\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx\) [79]

Optimal result

Integrand size = 36, antiderivative size = 162 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\frac {(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 (7 A-2 B) \tan (e+f x)}{21 a^3 c^5 f}+\frac {4 (7 A-2 B) \tan ^3(e+f x)}{63 a^3 c^5 f}+\frac {2 (7 A-2 B) \tan ^5(e+f x)}{105 a^3 c^5 f} \]

[Out]

1/9*(A+B)*sec(f*x+e)^5/a^3/c^3/f/(c-c*sin(f*x+e))^2+1/63*(7*A-2*B)*sec(f*x+e)^5/a^3/f/(c^5-c^5*sin(f*x+e))+2/2
1*(7*A-2*B)*tan(f*x+e)/a^3/c^5/f+4/63*(7*A-2*B)*tan(f*x+e)^3/a^3/c^5/f+2/105*(7*A-2*B)*tan(f*x+e)^5/a^3/c^5/f

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2938, 2751, 3852} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\frac {2 (7 A-2 B) \tan ^5(e+f x)}{105 a^3 c^5 f}+\frac {4 (7 A-2 B) \tan ^3(e+f x)}{63 a^3 c^5 f}+\frac {2 (7 A-2 B) \tan (e+f x)}{21 a^3 c^5 f}+\frac {(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2} \]

[In]

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5),x]

[Out]

((A + B)*Sec[e + f*x]^5)/(9*a^3*c^3*f*(c - c*Sin[e + f*x])^2) + ((7*A - 2*B)*Sec[e + f*x]^5)/(63*a^3*f*(c^5 -
c^5*Sin[e + f*x])) + (2*(7*A - 2*B)*Tan[e + f*x])/(21*a^3*c^5*f) + (4*(7*A - 2*B)*Tan[e + f*x]^3)/(63*a^3*c^5*
f) + (2*(7*A - 2*B)*Tan[e + f*x]^5)/(105*a^3*c^5*f)

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[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[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 2938

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] + Dist[(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]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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] && I
ntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2} \, dx}{a^3 c^3} \\ & = \frac {(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {(7 A-2 B) \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{9 a^3 c^4} \\ & = \frac {(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {(2 (7 A-2 B)) \int \sec ^6(e+f x) \, dx}{21 a^3 c^5} \\ & = \frac {(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac {(2 (7 A-2 B)) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{21 a^3 c^5 f} \\ & = \frac {(A+B) \sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {(7 A-2 B) \sec ^5(e+f x)}{63 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 (7 A-2 B) \tan (e+f x)}{21 a^3 c^5 f}+\frac {4 (7 A-2 B) \tan ^3(e+f x)}{63 a^3 c^5 f}+\frac {2 (7 A-2 B) \tan ^5(e+f x)}{105 a^3 c^5 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(373\) vs. \(2(162)=324\).

Time = 3.75 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.30 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-184320 B+1125 (49 A+13 B) \cos (e+f x)-20480 (7 A-2 B) \cos (2 (e+f x))+23275 A \cos (3 (e+f x))+6175 B \cos (3 (e+f x))-114688 A \cos (4 (e+f x))+32768 B \cos (4 (e+f x))+1225 A \cos (5 (e+f x))+325 B \cos (5 (e+f x))-28672 A \cos (6 (e+f x))+8192 B \cos (6 (e+f x))-1225 A \cos (7 (e+f x))-325 B \cos (7 (e+f x))-322560 A \sin (e+f x)+92160 B \sin (e+f x)-24500 A \sin (2 (e+f x))-6500 B \sin (2 (e+f x))-136192 A \sin (3 (e+f x))+38912 B \sin (3 (e+f x))-19600 A \sin (4 (e+f x))-5200 B \sin (4 (e+f x))-7168 A \sin (5 (e+f x))+2048 B \sin (5 (e+f x))-4900 A \sin (6 (e+f x))-1300 B \sin (6 (e+f x))+7168 A \sin (7 (e+f x))-2048 B \sin (7 (e+f x)))}{1290240 a^3 c^5 f (-1+\sin (e+f x))^5 (1+\sin (e+f x))^3} \]

