3.259 \(\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx\)

Optimal. Leaf size=132 \[ \frac{3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7}+\frac{2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8} \]

[Out]

(a^3*c^3*Cos[e + f*x]^7)/(13*f*(c - c*Sin[e + f*x])^10) + (3*a^3*c^2*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x
])^9) + (2*a^3*c*Cos[e + f*x]^7)/(429*f*(c - c*Sin[e + f*x])^8) + (2*a^3*Cos[e + f*x]^7)/(3003*f*(c - c*Sin[e
+ f*x])^7)

________________________________________________________________________________________

Rubi [A]  time = 0.229953, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2672, 2671} \[ \frac{3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7}+\frac{2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^7,x]

[Out]

(a^3*c^3*Cos[e + f*x]^7)/(13*f*(c - c*Sin[e + f*x])^10) + (3*a^3*c^2*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x
])^9) + (2*a^3*c*Cos[e + f*x]^7)/(429*f*(c - c*Sin[e + f*x])^8) + (2*a^3*Cos[e + f*x]^7)/(3003*f*(c - c*Sin[e
+ f*x])^7)

Rule 2736

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 2672

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] + 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 2671

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*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^7} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{10}} \, dx\\ &=\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{1}{13} \left (3 a^3 c^2\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^9} \, dx\\ &=\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{1}{143} \left (6 a^3 c\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^8} \, dx\\ &=\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}+\frac{1}{429} \left (2 a^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac{a^3 c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac{3 a^3 c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^8}+\frac{2 a^3 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^7}\\ \end{align*}

Mathematica [A]  time = 1.8649, size = 157, normalized size = 1.19 \[ \frac{a^3 \left (16302 \sin \left (\frac{1}{2} (e+f x)\right )+9009 \sin \left (\frac{3}{2} (e+f x)\right )-2288 \sin \left (\frac{5}{2} (e+f x)\right )-78 \sin \left (\frac{9}{2} (e+f x)\right )+\sin \left (\frac{13}{2} (e+f x)\right )+18018 \cos \left (\frac{1}{2} (e+f x)\right )-10296 \cos \left (\frac{3}{2} (e+f x)\right )-3003 \cos \left (\frac{5}{2} (e+f x)\right )+286 \cos \left (\frac{7}{2} (e+f x)\right )-13 \cos \left (\frac{11}{2} (e+f x)\right )\right )}{48048 c^7 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^7,x]

[Out]

(a^3*(18018*Cos[(e + f*x)/2] - 10296*Cos[(3*(e + f*x))/2] - 3003*Cos[(5*(e + f*x))/2] + 286*Cos[(7*(e + f*x))/
2] - 13*Cos[(11*(e + f*x))/2] + 16302*Sin[(e + f*x)/2] + 9009*Sin[(3*(e + f*x))/2] - 2288*Sin[(5*(e + f*x))/2]
- 78*Sin[(9*(e + f*x))/2] + Sin[(13*(e + f*x))/2]))/(48048*c^7*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13)

________________________________________________________________________________________

Maple [A]  time = 0.129, size = 208, normalized size = 1.6 \begin{align*} 2\,{\frac{{a}^{3}}{f{c}^{7}} \left ( -{\frac{13112}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-{\frac{512}{13\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{13}}}-540\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-5}-9\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2}-256\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-12}-1148\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-6}-{\frac{8832}{11\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{11}}}-50\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-3}-192\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-4}-1600\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-10}-2352\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-8}- \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-1}-{\frac{6752}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x)

[Out]

2/f*a^3/c^7*(-13112/7/(tan(1/2*f*x+1/2*e)-1)^7-512/13/(tan(1/2*f*x+1/2*e)-1)^13-540/(tan(1/2*f*x+1/2*e)-1)^5-9
/(tan(1/2*f*x+1/2*e)-1)^2-256/(tan(1/2*f*x+1/2*e)-1)^12-1148/(tan(1/2*f*x+1/2*e)-1)^6-8832/11/(tan(1/2*f*x+1/2
*e)-1)^11-50/(tan(1/2*f*x+1/2*e)-1)^3-192/(tan(1/2*f*x+1/2*e)-1)^4-1600/(tan(1/2*f*x+1/2*e)-1)^10-2352/(tan(1/
2*f*x+1/2*e)-1)^8-1/(tan(1/2*f*x+1/2*e)-1)-6752/3/(tan(1/2*f*x+1/2*e)-1)^9)

________________________________________________________________________________________

Maxima [B]  time = 2.04692, size = 2805, normalized size = 21.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x, algorithm="maxima")

[Out]

