Integrand size = 36, antiderivative size = 151 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 B x}{c^4}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {2 a^3 B c \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^5}+\frac {2 a^3 B c^2 \cos ^3(e+f x)}{3 f \left (c^2-c^2 \sin (e+f x)\right )^3}-\frac {2 a^3 B \cos (e+f x)}{f \left (c^4-c^4 \sin (e+f x)\right )} \] Output:
a^3*B*x/c^4+1/7*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^7-2/5*a^3*B* c*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^5+2/3*a^3*B*c^2*cos(f*x+e)^3/f/(c^2-c^2* sin(f*x+e))^3-2*a^3*B*cos(f*x+e)/f/(c^4-c^4*sin(f*x+e))
Leaf count is larger than twice the leaf count of optimal. \(356\) vs. \(2(151)=302\).
Time = 11.88 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.36 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (120 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-12 (15 A+29 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+2 (45 A+199 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+105 B (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7+240 (A+B) \sin \left (\frac {1}{2} (e+f x)\right )-24 (15 A+29 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+4 (45 A+199 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {1}{2} (e+f x)\right )-2 (15 A+337 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3}{105 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^4} \] Input:
Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x ])^4,x]
Output:
(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(120*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 12*(15*A + 29*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2 ])^3 + 2*(45*A + 199*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 105*B*(e + f*x)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7 + 240*(A + B)*Sin[(e + f*x )/2] - 24*(15*A + 29*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f *x)/2] + 4*(45*A + 199*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[(e + f*x)/2] - 2*(15*A + 337*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6*Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)/(105*f*(Cos[(e + f*x)/2] + Sin[(e + f*x) /2])^6*(c - c*Sin[e + f*x])^4)
Time = 0.81 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {3042, 3446, 3042, 3338, 3042, 3159, 3042, 3159, 3042, 3159, 24}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {B \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^6}dx}{c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {B \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^6}dx}{c}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {B \left (\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^5}-\frac {\int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^4}dx}{c^2}\right )}{c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {B \left (\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^5}-\frac {\int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^4}dx}{c^2}\right )}{c}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {B \left (\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^5}-\frac {\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^3}-\frac {\int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^2}dx}{c^2}}{c^2}\right )}{c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {B \left (\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^5}-\frac {\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^3}-\frac {\int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^2}dx}{c^2}}{c^2}\right )}{c}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {B \left (\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^5}-\frac {\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^3}-\frac {\frac {2 \cos (e+f x)}{f \left (c^2-c^2 \sin (e+f x)\right )}-\frac {\int 1dx}{c^2}}{c^2}}{c^2}\right )}{c}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}-\frac {B \left (\frac {2 \cos ^5(e+f x)}{5 c f (c-c \sin (e+f x))^5}-\frac {\frac {2 \cos ^3(e+f x)}{3 c f (c-c \sin (e+f x))^3}-\frac {\frac {2 \cos (e+f x)}{f \left (c^2-c^2 \sin (e+f x)\right )}-\frac {x}{c^2}}{c^2}}{c^2}\right )}{c}\right )\) |
Input:
Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^4,x ]
Output:
a^3*c^3*(((A + B)*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^7) - (B*((2*Co s[e + f*x]^5)/(5*c*f*(c - c*Sin[e + f*x])^5) - ((2*Cos[e + f*x]^3)/(3*c*f* (c - c*Sin[e + f*x])^3) - (-(x/c^2) + (2*Cos[e + f*x])/(f*(c^2 - c^2*Sin[e + f*x])))/c^2)/c^2))/c)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
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]))
Time = 1.86 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.17
| method | result | size |
| derivativedivides | \(\frac {2 a^{3} \left (-\frac {A -B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {12 A +4 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {60 A +20 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {64 A +64 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {160 A +96 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {192 A +192 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {240 A +208 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}+B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{4}}\) | \(177\) |
| default | \(\frac {2 a^{3} \left (-\frac {A -B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {12 A +4 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {60 A +20 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {64 A +64 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {160 A +96 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {192 A +192 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {240 A +208 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}+B \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{4}}\) | \(177\) |
