\(\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx\) [288]

Optimal result

Integrand size = 26, antiderivative size = 131 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f} \]

[Out]

1/9*sec(f*x+e)^5/a^3/c^3/f/(c-c*sin(f*x+e))^2+1/9*sec(f*x+e)^5/a^3/f/(c^5-c^5*sin(f*x+e))+2/3*tan(f*x+e)/a^3/c
^5/f+4/9*tan(f*x+e)^3/a^3/c^5/f+2/15*tan(f*x+e)^5/a^3/c^5/f

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751, 3852} \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2} \]

[In]

Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5),x]

[Out]

Sec[e + f*x]^5/(9*a^3*c^3*f*(c - c*Sin[e + f*x])^2) + Sec[e + f*x]^5/(9*a^3*f*(c^5 - c^5*Sin[e + f*x])) + (2*T
an[e + f*x])/(3*a^3*c^5*f) + (4*Tan[e + f*x]^3)/(9*a^3*c^5*f) + (2*Tan[e + f*x]^5)/(15*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 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))^2} \, dx}{a^3 c^3} \\ & = \frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {7 \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{9 a^3 c^4} \\ & = \frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \int \sec ^6(e+f x) \, dx}{3 a^3 c^5} \\ & = \frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac {2 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{3 a^3 c^5 f} \\ & = \frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.52 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (19843425 \cos (e+f x)+1310720 \cos (2 (e+f x))+8378335 \cos (3 (e+f x))+1048576 \cos (4 (e+f x))+440965 \cos (5 (e+f x))+262144 \cos (6 (e+f x))-440965 \cos (7 (e+f x))+2949120 \sin (e+f x)-8819300 \sin (2 (e+f x))+1245184 \sin (3 (e+f x))-7055440 \sin (4 (e+f x))+65536 \sin (5 (e+f x))-1763860 \sin (6 (e+f x))-65536 \sin (7 (e+f x)))}{11796480 a^3 c^5 f (-1+\sin (e+f x))^5 (1+\sin (e+f x))^3} \]

[In]

Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5),x]

[Out]

-1/11796480*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(19843425*Cos[e + f*x
] + 1310720*Cos[2*(e + f*x)] + 8378335*Cos[3*(e + f*x)] + 1048576*Cos[4*(e + f*x)] + 440965*Cos[5*(e + f*x)] +
262144*Cos[6*(e + f*x)] - 440965*Cos[7*(e + f*x)] + 2949120*Sin[e + f*x] - 8819300*Sin[2*(e + f*x)] + 1245184
*Sin[3*(e + f*x)] - 7055440*Sin[4*(e + f*x)] + 65536*Sin[5*(e + f*x)] - 1763860*Sin[6*(e + f*x)] - 65536*Sin[7
*(e + f*x)]))/(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 = 3.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {32 i \left (-20 i {\mathrm e}^{5 i \left (f x +e \right )}+45 \,{\mathrm e}^{6 i \left (f x +e \right )}-16 i {\mathrm e}^{3 i \left (f x +e \right )}+19 \,{\mathrm e}^{4 i \left (f x +e \right )}-4 i {\mathrm e}^{i \left (f x +e \right )}+{\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{45 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} f \,c^{5} a^{3}}\)	\(110\)
parallelrisch	\(\frac {-\frac {4}{9}-\frac {32 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {538 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{45}+4 \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {4 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {46 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}+\frac {236 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}-\frac {20 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9}-\frac {64 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}-\frac {344 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}-\frac {76 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{9}+\frac {32 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9}-2 \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f \,c^{5} a^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(207\)
derivativedivides	\(\frac {-\frac {4}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {49}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {49}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {35}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {49}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {51}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {99}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {13}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {9}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {29}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{3} c^{5} f}\)	\(223\)
default	\(\frac {-\frac {4}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {49}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {49}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {35}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {49}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {51}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {99}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {13}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {9}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {29}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{3} c^{5} f}\)	\(223\)
norman	\(\frac {\frac {4 \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {344 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}-\frac {4}{9 a c f}-\frac {2 \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {32 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9 a c f}-\frac {4 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{9 a c f}-\frac {76 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9 a c f}-\frac {32 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}+\frac {46 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}-\frac {20 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9 a c f}+\frac {538 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{45 a c f}-\frac {64 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}+\frac {236 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(330\)