-2/15015*(2*a^3*(4771*sin(f*x + e)/(cos(f*x + e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*sin(
f*x + e)^3/(cos(f*x + e) + 1)^3 - 187330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 265122*sin(f*x + e)^5/(cos(f*x
+ e) + 1)^5 - 353496*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 276276*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 207207
*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 30030*sin(f*x + e)^10/(cos(
f*x + e) + 1)^10 - 367)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*si
n(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(co
s(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9
+ 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f
*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) + 5*a^3*(3796*sin(f*x + e)/(cos(
f*x + e) + 1) - 22776*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 77506*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 193765
*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 339768*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 453024*sin(f*x + e)^6/(cos
(f*x + e) + 1)^6 + 444444*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 333333*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1
80180*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 72072*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 18018*sin(f*x + e)^1
1/(cos(f*x + e) + 1)^11 - 3003*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 523)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*
x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c
^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^
6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e)
+ 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*
sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(co
s(f*x + e) + 1)^13) - 35*a^3*(611*sin(f*x + e)/(cos(f*x + e) + 1) - 2379*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 +
8723*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 18590*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33462*sin(f*x + e)^5/(
cos(f*x + e) + 1)^5 - 40326*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 40326*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 -
27027*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 15015*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 4719*sin(f*x + e)^10/(
cos(f*x + e) + 1)^10 + 1287*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 47)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x +
e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*s
in(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)
^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(
f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*
x + e) + 1)^13) - 154*a^3*(13*sin(f*x + e)/(cos(f*x + e) + 1) - 78*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 286*s
in(f*x + e)^3/(cos(f*x + e) + 1)^3 - 520*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 936*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 - 858*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 858*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 351*sin(f*x + e
)^8/(cos(f*x + e) + 1)^8 + 195*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x +
e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*s
in(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)
^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(
f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*
x + e) + 1)^13))/f

________________________________________________________________________________________

Fricas [B]  time = 1.46854, size = 976, normalized size = 7.39 \begin{align*} -\frac{2 \, a^{3} \cos \left (f x + e\right )^{7} - 12 \, a^{3} \cos \left (f x + e\right )^{6} - 49 \, a^{3} \cos \left (f x + e\right )^{5} + 70 \, a^{3} \cos \left (f x + e\right )^{4} - 567 \, a^{3} \cos \left (f x + e\right )^{3} - 1596 \, a^{3} \cos \left (f x + e\right )^{2} + 924 \, a^{3} \cos \left (f x + e\right ) + 1848 \, a^{3} +{\left (2 \, a^{3} \cos \left (f x + e\right )^{6} + 14 \, a^{3} \cos \left (f x + e\right )^{5} - 35 \, a^{3} \cos \left (f x + e\right )^{4} - 105 \, a^{3} \cos \left (f x + e\right )^{3} - 672 \, a^{3} \cos \left (f x + e\right )^{2} + 924 \, a^{3} \cos \left (f x + e\right ) + 1848 \, a^{3}\right )} \sin \left (f x + e\right )}{3003 \,{\left (c^{7} f \cos \left (f x + e\right )^{7} + 7 \, c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} - 56 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} + 112 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f -{\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} - 24 \, c^{7} f \cos \left (f x + e\right )^{4} + 32 \, c^{7} f \cos \left (f x + e\right )^{3} + 80 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x, algorithm="fricas")

[Out]

-1/3003*(2*a^3*cos(f*x + e)^7 - 12*a^3*cos(f*x + e)^6 - 49*a^3*cos(f*x + e)^5 + 70*a^3*cos(f*x + e)^4 - 567*a^
3*cos(f*x + e)^3 - 1596*a^3*cos(f*x + e)^2 + 924*a^3*cos(f*x + e) + 1848*a^3 + (2*a^3*cos(f*x + e)^6 + 14*a^3*
cos(f*x + e)^5 - 35*a^3*cos(f*x + e)^4 - 105*a^3*cos(f*x + e)^3 - 672*a^3*cos(f*x + e)^2 + 924*a^3*cos(f*x + e
) + 1848*a^3)*sin(f*x + e))/(c^7*f*cos(f*x + e)^7 + 7*c^7*f*cos(f*x + e)^6 - 18*c^7*f*cos(f*x + e)^5 - 56*c^7*
f*cos(f*x + e)^4 + 48*c^7*f*cos(f*x + e)^3 + 112*c^7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f - (c^
7*f*cos(f*x + e)^6 - 6*c^7*f*cos(f*x + e)^5 - 24*c^7*f*cos(f*x + e)^4 + 32*c^7*f*cos(f*x + e)^3 + 80*c^7*f*cos
(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f)*sin(f*x + e))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**7,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.29861, size = 311, normalized size = 2.36 \begin{align*} -\frac{2 \,{\left (3003 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{12} - 9009 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} + 51051 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 99099 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 216216 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 246246 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 285714 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 182754 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 122551 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 37609 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 15171 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1027 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 310 \, a^{3}\right )}}{3003 \, c^{7} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x, algorithm="giac")

[Out]

-2/3003*(3003*a^3*tan(1/2*f*x + 1/2*e)^12 - 9009*a^3*tan(1/2*f*x + 1/2*e)^11 + 51051*a^3*tan(1/2*f*x + 1/2*e)^
10 - 99099*a^3*tan(1/2*f*x + 1/2*e)^9 + 216216*a^3*tan(1/2*f*x + 1/2*e)^8 - 246246*a^3*tan(1/2*f*x + 1/2*e)^7
+ 285714*a^3*tan(1/2*f*x + 1/2*e)^6 - 182754*a^3*tan(1/2*f*x + 1/2*e)^5 + 122551*a^3*tan(1/2*f*x + 1/2*e)^4 -
37609*a^3*tan(1/2*f*x + 1/2*e)^3 + 15171*a^3*tan(1/2*f*x + 1/2*e)^2 - 1027*a^3*tan(1/2*f*x + 1/2*e) + 310*a^3)
/(c^7*f*(tan(1/2*f*x + 1/2*e) - 1)^13)