\(\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx\) [289]
Optimal result
Integrand size = 26, antiderivative size = 167 \[
\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f}
\]
[Out]
1/11*sec(f*x+e)^5/a^3/f/(c^2-c^2*sin(f*x+e))^3+8/99*sec(f*x+e)^5/a^3/f/(c^3-c^3*sin(f*x+e))^2+8/99*sec(f*x+e)^
5/a^3/f/(c^6-c^6*sin(f*x+e))+16/33*tan(f*x+e)/a^3/c^6/f+32/99*tan(f*x+e)^3/a^3/c^6/f+16/165*tan(f*x+e)^5/a^3/c
^6/f
Rubi [A] (verified)
Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751, 3852}
\[
\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}
\]
[In]
Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]
[Out]
Sec[e + f*x]^5/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^3 - c^3*Sin[e + f*x])^2
) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (16*Tan[e + f*x])/(33*a^3*c^6*f) + (32*Tan[e + f*
x]^3)/(99*a^3*c^6*f) + (16*Tan[e + f*x]^5)/(165*a^3*c^6*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 2815
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((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)}{(c-c \sin (e+f x))^3} \, dx}{a^3 c^3} \\ & = \frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{11 a^3 c^4} \\ & = \frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {56 \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{99 a^3 c^5} \\ & = \frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \int \sec ^6(e+f x) \, dx}{33 a^3 c^6} \\ & = \frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}-\frac {16 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{33 a^3 c^6 f} \\ & = \frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f} \\
\end{align*}
Mathematica [A] (verified)
Time = 3.30 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.40
\[
\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (420053810 \cos (e+f x)+17301504 \cos (2 (e+f x))+129834814 \cos (3 (e+f x))+13107200 \cos (4 (e+f x))-38186710 \cos (5 (e+f x))+2621440 \cos (6 (e+f x))-22912026 \cos (7 (e+f x))-262144 \cos (8 (e+f x))+28835840 \sin (e+f x)-252032286 \sin (2 (e+f x))+8912896 \sin (3 (e+f x))-190933550 \sin (4 (e+f x))-2621440 \sin (5 (e+f x))-38186710 \sin (6 (e+f x))-1572864 \sin (7 (e+f x))+3818671 \sin (8 (e+f x)))}{129761280 a^3 c^6 f (-1+\sin (e+f x))^6 (1+\sin (e+f x))^3}
\]
[In]
Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]
[Out]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(420053810*Cos[e + f*x] + 1730150
4*Cos[2*(e + f*x)] + 129834814*Cos[3*(e + f*x)] + 13107200*Cos[4*(e + f*x)] - 38186710*Cos[5*(e + f*x)] + 2621
440*Cos[6*(e + f*x)] - 22912026*Cos[7*(e + f*x)] - 262144*Cos[8*(e + f*x)] + 28835840*Sin[e + f*x] - 252032286
*Sin[2*(e + f*x)] + 8912896*Sin[3*(e + f*x)] - 190933550*Sin[4*(e + f*x)] - 2621440*Sin[5*(e + f*x)] - 3818671
0*Sin[6*(e + f*x)] - 1572864*Sin[7*(e + f*x)] + 3818671*Sin[8*(e + f*x)]))/(129761280*a^3*c^6*f*(-1 + Sin[e +
f*x])^6*(1 + Sin[e + f*x])^3)
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 4.39 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.74
| | |
| method | result | size |
| | |
| risch |
\(-\frac {256 \left (-10 \,{\mathrm e}^{3 i \left (f x +e \right )}+i+34 \,{\mathrm e}^{5 i \left (f x +e \right )}+110 \,{\mathrm e}^{7 i \left (f x +e \right )}-6 \,{\mathrm e}^{i \left (f x +e \right )}-50 i {\mathrm e}^{4 i \left (f x +e \right )}-66 i {\mathrm e}^{6 i \left (f x +e \right )}-10 i {\mathrm e}^{2 i \left (f x +e \right )}\right )}{495 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} f \,c^{6} a^{3}}\) |
\(123\) |
| parallelrisch |
\(\frac {-\frac {50}{99}+\frac {2 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}+\frac {1334 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{495}-10 \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {298 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}-2 \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {142 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {106 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {1510 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99}-\frac {166 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {74 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9}-\frac {1226 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{33}+\frac {94 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99}-\frac {22 \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+6 \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f \,c^{6} a^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) |
\(233\) |
| derivativedivides |
\(\frac {-\frac {1}{40 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {7}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {5}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {37}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {106}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {23}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {33}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {217}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {623}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {169}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {365}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {303}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {219}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{3} c^{6} f}\) |
\(253\) |
| default |
\(\frac {-\frac {1}{40 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {7}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {5}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {37}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {106}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {23}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {33}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {217}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {623}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {169}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {365}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {303}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {219}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a^{3} c^{6} f}\) |
\(253\) |
| norman |
\(\frac {\frac {106 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {50}{99 a c f}-\frac {2 \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {10 \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {6 \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {22 \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{33 a c f}-\frac {142 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {166 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}+\frac {298 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}+\frac {94 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99 a c f}+\frac {2 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}+\frac {74 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9 a c f}+\frac {1510 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99 a c f}-\frac {1226 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99 a c f}+\frac {1334 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{495 a c f}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) |
\(374\) |
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[In]
int(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x,method=_RETURNVERBOSE)
[Out]
-256/495*(-10*exp(3*I*(f*x+e))+I+34*exp(5*I*(f*x+e))+110*exp(7*I*(f*x+e))-6*exp(I*(f*x+e))-50*I*exp(4*I*(f*x+e