[In]

Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5),x]

[Out]

((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-184320*B + 1125*(49*A + 13*B)*C
os[e + f*x] - 20480*(7*A - 2*B)*Cos[2*(e + f*x)] + 23275*A*Cos[3*(e + f*x)] + 6175*B*Cos[3*(e + f*x)] - 114688
*A*Cos[4*(e + f*x)] + 32768*B*Cos[4*(e + f*x)] + 1225*A*Cos[5*(e + f*x)] + 325*B*Cos[5*(e + f*x)] - 28672*A*Co
s[6*(e + f*x)] + 8192*B*Cos[6*(e + f*x)] - 1225*A*Cos[7*(e + f*x)] - 325*B*Cos[7*(e + f*x)] - 322560*A*Sin[e +
f*x] + 92160*B*Sin[e + f*x] - 24500*A*Sin[2*(e + f*x)] - 6500*B*Sin[2*(e + f*x)] - 136192*A*Sin[3*(e + f*x)]
+ 38912*B*Sin[3*(e + f*x)] - 19600*A*Sin[4*(e + f*x)] - 5200*B*Sin[4*(e + f*x)] - 7168*A*Sin[5*(e + f*x)] + 20
48*B*Sin[5*(e + f*x)] - 4900*A*Sin[6*(e + f*x)] - 1300*B*Sin[6*(e + f*x)] + 7168*A*Sin[7*(e + f*x)] - 2048*B*S
in[7*(e + f*x)]))/(1290240*a^3*c^5*f*(-1 + Sin[e + f*x])^5*(1 + Sin[e + f*x])^3)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.19 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.30

method	result	size
risch	\(-\frac {32 \left (-2 i B \,{\mathrm e}^{2 i \left (f x +e \right )}+2 i B +315 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+133 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-38 i B \,{\mathrm e}^{4 i \left (f x +e \right )}-90 i B \,{\mathrm e}^{6 i \left (f x +e \right )}+28 A \,{\mathrm e}^{i \left (f x +e \right )}+180 B \,{\mathrm e}^{7 i \left (f x +e \right )}-8 B \,{\mathrm e}^{i \left (f x +e \right )}+7 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+112 A \,{\mathrm e}^{3 i \left (f x +e \right )}+140 A \,{\mathrm e}^{5 i \left (f x +e \right )}-40 B \,{\mathrm e}^{5 i \left (f x +e \right )}-32 B \,{\mathrm e}^{3 i \left (f x +e \right )}-7 i A \right )}{315 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} f \,c^{5} a^{3}}\)	\(211\)
parallelrisch	\(\frac {-630 A \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (1260 A -630 B \right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-420 A +840 B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-3360 A -840 B \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (966 A -1176 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (4956 A -966 B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-7224 A +2064 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-1344 A -3216 B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3766 A -176 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-700 A +110 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-2660 A +40 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (1120 A -680 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-70 A +200 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-140 A -50 B}{315 f \,c^{5} a^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(285\)
derivativedivides	\(\frac {-\frac {2 \left (2 A +2 B \right )}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {8 A +8 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {\frac {35 A}{2}+12 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {2 \left (\frac {35 A}{2}+\frac {33 B}{2}\right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {\frac {49 A}{2}+\frac {43 B}{2}}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {\frac {51 A}{16}+\frac {21 B}{16}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {49 A}{2}+\frac {77 B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 \left (\frac {99 A}{128}+\frac {15 B}{128}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {147 A}{16}+\frac {81 B}{16}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {-\frac {9 A}{32}+\frac {7 B}{32}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-\frac {A}{4}+\frac {B}{4}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (\frac {A}{8}-\frac {B}{8}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (\frac {13 A}{32}-\frac {11 B}{32}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {29 A}{128}-\frac {15 B}{128}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{3} c^{5} f}\)	\(321\)
default	\(\frac {-\frac {2 \left (2 A +2 B \right )}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {8 A +8 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {\frac {35 A}{2}+12 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {2 \left (\frac {35 A}{2}+\frac {33 B}{2}\right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {\frac {49 A}{2}+\frac {43 B}{2}}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {\frac {51 A}{16}+\frac {21 B}{16}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {49 A}{2}+\frac {77 B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 \left (\frac {99 A}{128}+\frac {15 B}{128}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {147 A}{16}+\frac {81 B}{16}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {-\frac {9 A}{32}+\frac {7 B}{32}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-\frac {A}{4}+\frac {B}{4}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (\frac {A}{8}-\frac {B}{8}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (\frac {13 A}{32}-\frac {11 B}{32}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {29 A}{128}-\frac {15 B}{128}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a^{3} c^{5} f}\)	\(321\)
norman	\(\frac {\frac {14 A -40 B}{252 c f a}+\frac {2 \left (602 A -697 B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f a}-\frac {A \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c f a}-\frac {\left (1036 A -296 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f a}+\frac {\left (476 A -136 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 c f a}-\frac {\left (56 A -16 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f a}+\frac {\left (6 A -4 B \right ) \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c f a}-\frac {\left (28 A -8 B \right ) \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f a}+\frac {\left (56 A -16 B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f a}+\frac {\left (14 A -28 B \right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 c f a}-\frac {\left (140 A -40 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{63 c f a}+\frac {\left (518 A -292 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{126 c f a}-\frac {\left (322 A +76 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{42 c f a}-\frac {\left (238 A +172 B \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{30 c f a}-\frac {\left (6608 A -1888 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 c f a}+\frac {\left (4102 A -6212 B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{630 c f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(495\)

