3.260 \(\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^8} \, dx\)

Optimal. Leaf size=166 \[ \frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac{8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9} \]

[Out]

(a^3*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (4*a^3*c^2*Cos[e + f*x]^7)/(195*f*(c - c*Sin[e + f*x
])^10) + (4*a^3*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (8*a^3*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e
+ f*x])^8) + (8*a^3*Cos[e + f*x]^7)/(45045*c*f*(c - c*Sin[e + f*x])^7)

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Rubi [A]  time = 0.287297, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2672, 2671} \[ \frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac{8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^8,x]

[Out]

(a^3*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (4*a^3*c^2*Cos[e + f*x]^7)/(195*f*(c - c*Sin[e + f*x
])^10) + (4*a^3*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (8*a^3*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e
+ f*x])^8) + (8*a^3*Cos[e + f*x]^7)/(45045*c*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))^8} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{11}} \, dx\\ &=\frac{a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{1}{15} \left (4 a^3 c^2\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)}{15 f (c-c \sin (e+f x))^{11}}+\frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{1}{65} \left (4 a^3 c\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)}{15 f (c-c \sin (e+f x))^{11}}+\frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac{1}{715} \left (8 a^3\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)}{15 f (c-c \sin (e+f x))^{11}}+\frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac{8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{\left (8 a^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx}{6435 c}\\ &=\frac{a^3 c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac{4 a^3 c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac{4 a^3 c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac{8 a^3 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac{8 a^3 \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}\\ \end{align*}

Mathematica [A]  time = 2.11265, size = 209, normalized size = 1.26 \[ \frac{(a \sin (e+f x)+a)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (109395 \sin \left (\frac{1}{2} (e+f x)\right )+60060 \sin \left (\frac{3}{2} (e+f x)\right )-15015 \sin \left (\frac{5}{2} (e+f x)\right )-455 \sin \left (\frac{9}{2} (e+f x)\right )+15 \sin \left (\frac{13}{2} (e+f x)\right )+115830 \cos \left (\frac{1}{2} (e+f x)\right )-65065 \cos \left (\frac{3}{2} (e+f x)\right )-18018 \cos \left (\frac{5}{2} (e+f x)\right )+1365 \cos \left (\frac{7}{2} (e+f x)\right )-105 \cos \left (\frac{11}{2} (e+f x)\right )+\cos \left (\frac{15}{2} (e+f x)\right )\right )}{360360 f (c-c \sin (e+f x))^8 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^8,x]

[Out]

((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a + a*Sin[e + f*x])^3*(115830*Cos[(e + f*x)/2] - 65065*Cos[(3*(e + f*x
))/2] - 18018*Cos[(5*(e + f*x))/2] + 1365*Cos[(7*(e + f*x))/2] - 105*Cos[(11*(e + f*x))/2] + Cos[(15*(e + f*x)
)/2] + 109395*Sin[(e + f*x)/2] + 60060*Sin[(3*(e + f*x))/2] - 15015*Sin[(5*(e + f*x))/2] - 455*Sin[(9*(e + f*x
))/2] + 15*Sin[(13*(e + f*x))/2]))/(360360*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(c - c*Sin[e + f*x])^8)

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Maple [A]  time = 0.139, size = 238, normalized size = 1.4 \begin{align*} 2\,{\frac{{a}^{3}}{f{c}^{8}} \left ( -276\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-4}-{\frac{24320}{13\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{13}}}-{\frac{32288}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-512\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-14}-{\frac{188}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-{\frac{13184}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{12}}}-{\frac{47072}{5\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{10}}}-{\frac{81344}{11\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{11}}}-{\frac{4536}{5\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}}}-2304\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-6}-10\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2}-7352\, \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{84112}{9\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-{\frac{1024}{15\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{15}}} \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))^8,x)

[Out]