[In]

int(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

-32/45*I*(-20*I*exp(5*I*(f*x+e))+45*exp(6*I*(f*x+e))-16*I*exp(3*I*(f*x+e))+19*exp(4*I*(f*x+e))-4*I*exp(I*(f*x+
e))+exp(2*I*(f*x+e))-1)/(exp(I*(f*x+e))-I)^9/(exp(I*(f*x+e))+I)^5/f/c^5/a^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=-\frac {32 \, \cos \left (f x + e\right )^{6} - 16 \, \cos \left (f x + e\right )^{4} - 4 \, \cos \left (f x + e\right )^{2} - {\left (16 \, \cos \left (f x + e\right )^{6} - 24 \, \cos \left (f x + e\right )^{4} - 10 \, \cos \left (f x + e\right )^{2} - 7\right )} \sin \left (f x + e\right ) - 2}{45 \, {\left (a^{3} c^{5} f \cos \left (f x + e\right )^{7} + 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) - 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5}\right )}} \]

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

-1/45*(32*cos(f*x + e)^6 - 16*cos(f*x + e)^4 - 4*cos(f*x + e)^2 - (16*cos(f*x + e)^6 - 24*cos(f*x + e)^4 - 10*
cos(f*x + e)^2 - 7)*sin(f*x + e) - 2)/(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. 4335 vs. \(2 (117) = 234\).

Time = 36.76 (sec) , antiderivative size = 4335, normalized size of antiderivative = 33.09 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**5,x)

[Out]

Piecewise((-90*tan(e/2 + f*x/2)**13/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**1
3 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x
/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/
2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*ta
n(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) +
180*tan(e/2 + f*x/2)**12/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3
*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 9
00*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)*
*6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x
/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 60*tan(e/2
+ f*x/2)**11/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(
e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5
*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**
3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*
a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 480*tan(e/2 + f*x/2)**1
0/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2
)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2
+ f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*ta
n(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f
*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) + 138*tan(e/2 + f*x/2)**9/(45*a**3*c
**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*
a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9
+ 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/
2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f
*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) + 708*tan(e/2 + f*x/2)**8/(45*a**3*c**5*f*tan(e/
2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*
tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*
c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*
a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1
80*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 1032*tan(e/2 + f*x/2)**7/(45*a**3*c**5*f*tan(e/2 + f*x/2)*
*14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f
*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(
e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f
*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**
5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 192*tan(e/2 + f*x/2)**6/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a*
*3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 -
855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)
**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f
*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2
+ f*x/2) - 45*a**3*c**5*f) + 538*tan(e/2 + f*x/2)**5/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*ta
n(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**
5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a
**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 7
20*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 4
5*a**3*c**5*f) - 100*tan(e/2 + f*x/2)**4/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/
2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2
+ f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*t
an(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5
*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*
f) - 380*tan(e/2 + f*x/2)**3/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*
a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10
- 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x
/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 +
f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) + 160*tan
(e/2 + f*x/2)**2/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*
tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*
c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900
*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 -
45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 10*tan(e/2 + f*x/2)
/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)
**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 +
f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan
(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*
tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 20/(45*a**3*c**5*f*tan(e/2 + f*x/2)
**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 +
f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan
(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*
f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c*
*5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f), Ne(f, 0)), (x/((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. 610 vs. \(2 (123) = 246\).

