Integrand size = 26, antiderivative size = 167 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f} \] Output:
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
Time = 11.50 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-411950 \cos (e+f x)+1081344 \cos (2 (e+f x))-127330 \cos (3 (e+f x))+819200 \cos (4 (e+f x))+37450 \cos (5 (e+f x))+163840 \cos (6 (e+f x))+22470 \cos (7 (e+f x))-16384 \cos (8 (e+f x))+1802240 \sin (e+f x)+247170 \sin (2 (e+f x))+557056 \sin (3 (e+f x))+187250 \sin (4 (e+f x))-163840 \sin (5 (e+f x))+37450 \sin (6 (e+f x))-98304 \sin (7 (e+f x))-3745 \sin (8 (e+f x)))}{8110080 f (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \] Input:
Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]
Output:
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 ])*(-411950*Cos[e + f*x] + 1081344*Cos[2*(e + f*x)] - 127330*Cos[3*(e + f* x)] + 819200*Cos[4*(e + f*x)] + 37450*Cos[5*(e + f*x)] + 163840*Cos[6*(e + f*x)] + 22470*Cos[7*(e + f*x)] - 16384*Cos[8*(e + f*x)] + 1802240*Sin[e + f*x] + 247170*Sin[2*(e + f*x)] + 557056*Sin[3*(e + f*x)] + 187250*Sin[4*( e + f*x)] - 163840*Sin[5*(e + f*x)] + 37450*Sin[6*(e + f*x)] - 98304*Sin[7 *(e + f*x)] - 3745*Sin[8*(e + f*x)]))/(8110080*f*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6)
Time = 0.68 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3042, 3215, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 4254, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^3}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\cos (e+f x)^6 (c-c \sin (e+f x))^3}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {8 \int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^2}dx}{11 c}+\frac {\sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 \int \frac {1}{\cos (e+f x)^6 (c-c \sin (e+f x))^2}dx}{11 c}+\frac {\sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {8 \left (\frac {7 \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)}dx}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {\sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 \left (\frac {7 \int \frac {1}{\cos (e+f x)^6 (c-c \sin (e+f x))}dx}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {\sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {8 \left (\frac {7 \left (\frac {6 \int \sec ^6(e+f x)dx}{7 c}+\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}\right )}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {\sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 \left (\frac {7 \left (\frac {6 \int \csc \left (e+f x+\frac {\pi }{2}\right )^6dx}{7 c}+\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}\right )}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {\sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {8 \left (\frac {7 \left (\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}-\frac {6 \int \left (\tan ^4(e+f x)+2 \tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{7 c f}\right )}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {\sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}+\frac {8 \left (\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}+\frac {7 \left (\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}-\frac {6 \left (-\frac {1}{5} \tan ^5(e+f x)-\frac {2}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{7 c f}\right )}{9 c}\right )}{11 c}}{a^3 c^3}\) |
Input:
Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]
Output:
(Sec[e + f*x]^5/(11*f*(c - c*Sin[e + f*x])^3) + (8*(Sec[e + f*x]^5/(9*f*(c - c*Sin[e + f*x])^2) + (7*(Sec[e + f*x]^5/(7*f*(c - c*Sin[e + f*x])) - (6 *(-Tan[e + f*x] - (2*Tan[e + f*x]^3)/3 - Tan[e + f*x]^5/5))/(7*c*f)))/(9*c )))/(11*c))/(a^3*c^3)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[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[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[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] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains complex when optimal does not.
Time = 4.75 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {256 \left (110 \,{\mathrm e}^{7 i \left (f x +e \right )}+34 \,{\mathrm e}^{5 i \left (f x +e \right )}-10 \,{\mathrm e}^{3 i \left (f x +e \right )}+i-6 \,{\mathrm e}^{i \left (f x +e \right )}-50 i {\mathrm e}^{4 i \left (f x +e \right )}-66 i {\mathrm e}^{6 i \left (f x +e \right )}-10 i {\mathrm e}^{2 i \left (f x +e \right )}\right )}{495 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} f \,a^{3} c^{6}}\) | \(123\) |
| parallelrisch | \(\frac {-\frac {50}{99}-\frac {22 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{3}+\frac {298 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{15}-10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}-\frac {1226 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{99}+\frac {1510 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{99}-\frac {166 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{5}+\frac {74 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{9}-\frac {142 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{3}+\frac {1334 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{495}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{15}+\frac {106 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3}+6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{33}+\frac {94 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{99}}{f \,a^{3} c^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) | \(233\) |
| 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 | \(\frac {\frac {106 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3 a f c}-\frac {50}{99 a c f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{f c a}-\frac {10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{f c a}+\frac {6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{f c a}-\frac {22 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{3 f c a}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{33 a c f}-\frac {142 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{3 a f c}-\frac {166 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{5 a f c}+\frac {298 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{15 a f c}+\frac {94 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{99 a c f}+\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{15 a f c}+\frac {74 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{9 a f c}+\frac {1510 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{99 a f c}-\frac {1226 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{99 a c f}+\frac {1334 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{495 a f c}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) | \(374\) |
Input:
int(1/(a+sin(f*x+e)*a)^3/(c-c*sin(f*x+e))^6,x,method=_RETURNVERBOSE)
Output:
-256/495*(110*exp(7*I*(f*x+e))+34*exp(5*I*(f*x+e))-10*exp(3*I*(f*x+e))+I-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/a^3/c^6
Time = 0.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {128 \, \cos \left (f x + e\right )^{8} - 576 \, \cos \left (f x + e\right )^{6} + 240 \, \cos \left (f x + e\right )^{4} + 56 \, \cos \left (f x + e\right )^{2} + 8 \, {\left (48 \, \cos \left (f x + e\right )^{6} - 40 \, \cos \left (f x + e\right )^{4} - 14 \, \cos \left (f x + e\right )^{2} - 9\right )} \sin \left (f x + e\right ) + 27}{495 \, {\left (3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5} - {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5}\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="fricas")
Output:
1/495*(128*cos(f*x + e)^8 - 576*cos(f*x + e)^6 + 240*cos(f*x + e)^4 + 56*c os(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))
Leaf count of result is larger than twice the leaf count of optimal. 5661 vs. \(2 (151) = 302\).