2/f*a^3/c^8*(-276/(tan(1/2*f*x+1/2*e)-1)^4-24320/13/(tan(1/2*f*x+1/2*e)-1)^13-32288/7/(tan(1/2*f*x+1/2*e)-1)^7
-512/(tan(1/2*f*x+1/2*e)-1)^14-188/3/(tan(1/2*f*x+1/2*e)-1)^3-13184/3/(tan(1/2*f*x+1/2*e)-1)^12-47072/5/(tan(1
/2*f*x+1/2*e)-1)^10-81344/11/(tan(1/2*f*x+1/2*e)-1)^11-4536/5/(tan(1/2*f*x+1/2*e)-1)^5-2304/(tan(1/2*f*x+1/2*e
)-1)^6-10/(tan(1/2*f*x+1/2*e)-1)^2-7352/(tan(1/2*f*x+1/2*e)-1)^8-1/(tan(1/2*f*x+1/2*e)-1)-84112/9/(tan(1/2*f*x
+1/2*e)-1)^9-1024/15/(tan(1/2*f*x+1/2*e)-1)^15)

________________________________________________________________________________________

Maxima [B]  time = 1.94779, size = 3270, normalized size = 19.7 \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))^8,x, algorithm="maxima")

[Out]

2/45045*(3*a^3*(17715*sin(f*x + e)/(cos(f*x + e) + 1) - 78960*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 342160*sin
(f*x + e)^3/(cos(f*x + e) + 1)^3 - 891345*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1960959*sin(f*x + e)^5/(cos(f*
x + e) + 1)^5 - 3043040*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3912480*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 36
87255*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2867865*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1585584*sin(f*x + e)
^10/(cos(f*x + e) + 1)^10 + 720720*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 195195*sin(f*x + e)^12/(cos(f*x + e
) + 1)^12 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 1181)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1)
+ 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x
+ e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x
+ e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5
005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*
x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos
(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15)
- 7*a^3*(7845*sin(f*x + e)/(cos(f*x + e) + 1) - 54915*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 222950*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 - 668850*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1444443*sin(f*x + e)^5/(cos(f*x + e) +
1)^5 - 2407405*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3063060*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3063060*si
n(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2357355*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1414413*sin(f*x + e)^10/(cos
(f*x + e) + 1)^10 + 630630*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 210210*sin(f*x + e)^12/(cos(f*x + e) + 1)^1
2 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 6435*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - 952)/(c^8 - 15*
c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(co
s(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^
5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin
(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(co
s(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) +
1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin
(f*x + e)^15/(cos(f*x + e) + 1)^15) - 12*a^3*(1740*sin(f*x + e)/(cos(f*x + e) + 1) - 12180*sin(f*x + e)^2/(cos
(f*x + e) + 1)^2 + 37765*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 113295*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 20
4204*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 340340*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 373230*sin(f*x + e)^7/
(cos(f*x + e) + 1)^7 - 373230*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 240240*sin(f*x + e)^9/(cos(f*x + e) + 1)^9
- 144144*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 45045*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 15015*sin(f*x
+ e)^12/(cos(f*x + e) + 1)^12 - 116)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^
4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin
(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos
(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1
)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*
sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15) + 6*a^3*(675*sin(f*x + e)/(
cos(f*x + e) + 1) - 4725*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 20475*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 464
10*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 102102*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 130130*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 + 167310*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 122265*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 +
95095*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 33033*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 15015*sin(f*x + e)^
11/(cos(f*x + e) + 1)^11 - 45)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x
+ e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 30
03*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x +
e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x +
e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 +
455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*
x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15))/f

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Fricas [B]  time = 1.44161, size = 1138, normalized size = 6.86 \begin{align*} \frac{8 \, a^{3} \cos \left (f x + e\right )^{8} + 64 \, a^{3} \cos \left (f x + e\right )^{7} - 196 \, a^{3} \cos \left (f x + e\right )^{6} - 672 \, a^{3} \cos \left (f x + e\right )^{5} + 735 \, a^{3} \cos \left (f x + e\right )^{4} - 7161 \, a^{3} \cos \left (f x + e\right )^{3} - 20328 \, a^{3} \cos \left (f x + e\right )^{2} + 12012 \, a^{3} \cos \left (f x + e\right ) + 24024 \, a^{3} -{\left (8 \, a^{3} \cos \left (f x + e\right )^{7} - 56 \, a^{3} \cos \left (f x + e\right )^{6} - 252 \, a^{3} \cos \left (f x + e\right )^{5} + 420 \, a^{3} \cos \left (f x + e\right )^{4} + 1155 \, a^{3} \cos \left (f x + e\right )^{3} + 8316 \, a^{3} \cos \left (f x + e\right )^{2} - 12012 \, a^{3} \cos \left (f x + e\right ) - 24024 \, a^{3}\right )} \sin \left (f x + e\right )}{45045 \,{\left (c^{8} f \cos \left (f x + e\right )^{8} - 7 \, c^{8} f \cos \left (f x + e\right )^{7} - 32 \, c^{8} f \cos \left (f x + e\right )^{6} + 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 160 \, c^{8} f \cos \left (f x + e\right )^{4} - 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 256 \, c^{8} f \cos \left (f x + e\right )^{2} + 64 \, c^{8} f \cos \left (f x + e\right ) + 128 \, c^{8} f +{\left (c^{8} f \cos \left (f x + e\right )^{7} + 8 \, c^{8} f \cos \left (f x + e\right )^{6} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} - 80 \, c^{8} f \cos \left (f x + e\right )^{4} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} + 192 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} f \cos \left (f x + e\right ) - 128 \, c^{8} 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))^8,x, algorithm="fricas")