Time = 0.21 (sec) , antiderivative size = 610, normalized size of antiderivative = 4.66 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=\frac {2 \, {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {80 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {190 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {50 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {269 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {96 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {516 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {354 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {69 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {240 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {30 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {90 \, \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} + \frac {45 \, \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} + 10\right )}}{45 \, {\left (a^{3} c^{5} - \frac {4 \, a^{3} c^{5} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a^{3} c^{5} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {16 \, a^{3} c^{5} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {19 \, a^{3} c^{5} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {20 \, a^{3} c^{5} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {45 \, a^{3} c^{5} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {45 \, a^{3} c^{5} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {20 \, a^{3} c^{5} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {19 \, a^{3} c^{5} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {16 \, a^{3} c^{5} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {a^{3} c^{5} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} + \frac {4 \, a^{3} c^{5} \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} - \frac {a^{3} c^{5} \sin \left (f x + e\right )^{14}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{14}}\right )} f} \]

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

2/45*(5*sin(f*x + e)/(cos(f*x + e) + 1) - 80*sin(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/(co
s(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 [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=-\frac {\frac {3 \, {\left (435 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1470 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2060 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1330 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 353\right )}}{a^{3} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {4455 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 26460 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 78120 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 137340 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 157374 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 118356 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 57744 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 16596 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2339}{a^{3} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{2880 \, f} \]

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

-1/2880*(3*(435*tan(1/2*f*x + 1/2*e)^4 + 1470*tan(1/2*f*x + 1/2*e)^3 + 2060*tan(1/2*f*x + 1/2*e)^2 + 1330*tan(
1/2*f*x + 1/2*e) + 353)/(a^3*c^5*(tan(1/2*f*x + 1/2*e) + 1)^5) + (4455*tan(1/2*f*x + 1/2*e)^8 - 26460*tan(1/2*
f*x + 1/2*e)^7 + 78120*tan(1/2*f*x + 1/2*e)^6 - 137340*tan(1/2*f*x + 1/2*e)^5 + 157374*tan(1/2*f*x + 1/2*e)^4
- 118356*tan(1/2*f*x + 1/2*e)^3 + 57744*tan(1/2*f*x + 1/2*e)^2 - 16596*tan(1/2*f*x + 1/2*e) + 2339)/(a^3*c^5*(
tan(1/2*f*x + 1/2*e) - 1)^9))/f

Mupad [B] (verification not implemented)

Time = 8.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx=-\frac {\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {65\,\cos \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}{32}-\frac {225\,\cos \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}{32}-5\,\cos \left (\frac {7\,e}{2}+\frac {7\,f\,x}{2}\right )+\cos \left (\frac {9\,e}{2}+\frac {9\,f\,x}{2}\right )-\frac {37\,\cos \left (\frac {11\,e}{2}+\frac {11\,f\,x}{2}\right )}{32}+\frac {5\,\cos \left (\frac {13\,e}{2}+\frac {13\,f\,x}{2}\right )}{32}-\frac {89\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}+11\,\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )-\frac {63\,\sin \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}{8}+\frac {25\,\sin \left (\frac {7\,e}{2}+\frac {7\,f\,x}{2}\right )}{8}-\frac {5\,\sin \left (\frac {9\,e}{2}+\frac {9\,f\,x}{2}\right )}{8}+\frac {3\,\sin \left (\frac {11\,e}{2}+\frac {11\,f\,x}{2}\right )}{8}+\frac {\sin \left (\frac {13\,e}{2}+\frac {13\,f\,x}{2}\right )}{4}\right )}{2880\,a^3\,c^5\,f\,{\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}^5\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \]

[In]

int(1/((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^5),x)

[Out]

-(cos(e/2 + (f*x)/2)*((65*cos((5*e)/2 + (5*f*x)/2))/32 - (225*cos((3*e)/2 + (3*f*x)/2))/32 - 5*cos((7*e)/2 + (
7*f*x)/2) + cos((9*e)/2 + (9*f*x)/2) - (37*cos((11*e)/2 + (11*f*x)/2))/32 + (5*cos((13*e)/2 + (13*f*x)/2))/32
- (89*sin(e/2 + (f*x)/2))/4 + 11*sin((3*e)/2 + (3*f*x)/2) - (63*sin((5*e)/2 + (5*f*x)/2))/8 + (25*sin((7*e)/2
+ (7*f*x)/2))/8 - (5*sin((9*e)/2 + (9*f*x)/2))/8 + (3*sin((11*e)/2 + (11*f*x)/2))/8 + sin((13*e)/2 + (13*f*x)/
2)/4))/(2880*a^3*c^5*f*cos(e/2 - pi/4 + (f*x)/2)^5*cos(e/2 + pi/4 + (f*x)/2)^9)