Time = 66.83 (sec) , antiderivative size = 5661, normalized size of antiderivative = 33.90 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**6,x)
Output:
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 - 1683 0*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*tan(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*t an(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)**1...
Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (158) = 316\).
Time = 0.06 (sec) , antiderivative size = 703, normalized size of antiderivative = 4.21 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="maxima")
Output:
-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/(cos(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*s in(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)^1 2 - 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)
Time = 0.17 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=-\frac {\frac {33 \, {\left (555 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1920 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2710 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1760 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 463\right )}}{a^{3} c^{6} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {108405 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 784080 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 2901195 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 6652800 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 10407474 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 11435424 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 8949270 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4899840 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1816265 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 411664 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 47279}{a^{3} c^{6} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}}}{63360 \, f} \] Input:
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm="giac")
Output:
-1/63360*(33*(555*tan(1/2*f*x + 1/2*e)^4 + 1920*tan(1/2*f*x + 1/2*e)^3 + 2 710*tan(1/2*f*x + 1/2*e)^2 + 1760*tan(1/2*f*x + 1/2*e) + 463)/(a^3*c^6*(ta n(1/2*f*x + 1/2*e) + 1)^5) + (108405*tan(1/2*f*x + 1/2*e)^10 - 784080*tan( 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 + 1 816265*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)^11))/f
Time = 18.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=-\frac {\frac {2\,\sin \left (e+f\,x\right )}{9}+\frac {2\,\cos \left (2\,e+2\,f\,x\right )}{15}+\frac {10\,\cos \left (4\,e+4\,f\,x\right )}{99}+\frac {2\,\cos \left (6\,e+6\,f\,x\right )}{99}-\frac {\cos \left (8\,e+8\,f\,x\right )}{495}+\frac {34\,\sin \left (3\,e+3\,f\,x\right )}{495}-\frac {2\,\sin \left (5\,e+5\,f\,x\right )}{99}-\frac {2\,\sin \left (7\,e+7\,f\,x\right )}{165}}{a^3\,c^6\,f\,\left (\frac {5\,\cos \left (5\,e+5\,f\,x\right )}{64}-\frac {17\,\cos \left (3\,e+3\,f\,x\right )}{64}-\frac {55\,\cos \left (e+f\,x\right )}{64}+\frac {3\,\cos \left (7\,e+7\,f\,x\right )}{64}+\frac {33\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {25\,\sin \left (4\,e+4\,f\,x\right )}{64}+\frac {5\,\sin \left (6\,e+6\,f\,x\right )}{64}-\frac {\sin \left (8\,e+8\,f\,x\right )}{128}\right )} \] Input:
int(1/((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^6),x)
Output:
-((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))/4 95 - (2*sin(5*e + 5*f*x))/99 - (2*sin(7*e + 7*f*x))/165)/(a^3*c^6*f*((5*co s(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))
Time = 0.17 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {-40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{7}+120 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6}-40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}-200 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}+200 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-120 \cos \left (f x +e \right ) \sin \left (f x +e \right )+40 \cos \left (f x +e \right )+128 \sin \left (f x +e \right )^{8}-384 \sin \left (f x +e \right )^{7}+64 \sin \left (f x +e \right )^{6}+832 \sin \left (f x +e \right )^{5}-720 \sin \left (f x +e \right )^{4}-400 \sin \left (f x +e \right )^{3}+680 \sin \left (f x +e \right )^{2}-120 \sin \left (f x +e \right )-125}{495 \cos \left (f x +e \right ) a^{3} c^{6} f \left (\sin \left (f x +e \right )^{7}-3 \sin \left (f x +e \right )^{6}+\sin \left (f x +e \right )^{5}+5 \sin \left (f x +e \right )^{4}-5 \sin \left (f x +e \right )^{3}-\sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )-1\right )} \] Input:
int(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x)
Output:
( - 40*cos(e + f*x)*sin(e + f*x)**7 + 120*cos(e + f*x)*sin(e + f*x)**6 - 4 0*cos(e + f*x)*sin(e + f*x)**5 - 200*cos(e + f*x)*sin(e + f*x)**4 + 200*co s(e + f*x)*sin(e + f*x)**3 + 40*cos(e + f*x)*sin(e + f*x)**2 - 120*cos(e + f*x)*sin(e + f*x) + 40*cos(e + f*x) + 128*sin(e + f*x)**8 - 384*sin(e + f *x)**7 + 64*sin(e + f*x)**6 + 832*sin(e + f*x)**5 - 720*sin(e + f*x)**4 - 400*sin(e + f*x)**3 + 680*sin(e + f*x)**2 - 120*sin(e + f*x) - 125)/(495*c os(e + f*x)*a**3*c**6*f*(sin(e + f*x)**7 - 3*sin(e + f*x)**6 + sin(e + f*x )**5 + 5*sin(e + f*x)**4 - 5*sin(e + f*x)**3 - sin(e + f*x)**2 + 3*sin(e + f*x) - 1))