))-66*I*exp(6*I*(f*x+e))-10*I*exp(2*I*(f*x+e)))/(exp(I*(f*x+e))-I)^11/(exp(I*(f*x+e))+I)^5/f/c^6/a^3
Fricas [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98
\[
\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {128 \, \cos \left (f x + e\right )^{8} - 576 \, \cos \left (f x + e\right )^{6} + 240 \, \cos \left (f x + e\right )^{4} + 56 \, \cos \left (f x + e\right )^{2} + 8 \, {\left (48 \, \cos \left (f x + e\right )^{6} - 40 \, \cos \left (f x + e\right )^{4} - 14 \, \cos \left (f x + e\right )^{2} - 9\right )} \sin \left (f x + e\right ) + 27}{495 \, {\left (3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5} - {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5}\right )} \sin \left (f x + e\right )\right )}}
\]
[In]
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="fricas")
[Out]
1/495*(128*cos(f*x + e)^8 - 576*cos(f*x + e)^6 + 240*cos(f*x + e)^4 + 56*cos(f*x + e)^2 + 8*(48*cos(f*x + e)^6
- 40*cos(f*x + e)^4 - 14*cos(f*x + e)^2 - 9)*sin(f*x + e) + 27)/(3*a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos
(f*x + e)^5 - (a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos(f*x + e)^5)*sin(f*x + e))
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5661 vs. \(2 (151) = 302\).
Time = 66.43 (sec) , antiderivative size = 5661, normalized size of antiderivative = 33.90
\[
\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**6,x)
[Out]
Piecewise((-990*tan(e/2 + f*x/2)**15/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)
**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e
/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3
*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 -
16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x
/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 2970*ta
n(e/2 + f*x/2)**14/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c
**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 +
16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 +
f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*
tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3
*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 3630*tan(e/2 + f*x/2)**13
/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*
x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*
tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*
a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**
5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 +
f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 4950*tan(e/2 + f*x/2)**12/(495*a**3*c**6*f*
tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a*
*3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**
11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/
2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**
6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a
**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 9834*tan(e/2 + f*x/2)**11/(495*a**3*c**6*f*tan(e/2 + f*x/2)**
16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2
+ f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c*
*6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32
670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/
2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2
+ f*x/2) - 495*a**3*c**6*f) + 66*tan(e/2 + f*x/2)**10/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*
f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750
*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2
)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(
e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c*
*6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3
*c**6*f) - 23430*tan(e/2 + f*x/2)**9/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)
**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e
/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3
*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 -
16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x
/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 17490*t
an(e/2 + f*x/2)**8/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c
**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 +
16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 +
f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*
tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3
*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 4070*tan(e/2 + f*x/2)**7/
(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x
/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*t
an(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a
**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5
+ 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f
*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 16434*tan(e/2 + f*x/2)**6/(495*a**3*c**6*f*t
an(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**
3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**1
1 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2
+ f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6
*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a*
*3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 1334*tan(e/2 + f*x/2)**5/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16
- 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 +
f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6
*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 3267
0*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)
**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 +
f*x/2) - 495*a**3*c**6*f) + 7550*tan(e/2 + f*x/2)**4/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f
*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*
a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)
**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e
/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**
6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*
c**6*f) - 6130*tan(e/2 + f*x/2)**3/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**
15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2
+ f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c
**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16
830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2
)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 470*tan(e
/2 + f*x/2)**2/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*
f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 1683
0*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/
2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(
e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**
6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 510*tan(e/2 + f*x/2)/(495*a**
3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14
+ 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 +
f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6
*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750
*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2
+ 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 250/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**
3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13
- 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2
+ f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6
*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*
a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 4
95*a**3*c**6*f), Ne(f, 0)), (x/((a*sin(e) + a)**3*(-c*sin(e) + c)**6), True))
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (158) = 316\).