| parallelrisch | \(-\frac {2 \left (-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} x f B}{2}+\left (\frac {7}{2} f x B +A -B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+B \left (-\frac {21 f x}{2}+8\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (\frac {35}{2} f x B +5 A -\frac {55}{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+B \left (-\frac {35 f x}{2}+\frac {112}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (\frac {21}{2} f x B +3 A -\frac {127}{5} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+B \left (-\frac {7 f x}{2}+\frac {152}{15}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {f x B}{2}+\frac {A}{7}-\frac {167 B}{105}\right ) a^{3}}{f \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(178\) |
| risch | \(\frac {a^{3} B x}{c^{4}}-\frac {2 \left (-15 a^{3} A -337 a^{3} B -1624 i B \,a^{3} {\mathrm e}^{i \left (f x +e \right )}-2520 i B \,a^{3} {\mathrm e}^{5 i \left (f x +e \right )}+6160 i B \,a^{3} {\mathrm e}^{3 i \left (f x +e \right )}+105 A \,a^{3} {\mathrm e}^{6 i \left (f x +e \right )}+735 B \,a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-525 A \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-5635 B \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+315 A \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}+4557 B \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}\right )}{105 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,c^{4}}\) | \(184\) |
| norman | \(\frac {-\frac {a^{3} x B}{c}-\frac {\left (150 a^{3} A -1334 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{21 f c}-\frac {\left (750 a^{3} A -5438 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{21 f c}-\frac {\left (1662 a^{3} A -9790 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{21 f c}-\frac {\left (1938 a^{3} A -9122 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{21 f c}-\frac {\left (870 a^{3} A -3142 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{15 f c}-\frac {\left (54 a^{3} A -134 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{3 f c}-\frac {\left (2 a^{3} A -2 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{f c}-\frac {6544 a^{3} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15 f c}-\frac {8896 a^{3} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{15 f c}-\frac {6224 a^{3} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{15 f c}-\frac {416 a^{3} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{3 f c}-\frac {16 a^{3} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{f c}-\frac {30 a^{3} A -334 a^{3} B}{105 f c}+\frac {7 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c}-\frac {25 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{c}+\frac {63 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{c}-\frac {125 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{c}+\frac {203 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{c}-\frac {277 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{c}+\frac {323 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{c}-\frac {323 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{c}+\frac {277 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{c}-\frac {203 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{c}+\frac {125 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{c}-\frac {63 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{c}+\frac {25 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{c}-\frac {7 a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{c}+\frac {a^{3} x B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{c}-\frac {304 a^{3} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 f c}-\frac {2336 a^{3} B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{15 f c}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(760\) |
Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^4,x,method=_RETUR NVERBOSE)
Output:
2/f*a^3/c^4*(-(A-B)/(tan(1/2*f*x+1/2*e)-1)-1/2*(12*A+4*B)/(tan(1/2*f*x+1/2 *e)-1)^2-1/3*(60*A+20*B)/(tan(1/2*f*x+1/2*e)-1)^3-1/7*(64*A+64*B)/(tan(1/2 *f*x+1/2*e)-1)^7-1/4*(160*A+96*B)/(tan(1/2*f*x+1/2*e)-1)^4-1/6*(192*A+192* B)/(tan(1/2*f*x+1/2*e)-1)^6-1/5*(240*A+208*B)/(tan(1/2*f*x+1/2*e)-1)^5+B*a rctan(tan(1/2*f*x+1/2*e)))
Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (149) = 298\).
Time = 0.09 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.40 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {840 \, B a^{3} f x + {\left (105 \, B a^{3} f x + {\left (15 \, A + 337 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{4} + 120 \, {\left (A + B\right )} a^{3} - {\left (315 \, B a^{3} f x + {\left (45 \, A - 613 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{3} - 24 \, {\left (35 \, B a^{3} f x + {\left (5 \, A + 26 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{2} + 60 \, {\left (7 \, B a^{3} f x + {\left (A - 13 \, B\right )} a^{3}\right )} \cos \left (f x + e\right ) - {\left (840 \, B a^{3} f x - 120 \, {\left (A + B\right )} a^{3} - {\left (105 \, B a^{3} f x - {\left (15 \, A + 337 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{3} - 12 \, {\left (35 \, B a^{3} f x - {\left (5 \, A - 23 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{2} + 60 \, {\left (7 \, B a^{3} f x - {\left (A + 15 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^4,x, algori thm="fricas")
Output:
1/105*(840*B*a^3*f*x + (105*B*a^3*f*x + (15*A + 337*B)*a^3)*cos(f*x + e)^4 + 120*(A + B)*a^3 - (315*B*a^3*f*x + (45*A - 613*B)*a^3)*cos(f*x + e)^3 - 24*(35*B*a^3*f*x + (5*A + 26*B)*a^3)*cos(f*x + e)^2 + 60*(7*B*a^3*f*x + ( A - 13*B)*a^3)*cos(f*x + e) - (840*B*a^3*f*x - 120*(A + B)*a^3 - (105*B*a^ 3*f*x - (15*A + 337*B)*a^3)*cos(f*x + e)^3 - 12*(35*B*a^3*f*x - (5*A - 23* B)*a^3)*cos(f*x + e)^2 + 60*(7*B*a^3*f*x - (A + 15*B)*a^3)*cos(f*x + e))*s in(f*x + e))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos( f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^ 4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 2951 vs. \(2 (141) = 282\).