[In]

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

-32/315*(-2*I*B*exp(2*I*(f*x+e))+2*I*B+315*I*A*exp(6*I*(f*x+e))+133*I*A*exp(4*I*(f*x+e))-38*I*B*exp(4*I*(f*x+e
))-90*I*B*exp(6*I*(f*x+e))+28*A*exp(I*(f*x+e))+180*B*exp(7*I*(f*x+e))-8*B*exp(I*(f*x+e))+7*I*A*exp(2*I*(f*x+e)
)+112*A*exp(3*I*(f*x+e))+140*A*exp(5*I*(f*x+e))-40*B*exp(5*I*(f*x+e))-32*B*exp(3*I*(f*x+e))-7*I*A)/(exp(I*(f*x
+e))+I)^5/(exp(I*(f*x+e))-I)^9/f/c^5/a^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.14 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=-\frac {32 \, {\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{6} - 16 \, {\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{4} - 4 \, {\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{2} - {\left (16 \, {\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{6} - 24 \, {\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{4} - 10 \, {\left (7 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{2} - 49 \, A + 14 \, B\right )} \sin \left (f x + e\right ) - 14 \, A + 49 \, B}{315 \, {\left (a^{3} c^{5} f \cos \left (f x + e\right )^{7} + 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) - 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5}\right )}} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

-1/315*(32*(7*A - 2*B)*cos(f*x + e)^6 - 16*(7*A - 2*B)*cos(f*x + e)^4 - 4*(7*A - 2*B)*cos(f*x + e)^2 - (16*(7*
A - 2*B)*cos(f*x + e)^6 - 24*(7*A - 2*B)*cos(f*x + e)^4 - 10*(7*A - 2*B)*cos(f*x + e)^2 - 49*A + 14*B)*sin(f*x
+ e) - 14*A + 49*B)/(a^3*c^5*f*cos(f*x + e)^7 + 2*a^3*c^5*f*cos(f*x + e)^5*sin(f*x + e) - 2*a^3*c^5*f*cos(f*x
+ e)^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8396 vs. \(2 (150) = 300\).

