 1/2*e)^2 + 480*A*tan(1/2*f*x + 1/2*e ) - 170*B*tan(1/2*f*x + 1/2*e) + 129*A - 49*B)/(a^3*c^4*(tan(1/2*f*x + 1/2 *e) + 1)^5) + (2205*A*tan(1/2*f*x + 1/2*e)^6 + 525*B*tan(1/2*f*x + 1/2*e)^ 6 - 10080*A*tan(1/2*f*x + 1/2*e)^5 - 1470*B*tan(1/2*f*x + 1/2*e)^5 + 21945 *A*tan(1/2*f*x + 1/2*e)^4 + 2555*B*tan(1/2*f*x + 1/2*e)^4 - 26460*A*tan(1/ 2*f*x + 1/2*e)^3 - 2240*B*tan(1/2*f*x + 1/2*e)^3 + 18963*A*tan(1/2*f*x + 1 /2*e)^2 + 1407*B*tan(1/2*f*x + 1/2*e)^2 - 7476*A*tan(1/2*f*x + 1/2*e) - 43 4*B*tan(1/2*f*x + 1/2*e) + 1383*A + 137*B)/(a^3*c^4*(tan(1/2*f*x + 1/2*e) - 1)^7))/f
Time = 37.45 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.79 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\frac {\left (\frac {16\,A}{35}-\frac {8\,B}{105}-\frac {32\,A\,\sin \left (e+f\,x\right )}{35}+\frac {16\,B\,\sin \left (e+f\,x\right )}{105}\right )\,{\cos \left (e+f\,x\right )}^4+\left (\frac {4\,A}{35}-\frac {2\,B}{105}-\frac {16\,A\,\sin \left (e+f\,x\right )}{35}+\frac {8\,B\,\sin \left (e+f\,x\right )}{105}\right )\,{\cos \left (e+f\,x\right )}^2+\frac {2\,A}{35}-\frac {12\,B}{35}-\frac {12\,A\,\sin \left (e+f\,x\right )}{35}+\frac {2\,B\,\sin \left (e+f\,x\right )}{35}}{a^3\,c^4\,f\,\left (2\,{\cos \left (e+f\,x\right )}^5\,\sin \left (e+f\,x\right )-2\,{\cos \left (e+f\,x\right )}^5\right )}-\frac {\frac {2\,A}{7}+\frac {2\,B}{7}-\frac {2\,A\,\sin \left (e+f\,x\right )}{7}-\frac {2\,B\,\sin \left (e+f\,x\right )}{7}}{a^3\,c^4\,f\,\left (2\,\sin \left (e+f\,x\right )-2\right )}-\frac {\cos \left (e+f\,x\right )\,\left (\frac {32\,A}{35}-\frac {16\,B}{105}\right )}{a^3\,c^4\,f\,\left (2\,\sin \left (e+f\,x\right )-2\right )} \] Input:
int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^4),x )
Output:
((2*A)/35 - (12*B)/35 - (12*A*sin(e + f*x))/35 + (2*B*sin(e + f*x))/35 + c os(e + f*x)^2*((4*A)/35 - (2*B)/105 - (16*A*sin(e + f*x))/35 + (8*B*sin(e + f*x))/105) + cos(e + f*x)^4*((16*A)/35 - (8*B)/105 - (32*A*sin(e + f*x)) /35 + (16*B*sin(e + f*x))/105))/(a^3*c^4*f*(2*cos(e + f*x)^5*sin(e + f*x) - 2*cos(e + f*x)^5)) - ((2*A)/7 + (2*B)/7 - (2*A*sin(e + f*x))/7 - (2*B*si n(e + f*x))/7)/(a^3*c^4*f*(2*sin(e + f*x) - 2)) - (cos(e + f*x)*((32*A)/35 - (16*B)/105))/(a^3*c^4*f*(2*sin(e + f*x) - 2))
Time = 0.19 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.19 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx=\frac {-15 a -15 b +48 \sin \left (f x +e \right )^{6} a -15 \cos \left (f x +e \right ) b -8 \sin \left (f x +e \right )^{6} b -48 \sin \left (f x +e \right )^{5} a -90 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} a +15 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b +90 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a +30 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b -90 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a -90 a \sin \left (f x +e \right )-30 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -180 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a -15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +180 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a +20 \sin \left (f x +e \right )^{4} b +120 \sin \left (f x +e \right )^{3} a +8 \sin \left (f x +e \right )^{5} b -120 \sin \left (f x +e \right )^{4} a +90 \cos \left (f x +e \right ) a -20 \sin \left (f x +e \right )^{3} b +90 \sin \left (f x +e \right )^{2} a -15 \sin \left (f x +e \right )^{2} b +15 \sin \left (f x +e \right ) b}{105 \cos \left (f x +e \right ) a^{3} c^{4} f \left (\sin \left (f x +e \right )^{5}-\sin \left (f x +e \right )^{4}-2 \sin \left (f x +e \right )^{3}+2 \sin \left (f x +e \right )^{2}+\sin \left (f x +e \right )-1\right )} \] Input:
int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x)
Output:
( - 90*cos(e + f*x)*sin(e + f*x)**5*a + 15*cos(e + f*x)*sin(e + f*x)**5*b + 90*cos(e + f*x)*sin(e + f*x)**4*a - 15*cos(e + f*x)*sin(e + f*x)**4*b + 180*cos(e + f*x)*sin(e + f*x)**3*a - 30*cos(e + f*x)*sin(e + f*x)**3*b - 1 80*cos(e + f*x)*sin(e + f*x)**2*a + 30*cos(e + f*x)*sin(e + f*x)**2*b - 90 *cos(e + f*x)*sin(e + f*x)*a + 15*cos(e + f*x)*sin(e + f*x)*b + 90*cos(e + f*x)*a - 15*cos(e + f*x)*b + 48*sin(e + f*x)**6*a - 8*sin(e + f*x)**6*b - 48*sin(e + f*x)**5*a + 8*sin(e + f*x)**5*b - 120*sin(e + f*x)**4*a + 20*s in(e + f*x)**4*b + 120*sin(e + f*x)**3*a - 20*sin(e + f*x)**3*b + 90*sin(e + f*x)**2*a - 15*sin(e + f*x)**2*b - 90*sin(e + f*x)*a + 15*sin(e + f*x)* b - 15*a - 15*b)/(105*cos(e + f*x)*a**3*c**4*f*(sin(e + f*x)**5 - sin(e + f*x)**4 - 2*sin(e + f*x)**3 + 2*sin(e + f*x)**2 + sin(e + f*x) - 1))