\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx\) [69]
Optimal result
Integrand size = 36, antiderivative size = 175 \[
\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {4 (2 A-B) \tan (e+f x)}{21 a^2 c^5 f}+\frac {4 (2 A-B) \tan ^3(e+f x)}{63 a^2 c^5 f}
\]
[Out]
1/9*(A+B)*sec(f*x+e)^3/a^2/c^2/f/(c-c*sin(f*x+e))^3+1/21*(2*A-B)*sec(f*x+e)^3/a^2/c^3/f/(c-c*sin(f*x+e))^2+1/2
1*(2*A-B)*sec(f*x+e)^3/a^2/f/(c^5-c^5*sin(f*x+e))+4/21*(2*A-B)*tan(f*x+e)/a^2/c^5/f+4/63*(2*A-B)*tan(f*x+e)^3/
a^2/c^5/f
Rubi [A] (verified)
Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2938, 2751, 3852}
\[
\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {4 (2 A-B) \tan ^3(e+f x)}{63 a^2 c^5 f}+\frac {4 (2 A-B) \tan (e+f x)}{21 a^2 c^5 f}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}
\]
[In]
Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5),x]
[Out]
((A + B)*Sec[e + f*x]^3)/(9*a^2*c^2*f*(c - c*Sin[e + f*x])^3) + ((2*A - B)*Sec[e + f*x]^3)/(21*a^2*c^3*f*(c -
c*Sin[e + f*x])^2) + ((2*A - B)*Sec[e + f*x]^3)/(21*a^2*f*(c^5 - c^5*Sin[e + f*x])) + (4*(2*A - B)*Tan[e + f*x
])/(21*a^2*c^5*f) + (4*(2*A - B)*Tan[e + f*x]^3)/(63*a^2*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 ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx}{a^2 c^2} \\ & = \frac {(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {(2 A-B) \int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{3 a^2 c^3} \\ & = \frac {(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {(5 (2 A-B)) \int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)} \, dx}{21 a^2 c^4} \\ & = \frac {(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {(4 (2 A-B)) \int \sec ^4(e+f x) \, dx}{21 a^2 c^5} \\ & = \frac {(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac {(4 (2 A-B)) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{21 a^2 c^5 f} \\ & = \frac {(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {4 (2 A-B) \tan (e+f x)}{21 a^2 c^5 f}+\frac {4 (2 A-B) \tan ^3(e+f x)}{63 a^2 c^5 f} \\
\end{align*}
Mathematica [A] (verified)
Time = 3.15 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.88
\[
\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-10752 B+180 (31 A-5 B) \cos (e+f x)-6912 (2 A-B) \cos (2 (e+f x))+310 A \cos (3 (e+f x))-50 B \cos (3 (e+f x))-6144 A \cos (4 (e+f x))+3072 B \cos (4 (e+f x))-930 A \cos (5 (e+f x))+150 B \cos (5 (e+f x))+512 A \cos (6 (e+f x))-256 B \cos (6 (e+f x))-18432 A \sin (e+f x)+9216 B \sin (e+f x)-4185 A \sin (2 (e+f x))+675 B \sin (2 (e+f x))-1024 A \sin (3 (e+f x))+512 B \sin (3 (e+f x))-1860 A \sin (4 (e+f x))+300 B \sin (4 (e+f x))+3072 A \sin (5 (e+f x))-1536 B \sin (5 (e+f x))+155 A \sin (6 (e+f x))-25 B \sin (6 (e+f x)))}{64512 a^2 c^5 f (-1+\sin (e+f x))^5 (1+\sin (e+f x))^2}
\]
[In]
Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(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])*(-10752*B + 180*(31*A - 5*B)*Cos[
e + f*x] - 6912*(2*A - B)*Cos[2*(e + f*x)] + 310*A*Cos[3*(e + f*x)] - 50*B*Cos[3*(e + f*x)] - 6144*A*Cos[4*(e
+ f*x)] + 3072*B*Cos[4*(e + f*x)] - 930*A*Cos[5*(e + f*x)] + 150*B*Cos[5*(e + f*x)] + 512*A*Cos[6*(e + f*x)] -
256*B*Cos[6*(e + f*x)] - 18432*A*Sin[e + f*x] + 9216*B*Sin[e + f*x] - 4185*A*Sin[2*(e + f*x)] + 675*B*Sin[2*(
e + f*x)] - 1024*A*Sin[3*(e + f*x)] + 512*B*Sin[3*(e + f*x)] - 1860*A*Sin[4*(e + f*x)] + 300*B*Sin[4*(e + f*x)
] + 3072*A*Sin[5*(e + f*x)] - 1536*B*Sin[5*(e + f*x)] + 155*A*Sin[6*(e + f*x)] - 25*B*Sin[6*(e + f*x)]))/(6451
2*a^2*c^5*f*(-1 + Sin[e + f*x])^5*(1 + Sin[e + f*x])^2)
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 2.38 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.05
| | |
| method | result | size |
| | |
| risch |
\(\frac {16 i \left (72 i A \,{\mathrm e}^{5 i \left (f x +e \right )}-36 i B \,{\mathrm e}^{5 i \left (f x +e \right )}+42 B \,{\mathrm e}^{6 i \left (f x +e \right )}+4 i A \,{\mathrm e}^{3 i \left (f x +e \right )}+54 A \,{\mathrm e}^{4 i \left (f x +e \right )}-2 i B \,{\mathrm e}^{3 i \left (f x +e \right )}-27 B \,{\mathrm e}^{4 i \left (f x +e \right )}-12 i A \,{\mathrm e}^{i \left (f x +e \right )}+24 A \,{\mathrm e}^{2 i \left (f x +e \right )}+6 i B \,{\mathrm e}^{i \left (f x +e \right )}-12 B \,{\mathrm e}^{2 i \left (f x +e \right )}-2 A +B \right )}{63 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,c^{5} a^{2}}\) |
\(184\) |
| parallelrisch |
\(\frac {-126 A \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (378 A -126 B \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-546 A +252 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-126 A -378 B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+756 A \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-588 A +84 B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-612 A -72 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (900 A -324 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-470 A +256 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-78 A -150 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (102 A +12 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-38 A -2 B}{63 f \,c^{5} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) |
\(242\) |
| derivativedivides |