[Out]

1/45045*(8*a^3*cos(f*x + e)^8 + 64*a^3*cos(f*x + e)^7 - 196*a^3*cos(f*x + e)^6 - 672*a^3*cos(f*x + e)^5 + 735*
a^3*cos(f*x + e)^4 - 7161*a^3*cos(f*x + e)^3 - 20328*a^3*cos(f*x + e)^2 + 12012*a^3*cos(f*x + e) + 24024*a^3 -
(8*a^3*cos(f*x + e)^7 - 56*a^3*cos(f*x + e)^6 - 252*a^3*cos(f*x + e)^5 + 420*a^3*cos(f*x + e)^4 + 1155*a^3*co
s(f*x + e)^3 + 8316*a^3*cos(f*x + e)^2 - 12012*a^3*cos(f*x + e) - 24024*a^3)*sin(f*x + e))/(c^8*f*cos(f*x + e)
^8 - 7*c^8*f*cos(f*x + e)^7 - 32*c^8*f*cos(f*x + e)^6 + 56*c^8*f*cos(f*x + e)^5 + 160*c^8*f*cos(f*x + e)^4 - 1
12*c^8*f*cos(f*x + e)^3 - 256*c^8*f*cos(f*x + e)^2 + 64*c^8*f*cos(f*x + e) + 128*c^8*f + (c^8*f*cos(f*x + e)^7
+ 8*c^8*f*cos(f*x + e)^6 - 24*c^8*f*cos(f*x + e)^5 - 80*c^8*f*cos(f*x + e)^4 + 80*c^8*f*cos(f*x + e)^3 + 192*
c^8*f*cos(f*x + e)^2 - 64*c^8*f*cos(f*x + e) - 128*c^8*f)*sin(f*x + e))

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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))**8,x)

[Out]

Timed out

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Giac [A]  time = 2.20153, size = 356, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (45045 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{14} - 180180 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{13} + 1066065 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{12} - 2702700 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} + 6675669 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 10210200 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 14124825 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 13178880 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 11026015 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 6066060 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3088995 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 864500 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 265335 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 18600 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 4243 \, a^{3}\right )}}{45045 \, c^{8} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{15}} \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))^8,x, algorithm="giac")

[Out]

-2/45045*(45045*a^3*tan(1/2*f*x + 1/2*e)^14 - 180180*a^3*tan(1/2*f*x + 1/2*e)^13 + 1066065*a^3*tan(1/2*f*x + 1
/2*e)^12 - 2702700*a^3*tan(1/2*f*x + 1/2*e)^11 + 6675669*a^3*tan(1/2*f*x + 1/2*e)^10 - 10210200*a^3*tan(1/2*f*
x + 1/2*e)^9 + 14124825*a^3*tan(1/2*f*x + 1/2*e)^8 - 13178880*a^3*tan(1/2*f*x + 1/2*e)^7 + 11026015*a^3*tan(1/
2*f*x + 1/2*e)^6 - 6066060*a^3*tan(1/2*f*x + 1/2*e)^5 + 3088995*a^3*tan(1/2*f*x + 1/2*e)^4 - 864500*a^3*tan(1/
2*f*x + 1/2*e)^3 + 265335*a^3*tan(1/2*f*x + 1/2*e)^2 - 18600*a^3*tan(1/2*f*x + 1/2*e) + 4243*a^3)/(c^8*f*(tan(
1/2*f*x + 1/2*e) - 1)^15)