Time = 25.53 (sec) , antiderivative size = 2951, normalized size of antiderivative = 19.54 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**4,x)
Output:
Piecewise((-210*A*a**3*tan(e/2 + f*x/2)**6/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675 *c**4*f*tan(e/2 + f*x/2)**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4* f*tan(e/2 + f*x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f) - 1050*A *a**3*tan(e/2 + f*x/2)**4/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan (e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/2 + f*x/2)**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e/2 + f*x/2 )**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f) - 630*A*a**3*tan(e/2 + f* x/2)**2/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/2 + f*x/2)**4 + 3675* c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e/2 + f*x/2)**2 + 735*c**4*f* tan(e/2 + f*x/2) - 105*c**4*f) - 30*A*a**3/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan(e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675 *c**4*f*tan(e/2 + f*x/2)**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4* f*tan(e/2 + f*x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f) + 105*B* a**3*f*x*tan(e/2 + f*x/2)**7/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f* tan(e/2 + f*x/2)**6 + 2205*c**4*f*tan(e/2 + f*x/2)**5 - 3675*c**4*f*tan(e/ 2 + f*x/2)**4 + 3675*c**4*f*tan(e/2 + f*x/2)**3 - 2205*c**4*f*tan(e/2 + f* x/2)**2 + 735*c**4*f*tan(e/2 + f*x/2) - 105*c**4*f) - 735*B*a**3*f*x*tan(e /2 + f*x/2)**6/(105*c**4*f*tan(e/2 + f*x/2)**7 - 735*c**4*f*tan(e/2 + f...
Leaf count of result is larger than twice the leaf count of optimal. 2118 vs. \(2 (149) = 298\).
Time = 0.18 (sec) , antiderivative size = 2118, normalized size of antiderivative = 14.03 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^4,x, algori thm="maxima")
Output:
2/105*(5*B*a^3*((203*sin(f*x + e)/(cos(f*x + e) + 1) - 525*sin(f*x + e)^2/ (cos(f*x + e) + 1)^2 + 686*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 434*sin(f *x + e)^4/(cos(f*x + e) + 1)^4 + 147*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 21*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 32)/(c^4 - 7*c^4*sin(f*x + e)/(c os(f*x + e) + 1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^4*sin (f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1 )^4 - 21*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 7*c^4*sin(f*x + e)^6/(c os(f*x + e) + 1)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 21*arctan( sin(f*x + e)/(cos(f*x + e) + 1))/c^4) + 3*A*a^3*(91*sin(f*x + e)/(cos(f*x + e) + 1) - 168*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 280*sin(f*x + e)^3/( cos(f*x + e) + 1)^3 - 175*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105*sin(f* x + e)^5/(cos(f*x + e) + 1)^5 - 13)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e ) + 1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^4*sin(f*x + e)^ 3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 21*c ^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e ) + 1)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + B*a^3*(91*sin(f*x + e)/(cos(f*x + e) + 1) - 168*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 280*sin( f*x + e)^3/(cos(f*x + e) + 1)^3 - 175*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 13)/(c^4 - 7*c^4*sin(f*x + e)/ (cos(f*x + e) + 1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^...