\(\frac {-\frac {-\frac {A}{16}+\frac {B}{16}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{16}-\frac {B}{16}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {7 A}{64}-\frac {5 B}{64}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (4 A +4 B \right )}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {16 A +16 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {2 \left (34 A +32 B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46 A +40 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {\frac {9 A}{2}+\frac {13 B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {57 A}{64}+\frac {5 B}{64}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {\frac {59 A}{2}+\frac {39 B}{2}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {2 \left (\frac {57 A}{4}+\frac {59 B}{8}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2 \left (\frac {175 A}{4}+\frac {135 B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}}{a^{2} c^{5} f}\) |
\(277\) |
| default |
\(\frac {-\frac {-\frac {A}{16}+\frac {B}{16}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{16}-\frac {B}{16}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {7 A}{64}-\frac {5 B}{64}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (4 A +4 B \right )}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {16 A +16 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {2 \left (34 A +32 B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46 A +40 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {\frac {9 A}{2}+\frac {13 B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {57 A}{64}+\frac {5 B}{64}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {\frac {59 A}{2}+\frac {39 B}{2}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {2 \left (\frac {57 A}{4}+\frac {59 B}{8}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2 \left (\frac {175 A}{4}+\frac {135 B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}}{a^{2} c^{5} f}\) |
\(277\) |
| norman |
\(\frac {\frac {\left (4 A -8 B \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f c}+\frac {\left (6 A -2 B \right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f c}-\frac {38 A +2 B}{63 a f c}+\frac {8 \left (2 A -B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}-\frac {2 A \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f c}-\frac {4 \left (92 A -67 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 a f c}-\frac {4 \left (8 A -3 B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f c}+\frac {2 \left (5 A +6 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f c}+\frac {2 \left (17 A +2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{21 a f c}-\frac {\left (34 A +14 B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f c}+\frac {\left (274 A -158 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 a f c}+\frac {\left (104 A -80 B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 a f c}-\frac {\left (116 A +152 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 a f c}-\frac {2 \left (541 A -92 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 a f c}}{a \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) |
\(435\) |
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[In]
int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)
[Out]
16/63*I*(72*I*A*exp(5*I*(f*x+e))-36*I*B*exp(5*I*(f*x+e))+42*B*exp(6*I*(f*x+e))+4*I*A*exp(3*I*(f*x+e))+54*A*exp
(4*I*(f*x+e))-2*I*B*exp(3*I*(f*x+e))-27*B*exp(4*I*(f*x+e))-12*I*A*exp(I*(f*x+e))+24*A*exp(2*I*(f*x+e))+6*I*B*e
xp(I*(f*x+e))-12*B*exp(2*I*(f*x+e))-2*A+B)/(exp(I*(f*x+e))-I)^9/(exp(I*(f*x+e))+I)^3/f/c^5/a^2
Fricas [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07
\[
\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {8 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{6} - 36 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{4} + 15 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{2} + {\left (24 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{4} - 20 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{2} - 14 \, A + 7 \, B\right )} \sin \left (f x + e\right ) + 7 \, A - 14 \, B}{63 \, {\left (3 \, a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3} - {\left (a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )\right )}}
\]
[In]
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="fricas")
[Out]
1/63*(8*(2*A - B)*cos(f*x + e)^6 - 36*(2*A - B)*cos(f*x + e)^4 + 15*(2*A - B)*cos(f*x + e)^2 + (24*(2*A - B)*c
os(f*x + e)^4 - 20*(2*A - B)*cos(f*x + e)^2 - 14*A + 7*B)*sin(f*x + e) + 7*A - 14*B)/(3*a^2*c^5*f*cos(f*x + e)
^5 - 4*a^2*c^5*f*cos(f*x + e)^3 - (a^2*c^5*f*cos(f*x + e)^5 - 4*a^2*c^5*f*cos(f*x + e)^3)*sin(f*x + e))
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5868 vs. \(2 (160) = 320\).
Time = 33.91 (sec) , antiderivative size = 5868, normalized size of antiderivative = 33.53
\[
\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display}
\]
[In]
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**5,x)
[Out]
Piecewise((-126*A*tan(e/2 + f*x/2)**11/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)
**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 +
f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*ta
n(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*
f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) + 378*A*tan(e/2 + f*x/2)**10/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a
**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 -
1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/
2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 +
f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 546*A*tan(e/2 + f*x/2)**9/(63*a**2*c**5*f*ta
n(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c*
*5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*
a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 -