Time = 59.13 (sec) , antiderivative size = 8396, normalized size of antiderivative = 51.83 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**5,x)

[Out]

Piecewise((-630*A*tan(e/2 + f*x/2)**13/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/
2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e
/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c*
*5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*
a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 31
5*a**3*c**5*f) + 1260*A*tan(e/2 + f*x/2)**12/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2
+ f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f
*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a
**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 -
5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2
) - 315*a**3*c**5*f) - 420*A*tan(e/2 + f*x/2)**11/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan
(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c
**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14
175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)
**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 +
f*x/2) - 315*a**3*c**5*f) - 3360*A*tan(e/2 + f*x/2)**10/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5
*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*
a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**
8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 +
f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(
e/2 + f*x/2) - 315*a**3*c**5*f) + 966*A*tan(e/2 + f*x/2)**9/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*
c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5
985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/
2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/
2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*
tan(e/2 + f*x/2) - 315*a**3*c**5*f) + 4956*A*tan(e/2 + f*x/2)**8/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*
a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**1
1 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 +
f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*t
an(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c*
*5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 7224*A*tan(e/2 + f*x/2)**7/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 -
1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/
2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(
e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**
5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a*
*3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 1344*A*tan(e/2 + f*x/2)**6/(315*a**3*c**5*f*tan(e/2 + f*x/2)**
14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 +
f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f
*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**
3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 12
60*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) + 3766*A*tan(e/2 + f*x/2)**5/(315*a**3*c**5*f*tan(e/2 + f*x
/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(
e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c
**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 598
5*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2
+ 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 700*A*tan(e/2 + f*x/2)**4/(315*a**3*c**5*f*tan(e/2 +
f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*
tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a*
*3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 +
5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2
)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 2660*A*tan(e/2 + f*x/2)**3/(315*a**3*c**5*f*tan(
e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c*
*5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 141
75*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)
**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 +
f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) + 1120*A*tan(e/2 + f*x/2)**2/(315*a**3*c**5*f
*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a*
*3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9
+ 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f
*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e
/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 70*A*tan(e/2 + f*x/2)/(315*a**3*c**5*f
*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a*
*3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9
+ 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f
*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e
/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 140*A/(315*a**3*c**5*f*tan(e/2 + f*x/2
)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/
2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**
5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*
a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 +
1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 630*B*tan(e/2 + f*x/2)**12/(315*a**3*c**5*f*tan(e/2 +
f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*t
an(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**
3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 +
5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)
**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) + 840*B*tan(e/2 + f*x/2)**11/(315*a**3*c**5*f*tan(e
/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**
5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 1417
5*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)*
*5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f
*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 840*B*tan(e/2 + f*x/2)**10/(315*a**3*c**5*f*
tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 5040*a**
3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)**9 +
14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2 + f*
x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*tan(e/
2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 1176*B*tan(e/2 + f*x/2)**9/(315*a**3*c*
*5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 504
0*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*x/2)
**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan(e/2
+ f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5*f*t
an(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 966*B*tan(e/2 + f*x/2)**8/(315*a**
3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)**12 +
5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2 + f*
x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*f*tan
(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3*c**5
*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) + 2064*B*tan(e/2 + f*x/2)**7/(31
5*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x/2)*
*12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan(e/2
+ f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c**5*
f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*a**3
*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 3216*B*tan(e/2 + f*x/2)**
6/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*
x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*ta
n(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*
c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315
*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 176*B*tan(e/2 + f*x/
2)**5/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2
+ f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*
f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a
**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 -
315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) + 110*B*tan(e/2 +
f*x/2)**4/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(
e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c
**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 63
00*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)*
*3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) + 40*B*tan(e/2
+ f*x/2)**3/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*t
an(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**
3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 +
6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/
2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 680*B*tan
(e/2 + f*x/2)**2/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5
*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300
*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)*
*6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 +
f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) + 200*B
*tan(e/2 + f*x/2)/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**
5*f*tan(e/2 + f*x/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 630
0*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)
**6 + 6300*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 +
f*x/2)**3 - 315*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f) - 50*B
/(315*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 1260*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 315*a**3*c**5*f*tan(e/2 + f*x
/2)**12 + 5040*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 5985*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 6300*a**3*c**5*f*tan
(e/2 + f*x/2)**9 + 14175*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 14175*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 6300*a**3*c
**5*f*tan(e/2 + f*x/2)**5 + 5985*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 5040*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 315*
a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1260*a**3*c**5*f*tan(e/2 + f*x/2) - 315*a**3*c**5*f), Ne(f, 0)), (x*(A + B*s
in(e))/((a*sin(e) + a)**3*(-c*sin(e) + c)**5), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1201 vs. \(2 (154) = 308\).