Time = 0.29 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.34 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {\frac {105 \, {\left (f x + e\right )} B a^{3}}{c^{4}} - \frac {2 \, {\left (105 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 840 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 525 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1925 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 3920 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 315 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2667 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1064 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, A a^{3} - 167 \, B a^{3}\right )}}{c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{105 \, f} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^4,x, algori thm="giac")
Output:
1/105*(105*(f*x + e)*B*a^3/c^4 - 2*(105*A*a^3*tan(1/2*f*x + 1/2*e)^6 - 105 *B*a^3*tan(1/2*f*x + 1/2*e)^6 + 840*B*a^3*tan(1/2*f*x + 1/2*e)^5 + 525*A*a ^3*tan(1/2*f*x + 1/2*e)^4 - 1925*B*a^3*tan(1/2*f*x + 1/2*e)^4 + 3920*B*a^3 *tan(1/2*f*x + 1/2*e)^3 + 315*A*a^3*tan(1/2*f*x + 1/2*e)^2 - 2667*B*a^3*ta n(1/2*f*x + 1/2*e)^2 + 1064*B*a^3*tan(1/2*f*x + 1/2*e) + 15*A*a^3 - 167*B* a^3)/(c^4*(tan(1/2*f*x + 1/2*e) - 1)^7))/f
Time = 38.37 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.09 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {B\,a^3\,x}{c^4}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {a^3\,\left (1680\,B-2205\,B\,\left (e+f\,x\right )\right )}{105}+21\,B\,a^3\,\left (e+f\,x\right )\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {a^3\,\left (7840\,B-3675\,B\,\left (e+f\,x\right )\right )}{105}+35\,B\,a^3\,\left (e+f\,x\right )\right )+\frac {a^3\,\left (30\,A-334\,B+105\,B\,\left (e+f\,x\right )\right )}{105}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (210\,A-210\,B+735\,B\,\left (e+f\,x\right )\right )}{105}-7\,B\,a^3\,\left (e+f\,x\right )\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (630\,A-5334\,B+2205\,B\,\left (e+f\,x\right )\right )}{105}-21\,B\,a^3\,\left (e+f\,x\right )\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^3\,\left (1050\,A-3850\,B+3675\,B\,\left (e+f\,x\right )\right )}{105}-35\,B\,a^3\,\left (e+f\,x\right )\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {a^3\,\left (2128\,B-735\,B\,\left (e+f\,x\right )\right )}{105}+7\,B\,a^3\,\left (e+f\,x\right )\right )-B\,a^3\,\left (e+f\,x\right )}{c^4\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^7} \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^4,x )
Output:
(B*a^3*x)/c^4 - (tan(e/2 + (f*x)/2)^5*((a^3*(1680*B - 2205*B*(e + f*x)))/1 05 + 21*B*a^3*(e + f*x)) + tan(e/2 + (f*x)/2)^3*((a^3*(7840*B - 3675*B*(e + f*x)))/105 + 35*B*a^3*(e + f*x)) + (a^3*(30*A - 334*B + 105*B*(e + f*x)) )/105 + tan(e/2 + (f*x)/2)^6*((a^3*(210*A - 210*B + 735*B*(e + f*x)))/105 - 7*B*a^3*(e + f*x)) + tan(e/2 + (f*x)/2)^2*((a^3*(630*A - 5334*B + 2205*B *(e + f*x)))/105 - 21*B*a^3*(e + f*x)) + tan(e/2 + (f*x)/2)^4*((a^3*(1050* A - 3850*B + 3675*B*(e + f*x)))/105 - 35*B*a^3*(e + f*x)) + tan(e/2 + (f*x )/2)*((a^3*(2128*B - 735*B*(e + f*x)))/105 + 7*B*a^3*(e + f*x)) - B*a^3*(e + f*x))/(c^4*f*(tan(e/2 + (f*x)/2) - 1)^7)
Time = 0.18 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.36 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^4} \, dx=\frac {a^{3} \left (-30 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} a +105 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} b f x +30 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} b -735 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} b f x -630 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} a +2205 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} b f x -1050 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} b -3675 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} b f x +2800 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} b -1050 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} a +3675 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} b f x -6790 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} b -2205 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b f x +4704 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -210 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a +735 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b f x -1918 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b -105 b f x +304 b \right )}{105 c^{4} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+21 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-35 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+35 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-21 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )} \] Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^4,x)
Output:
(a**3*( - 30*tan((e + f*x)/2)**7*a + 105*tan((e + f*x)/2)**7*b*f*x + 30*ta n((e + f*x)/2)**7*b - 735*tan((e + f*x)/2)**6*b*f*x - 630*tan((e + f*x)/2) **5*a + 2205*tan((e + f*x)/2)**5*b*f*x - 1050*tan((e + f*x)/2)**5*b - 3675 *tan((e + f*x)/2)**4*b*f*x + 2800*tan((e + f*x)/2)**4*b - 1050*tan((e + f* x)/2)**3*a + 3675*tan((e + f*x)/2)**3*b*f*x - 6790*tan((e + f*x)/2)**3*b - 2205*tan((e + f*x)/2)**2*b*f*x + 4704*tan((e + f*x)/2)**2*b - 210*tan((e + f*x)/2)*a + 735*tan((e + f*x)/2)*b*f*x - 1918*tan((e + f*x)/2)*b - 105*b *f*x + 304*b))/(105*c**4*f*(tan((e + f*x)/2)**7 - 7*tan((e + f*x)/2)**6 + 21*tan((e + f*x)/2)**5 - 35*tan((e + f*x)/2)**4 + 35*tan((e + f*x)/2)**3 - 21*tan((e + f*x)/2)**2 + 7*tan((e + f*x)/2) - 1))