756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 126*A*tan(e/2 + f*
x/2)**8/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2
+ f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*t
an(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c*
*5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c*
*5*f) + 756*A*tan(e/2 + f*x/2)**7/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11
+ 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/
2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2
+ f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan
(e/2 + f*x/2) - 63*a**2*c**5*f) - 588*A*tan(e/2 + f*x/2)**6/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c*
*5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*
a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5
+ 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2
)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 612*A*tan(e/2 + f*x/2)**5/(63*a**2*c**5*f*tan(e/2
+ f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*t
an(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c
**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a
**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) + 900*A*tan(e/2 + f*x/2)**
4/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/
2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2
+ f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*t
an(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f)
- 470*A*tan(e/2 + f*x/2)**3/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*
a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8
+ 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x
/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 +
f*x/2) - 63*a**2*c**5*f) - 78*A*tan(e/2 + f*x/2)**2/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*ta
n(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c*
*5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*
a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 +
378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) + 102*A*tan(e/2 + f*x/2)/(63*a**2*c**5*f*tan(e/2 + f*x/2)**
12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f
*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(
e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f
*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 38*A/(63*a**2*c**5*f*tan(e/2 + f*x
/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/
2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f
*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c
**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 126*B*tan(e/2 + f*x/2)**10/(6
3*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**
10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f
*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e
/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) + 25
2*B*tan(e/2 + f*x/2)**9/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2
*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 22
68*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)*
*4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x
/2) - 63*a**2*c**5*f) - 378*B*tan(e/2 + f*x/2)**8/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e
/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*
f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**
2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378
*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) + 84*B*tan(e/2 + f*x/2)**6/(63*a**2*c**5*f*tan(e/2 + f*x/2)**1
2 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*
x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e
/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*
tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 72*B*tan(e/2 + f*x/2)**5/(63*a**2*c
**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126
*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7
- 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x
/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 324*B*tan(
e/2 + f*x/2)**4/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*
tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*
c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126
*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63
*a**2*c**5*f) + 256*B*tan(e/2 + f*x/2)**3/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x
/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/
2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f
*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c*
*5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 150*B*tan(e/2 + f*x/2)**2/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378
*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9
- 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*
x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2
+ f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) + 12*B*tan(e/2 + f*x/2)/(63*a**2*c**5*f*tan(
e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5
*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a*
*2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 7
56*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 2*B/(63*a**2*c**5*f*
tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*
c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 226
8*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3
- 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f), Ne(f, 0)), (x*(A
+ B*sin(e))/((a*sin(e) + a)**2*(-c*sin(e) + c)**5), True))
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (168) = 336\).