Time = 0.26 (sec) , antiderivative size = 1201, normalized size of antiderivative = 7.41 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

-2/315*(B*(100*sin(f*x + e)/(cos(f*x + e) + 1) - 340*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 20*sin(f*x + e)^3/(
cos(f*x + e) + 1)^3 + 55*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 88*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 1608*s
in(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1032*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 483*sin(f*x + e)^8/(cos(f*x +
e) + 1)^8 - 588*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 420*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 420*sin(f*x
+ e)^11/(cos(f*x + e) + 1)^11 - 315*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 25)/(a^3*c^5 - 4*a^3*c^5*sin(f*x +
e)/(cos(f*x + e) + 1) + a^3*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 16*a^3*c^5*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 - 19*a^3*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 20*a^3*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 45
*a^3*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 45*a^3*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 20*a^3*c^5*sin
(f*x + e)^9/(cos(f*x + e) + 1)^9 + 19*a^3*c^5*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 16*a^3*c^5*sin(f*x + e)^
11/(cos(f*x + e) + 1)^11 - a^3*c^5*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 + 4*a^3*c^5*sin(f*x + e)^13/(cos(f*x
+ e) + 1)^13 - a^3*c^5*sin(f*x + e)^14/(cos(f*x + e) + 1)^14) - 7*A*(5*sin(f*x + e)/(cos(f*x + e) + 1) - 80*si
n(f*x + e)^2/(cos(f*x + e) + 1)^2 + 190*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 50*sin(f*x + e)^4/(cos(f*x + e)
+ 1)^4 - 269*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 96*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 516*sin(f*x + e)^7
/(cos(f*x + e) + 1)^7 - 354*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 69*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 240
*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 30*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 90*sin(f*x + e)^12/(cos(f*
x + e) + 1)^12 + 45*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 10)/(a^3*c^5 - 4*a^3*c^5*sin(f*x + e)/(cos(f*x + e
) + 1) + a^3*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 16*a^3*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 19*a^3
*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 20*a^3*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 45*a^3*c^5*sin(f*x
+ e)^6/(cos(f*x + e) + 1)^6 - 45*a^3*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 20*a^3*c^5*sin(f*x + e)^9/(cos
(f*x + e) + 1)^9 + 19*a^3*c^5*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 16*a^3*c^5*sin(f*x + e)^11/(cos(f*x + e)
+ 1)^11 - a^3*c^5*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 + 4*a^3*c^5*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - a
^3*c^5*sin(f*x + e)^14/(cos(f*x + e) + 1)^14))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (154) = 308\).

Time = 0.37 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.40 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=-\frac {\frac {21 \, {\left (435 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 225 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1470 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 690 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2060 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 940 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1330 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 590 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 353 \, A - 163 \, B\right )}}{a^{3} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {31185 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 4725 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 185220 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 11340 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 546840 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 15120 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 961380 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3780 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1101618 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 24318 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 828492 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 33852 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 404208 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 19368 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 116172 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6732 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 16373 \, A - 223 \, B}{a^{3} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{20160 \, f} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