Time = 0.23 (sec) , antiderivative size = 703, normalized size of antiderivative = 4.21
\[
\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="maxima")
[Out]
-2/495*(255*sin(f*x + e)/(cos(f*x + e) + 1) + 235*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 3065*sin(f*x + e)^3/(c
os(f*x + e) + 1)^3 + 3775*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 667*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 8217
*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2035*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 8745*sin(f*x + e)^8/(cos(f*x
+ e) + 1)^8 - 11715*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 33*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 4917*sin
(f*x + e)^11/(cos(f*x + e) + 1)^11 - 2475*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 1815*sin(f*x + e)^13/(cos(f*
x + e) + 1)^13 + 1485*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - 495*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - 125)
/((a^3*c^6 - 6*a^3*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a
^3*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 50*a^3*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 34*a^3*c^6*sin(f
*x + e)^5/(cos(f*x + e) + 1)^5 + 66*a^3*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 110*a^3*c^6*sin(f*x + e)^7/(
cos(f*x + e) + 1)^7 + 110*a^3*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 66*a^3*c^6*sin(f*x + e)^10/(cos(f*x +
e) + 1)^10 - 34*a^3*c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 50*a^3*c^6*sin(f*x + e)^12/(cos(f*x + e) + 1)^
12 - 10*a^3*c^6*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 10*a^3*c^6*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 + 6*a
^3*c^6*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - a^3*c^6*sin(f*x + e)^16/(cos(f*x + e) + 1)^16)*f)
Giac [A] (verification not implemented)
none
Time = 0.36 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.37
\[
\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=-\frac {\frac {33 \, {\left (555 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1920 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2710 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1760 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 463\right )}}{a^{3} c^{6} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {108405 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 784080 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 2901195 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 6652800 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 10407474 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 11435424 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 8949270 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4899840 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1816265 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 411664 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 47279}{a^{3} c^{6} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}}}{63360 \, f}
\]
[In]
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="giac")
[Out]
-1/63360*(33*(555*tan(1/2*f*x + 1/2*e)^4 + 1920*tan(1/2*f*x + 1/2*e)^3 + 2710*tan(1/2*f*x + 1/2*e)^2 + 1760*ta
n(1/2*f*x + 1/2*e) + 463)/(a^3*c^6*(tan(1/2*f*x + 1/2*e) + 1)^5) + (108405*tan(1/2*f*x + 1/2*e)^10 - 784080*ta
n(1/2*f*x + 1/2*e)^9 + 2901195*tan(1/2*f*x + 1/2*e)^8 - 6652800*tan(1/2*f*x + 1/2*e)^7 + 10407474*tan(1/2*f*x
+ 1/2*e)^6 - 11435424*tan(1/2*f*x + 1/2*e)^5 + 8949270*tan(1/2*f*x + 1/2*e)^4 - 4899840*tan(1/2*f*x + 1/2*e)^3
+ 1816265*tan(1/2*f*x + 1/2*e)^2 - 411664*tan(1/2*f*x + 1/2*e) + 47279)/(a^3*c^6*(tan(1/2*f*x + 1/2*e) - 1)^1
1))/f
Mupad [B] (verification not implemented)
Time = 8.15 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.11
\[
\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=-\frac {\frac {2\,\sin \left (e+f\,x\right )}{9}+\frac {2\,\cos \left (2\,e+2\,f\,x\right )}{15}+\frac {10\,\cos \left (4\,e+4\,f\,x\right )}{99}+\frac {2\,\cos \left (6\,e+6\,f\,x\right )}{99}-\frac {\cos \left (8\,e+8\,f\,x\right )}{495}+\frac {34\,\sin \left (3\,e+3\,f\,x\right )}{495}-\frac {2\,\sin \left (5\,e+5\,f\,x\right )}{99}-\frac {2\,\sin \left (7\,e+7\,f\,x\right )}{165}}{a^3\,c^6\,f\,\left (\frac {5\,\cos \left (5\,e+5\,f\,x\right )}{64}-\frac {17\,\cos \left (3\,e+3\,f\,x\right )}{64}-\frac {55\,\cos \left (e+f\,x\right )}{64}+\frac {3\,\cos \left (7\,e+7\,f\,x\right )}{64}+\frac {33\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {25\,\sin \left (4\,e+4\,f\,x\right )}{64}+\frac {5\,\sin \left (6\,e+6\,f\,x\right )}{64}-\frac {\sin \left (8\,e+8\,f\,x\right )}{128}\right )}
\]
[In]
int(1/((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^6),x)
[Out]
-((2*sin(e + f*x))/9 + (2*cos(2*e + 2*f*x))/15 + (10*cos(4*e + 4*f*x))/99 + (2*cos(6*e + 6*f*x))/99 - cos(8*e
+ 8*f*x)/495 + (34*sin(3*e + 3*f*x))/495 - (2*sin(5*e + 5*f*x))/99 - (2*sin(7*e + 7*f*x))/165)/(a^3*c^6*f*((5*
cos(5*e + 5*f*x))/64 - (17*cos(3*e + 3*f*x))/64 - (55*cos(e + f*x))/64 + (3*cos(7*e + 7*f*x))/64 + (33*sin(2*e
+ 2*f*x))/64 + (25*sin(4*e + 4*f*x))/64 + (5*sin(6*e + 6*f*x))/64 - sin(8*e + 8*f*x)/128))