Time = 0.30 (sec) , antiderivative size = 998, normalized size of antiderivative = 5.70
\[
\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display}
\]
[In]
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="maxima")
[Out]
-2/63*(A*(51*sin(f*x + e)/(cos(f*x + e) + 1) - 39*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 235*sin(f*x + e)^3/(co
s(f*x + e) + 1)^3 + 450*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 306*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 294*si
n(f*x + e)^6/(cos(f*x + e) + 1)^6 + 378*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 63*sin(f*x + e)^8/(cos(f*x + e)
+ 1)^8 - 273*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 189*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 63*sin(f*x + e)
^11/(cos(f*x + e) + 1)^11 - 19)/(a^2*c^5 - 6*a^2*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^2*c^5*sin(f*x + e)
^2/(cos(f*x + e) + 1)^2 - 2*a^2*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 27*a^2*c^5*sin(f*x + e)^4/(cos(f*x +
e) + 1)^4 + 36*a^2*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 36*a^2*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 +
27*a^2*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2*a^2*c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 12*a^2*c^5*s
in(f*x + e)^10/(cos(f*x + e) + 1)^10 + 6*a^2*c^5*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - a^2*c^5*sin(f*x + e)^
12/(cos(f*x + e) + 1)^12) + B*(6*sin(f*x + e)/(cos(f*x + e) + 1) - 75*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 12
8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 162*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 36*sin(f*x + e)^5/(cos(f*x +
e) + 1)^5 + 42*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 189*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 126*sin(f*x +
e)^9/(cos(f*x + e) + 1)^9 - 63*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1)/(a^2*c^5 - 6*a^2*c^5*sin(f*x + e)/(c
os(f*x + e) + 1) + 12*a^2*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 2*a^2*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1
)^3 - 27*a^2*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 36*a^2*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 36*a^2
*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 27*a^2*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2*a^2*c^5*sin(f*x
+ e)^9/(cos(f*x + e) + 1)^9 - 12*a^2*c^5*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 6*a^2*c^5*sin(f*x + e)^11/(co
s(f*x + e) + 1)^11 - a^2*c^5*sin(f*x + e)^12/(cos(f*x + e) + 1)^12))/f
Giac [A] (verification not implemented)
none
Time = 0.53 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.90
\[
\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {\frac {21 \, {\left (21 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 24 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 19 \, A - 13 \, B\right )}}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {3591 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 315 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 19656 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 756 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 56196 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 4200 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 95760 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 11340 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 107730 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 14994 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 79464 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 13356 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 38484 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6768 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10944 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2196 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1615 \, A - 209 \, B}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{2016 \, f}