-1/20160*(21*(435*A*tan(1/2*f*x + 1/2*e)^4 - 225*B*tan(1/2*f*x + 1/2*e)^4 + 1470*A*tan(1/2*f*x + 1/2*e)^3 - 69
0*B*tan(1/2*f*x + 1/2*e)^3 + 2060*A*tan(1/2*f*x + 1/2*e)^2 - 940*B*tan(1/2*f*x + 1/2*e)^2 + 1330*A*tan(1/2*f*x
+ 1/2*e) - 590*B*tan(1/2*f*x + 1/2*e) + 353*A - 163*B)/(a^3*c^5*(tan(1/2*f*x + 1/2*e) + 1)^5) + (31185*A*tan(
1/2*f*x + 1/2*e)^8 + 4725*B*tan(1/2*f*x + 1/2*e)^8 - 185220*A*tan(1/2*f*x + 1/2*e)^7 - 11340*B*tan(1/2*f*x + 1
/2*e)^7 + 546840*A*tan(1/2*f*x + 1/2*e)^6 + 15120*B*tan(1/2*f*x + 1/2*e)^6 - 961380*A*tan(1/2*f*x + 1/2*e)^5 +
3780*B*tan(1/2*f*x + 1/2*e)^5 + 1101618*A*tan(1/2*f*x + 1/2*e)^4 - 24318*B*tan(1/2*f*x + 1/2*e)^4 - 828492*A*
tan(1/2*f*x + 1/2*e)^3 + 33852*B*tan(1/2*f*x + 1/2*e)^3 + 404208*A*tan(1/2*f*x + 1/2*e)^2 - 19368*B*tan(1/2*f*
x + 1/2*e)^2 - 116172*A*tan(1/2*f*x + 1/2*e) + 6732*B*tan(1/2*f*x + 1/2*e) + 16373*A - 223*B)/(a^3*c^5*(tan(1/
2*f*x + 1/2*e) - 1)^9))/f

Mupad [B] (verification not implemented)

Time = 13.69 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.43 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\frac {\left (\frac {128\,B}{315}-\frac {64\,A}{45}+\frac {32\,A\,\sin \left (e+f\,x\right )}{45}-\frac {64\,B\,\sin \left (e+f\,x\right )}{315}\right )\,{\cos \left (e+f\,x\right )}^6+\left (\frac {8\,A\,\sin \left (e+f\,x\right )}{9}-\frac {20\,B}{63}-\frac {8\,A}{9}+\frac {20\,B\,\sin \left (e+f\,x\right )}{63}-\frac {\left (4\,\sin \left (e+f\,x\right )-4\right )\,\left (\frac {4\,A}{9}+\frac {10\,B}{63}\right )}{2}\right )\,{\cos \left (e+f\,x\right )}^5+\left (\frac {32\,A}{45}-\frac {64\,B}{315}-\frac {16\,A\,\sin \left (e+f\,x\right )}{15}+\frac {32\,B\,\sin \left (e+f\,x\right )}{105}\right )\,{\cos \left (e+f\,x\right )}^4+\left (\frac {8\,A}{45}-\frac {16\,B}{315}-\frac {4\,A\,\sin \left (e+f\,x\right )}{9}+\frac {8\,B\,\sin \left (e+f\,x\right )}{63}\right )\,{\cos \left (e+f\,x\right )}^2+\frac {4\,A}{45}-\frac {14\,B}{45}-\frac {14\,A\,\sin \left (e+f\,x\right )}{45}+\frac {4\,B\,\sin \left (e+f\,x\right )}{45}}{a^3\,c^5\,f\,\left (4\,{\cos \left (e+f\,x\right )}^5\,\sin \left (e+f\,x\right )-4\,{\cos \left (e+f\,x\right )}^5+2\,{\cos \left (e+f\,x\right )}^7\right )} \]

[In]

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^5),x)

[Out]

((4*A)/45 - (14*B)/45 - (14*A*sin(e + f*x))/45 + (4*B*sin(e + f*x))/45 - cos(e + f*x)^5*((8*A)/9 + (20*B)/63 -
(8*A*sin(e + f*x))/9 - (20*B*sin(e + f*x))/63 + ((4*sin(e + f*x) - 4)*((4*A)/9 + (10*B)/63))/2) + cos(e + f*x
)^2*((8*A)/45 - (16*B)/315 - (4*A*sin(e + f*x))/9 + (8*B*sin(e + f*x))/63) + cos(e + f*x)^4*((32*A)/45 - (64*B
)/315 - (16*A*sin(e + f*x))/15 + (32*B*sin(e + f*x))/105) - cos(e + f*x)^6*((64*A)/45 - (128*B)/315 - (32*A*si
n(e + f*x))/45 + (64*B*sin(e + f*x))/315))/(a^3*c^5*f*(4*cos(e + f*x)^5*sin(e + f*x) - 4*cos(e + f*x)^5 + 2*co
s(e + f*x)^7))