\]
[In]
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="giac")
[Out]
-1/2016*(21*(21*A*tan(1/2*f*x + 1/2*e)^2 - 15*B*tan(1/2*f*x + 1/2*e)^2 + 36*A*tan(1/2*f*x + 1/2*e) - 24*B*tan(
1/2*f*x + 1/2*e) + 19*A - 13*B)/(a^2*c^5*(tan(1/2*f*x + 1/2*e) + 1)^3) + (3591*A*tan(1/2*f*x + 1/2*e)^8 + 315*
B*tan(1/2*f*x + 1/2*e)^8 - 19656*A*tan(1/2*f*x + 1/2*e)^7 + 756*B*tan(1/2*f*x + 1/2*e)^7 + 56196*A*tan(1/2*f*x
+ 1/2*e)^6 - 4200*B*tan(1/2*f*x + 1/2*e)^6 - 95760*A*tan(1/2*f*x + 1/2*e)^5 + 11340*B*tan(1/2*f*x + 1/2*e)^5
+ 107730*A*tan(1/2*f*x + 1/2*e)^4 - 14994*B*tan(1/2*f*x + 1/2*e)^4 - 79464*A*tan(1/2*f*x + 1/2*e)^3 + 13356*B*
tan(1/2*f*x + 1/2*e)^3 + 38484*A*tan(1/2*f*x + 1/2*e)^2 - 6768*B*tan(1/2*f*x + 1/2*e)^2 - 10944*A*tan(1/2*f*x
+ 1/2*e) + 2196*B*tan(1/2*f*x + 1/2*e) + 1615*A - 209*B)/(a^2*c^5*(tan(1/2*f*x + 1/2*e) - 1)^9))/f
Mupad [B] (verification not implemented)
Time = 13.10 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.93
\[
\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {2\,\left (7\,A-14\,B-14\,A\,\sin \left (e+f\,x\right )+7\,B\,\sin \left (e+f\,x\right )+30\,A\,{\cos \left (e+f\,x\right )}^2-76\,A\,{\cos \left (e+f\,x\right )}^3-72\,A\,{\cos \left (e+f\,x\right )}^4+57\,A\,{\cos \left (e+f\,x\right )}^5+16\,A\,{\cos \left (e+f\,x\right )}^6-15\,B\,{\cos \left (e+f\,x\right )}^2-4\,B\,{\cos \left (e+f\,x\right )}^3+36\,B\,{\cos \left (e+f\,x\right )}^4+3\,B\,{\cos \left (e+f\,x\right )}^5-8\,B\,{\cos \left (e+f\,x\right )}^6-40\,A\,{\cos \left (e+f\,x\right )}^2\,\sin \left (e+f\,x\right )+76\,A\,{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )+48\,A\,{\cos \left (e+f\,x\right )}^4\,\sin \left (e+f\,x\right )-19\,A\,{\cos \left (e+f\,x\right )}^5\,\sin \left (e+f\,x\right )+20\,B\,{\cos \left (e+f\,x\right )}^2\,\sin \left (e+f\,x\right )+4\,B\,{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )-24\,B\,{\cos \left (e+f\,x\right )}^4\,\sin \left (e+f\,x\right )-B\,{\cos \left (e+f\,x\right )}^5\,\sin \left (e+f\,x\right )\right )}{63\,a^2\,c^5\,f\,\left (8\,{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )-2\,{\cos \left (e+f\,x\right )}^5\,\sin \left (e+f\,x\right )-8\,{\cos \left (e+f\,x\right )}^3+6\,{\cos \left (e+f\,x\right )}^5\right )}
\]
[In]
int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^5),x)
[Out]
(2*(7*A - 14*B - 14*A*sin(e + f*x) + 7*B*sin(e + f*x) + 30*A*cos(e + f*x)^2 - 76*A*cos(e + f*x)^3 - 72*A*cos(e
+ f*x)^4 + 57*A*cos(e + f*x)^5 + 16*A*cos(e + f*x)^6 - 15*B*cos(e + f*x)^2 - 4*B*cos(e + f*x)^3 + 36*B*cos(e
+ f*x)^4 + 3*B*cos(e + f*x)^5 - 8*B*cos(e + f*x)^6 - 40*A*cos(e + f*x)^2*sin(e + f*x) + 76*A*cos(e + f*x)^3*si
n(e + f*x) + 48*A*cos(e + f*x)^4*sin(e + f*x) - 19*A*cos(e + f*x)^5*sin(e + f*x) + 20*B*cos(e + f*x)^2*sin(e +
f*x) + 4*B*cos(e + f*x)^3*sin(e + f*x) - 24*B*cos(e + f*x)^4*sin(e + f*x) - B*cos(e + f*x)^5*sin(e + f*x)))/(
63*a^2*c^5*f*(8*cos(e + f*x)^3*sin(e + f*x) - 2*cos(e + f*x)^5*sin(e + f*x) - 8*cos(e + f*x)^3 + 6*cos(e + f*x
)^5))