3.258 \(\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^6} \, dx\)
Optimal. Leaf size=101 \[ \frac{2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac{a^3 c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7} \]
[Out]
(a^3*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^9) + (2*a^3*c^2*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])
^8) + (2*a^3*c*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^7)
________________________________________________________________________________________
Rubi [A] time = 0.182151, antiderivative size = 101, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used =
{2736, 2672, 2671} \[ \frac{2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac{a^3 c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6,x]
[Out]
(a^3*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^9) + (2*a^3*c^2*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])
^8) + (2*a^3*c*Cos[e + f*x]^7)/(693*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))^6} \, dx &=\left (a^3 c^3\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)}{11 f (c-c \sin (e+f x))^9}+\frac{1}{11} \left (2 a^3 c^2\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)}{11 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac{1}{99} \left (2 a^3 c\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)}{11 f (c-c \sin (e+f x))^9}+\frac{2 a^3 c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac{2 a^3 c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7}\\ \end{align*}
Mathematica [A] time = 0.925097, size = 145, normalized size = 1.44 \[ -\frac{a^3 \left (-2079 \sin \left (\frac{1}{2} (e+f x)\right )-1155 \sin \left (\frac{3}{2} (e+f x)\right )+297 \sin \left (\frac{5}{2} (e+f x)\right )+11 \sin \left (\frac{9}{2} (e+f x)\right )-2541 \cos \left (\frac{1}{2} (e+f x)\right )+1485 \cos \left (\frac{3}{2} (e+f x)\right )+462 \cos \left (\frac{5}{2} (e+f x)\right )-55 \cos \left (\frac{7}{2} (e+f x)\right )+\cos \left (\frac{11}{2} (e+f x)\right )\right )}{5544 c^6 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6,x]
[Out]
-(a^3*(-2541*Cos[(e + f*x)/2] + 1485*Cos[(3*(e + f*x))/2] + 462*Cos[(5*(e + f*x))/2] - 55*Cos[(7*(e + f*x))/2]
+ Cos[(11*(e + f*x))/2] - 2079*Sin[(e + f*x)/2] - 1155*Sin[(3*(e + f*x))/2] + 297*Sin[(5*(e + f*x))/2] + 11*S
in[(9*(e + f*x))/2]))/(5544*c^6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11)
________________________________________________________________________________________
Maple [A] time = 0.119, size = 178, normalized size = 1.8 \begin{align*} 2\,{\frac{{a}^{3}}{f{c}^{6}} \left ( -8\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2}-{\frac{4272}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-128\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-10}-292\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-5}-544\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-8}-{\frac{256}{11\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{11}}}-126\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-4}-{\frac{116}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}- \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-1}-{\frac{3008}{9\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-{\frac{1480}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}}} \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))^6,x)
[Out]
2/f*a^3/c^6*(-8/(tan(1/2*f*x+1/2*e)-1)^2-4272/7/(tan(1/2*f*x+1/2*e)-1)^7-128/(tan(1/2*f*x+1/2*e)-1)^10-292/(ta
n(1/2*f*x+1/2*e)-1)^5-544/(tan(1/2*f*x+1/2*e)-1)^8-256/11/(tan(1/2*f*x+1/2*e)-1)^11-126/(tan(1/2*f*x+1/2*e)-1)
^4-116/3/(tan(1/2*f*x+1/2*e)-1)^3-1/(tan(1/2*f*x+1/2*e)-1)-3008/9/(tan(1/2*f*x+1/2*e)-1)^9-1480/3/(tan(1/2*f*x
+1/2*e)-1)^6)
________________________________________________________________________________________
Maxima [B] time = 1.89596, size = 2341, normalized size = 23.18 \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))^6,x, algorithm="maxima")
[Out]
-2/3465*(5*a^3*(913*sin(f*x + e)/(cos(f*x + e) + 1) - 4565*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 12540*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3 - 25080*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33726*sin(f*x + e)^5/(cos(f*x + e)
+ 1)^5 - 33726*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 23100*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 11550*sin(f*x
+ e)^8/(cos(f*x + e) + 1)^8 + 3465*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 693*sin(f*x + e)^10/(cos(f*x + e) +
1)^10 - 146)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*
c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5
/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1
)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*
x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 9*a^3*(671*sin(f*x + e)/(cos(f*
x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 6600*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 10890*sin(
f*x + e)^4/(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 12936*sin(f*x + e)^6/(cos(f*x +
e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1155*sin(f*x
+ e)^9/(cos(f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 -
462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x +
e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e)
+ 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 2*a^3*(34
1*sin(f*x + e)/(cos(f*x + e) + 1) - 1705*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5115*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3 - 6765*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9471*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 4851*sin(f*x
+ e)^6/(cos(f*x + e) + 1)^6 + 3465*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 31)/(c^6 - 11*c^6*sin(f*x + e)/(cos(
f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330
*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^
6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) +
1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x
+ e)^11/(cos(f*x + e) + 1)^11) + 12*a^3*(253*sin(f*x + e)/(cos(f*x + e) + 1) - 1265*sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 2640*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5280*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5313*sin(f*
x + e)^5/(cos(f*x + e) + 1)^5 - 5313*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2310*sin(f*x + e)^7/(cos(f*x + e) +
1)^7 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 23)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*
sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(co
s(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6
- 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x +
e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e)
+ 1)^11))/f
________________________________________________________________________________________
Fricas [B] time = 1.34543, size = 832, normalized size = 8.24 \begin{align*} \frac{2 \, a^{3} \cos \left (f x + e\right )^{6} + 12 \, a^{3} \cos \left (f x + e\right )^{5} - 25 \, a^{3} \cos \left (f x + e\right )^{4} + 161 \, a^{3} \cos \left (f x + e\right )^{3} + 448 \, a^{3} \cos \left (f x + e\right )^{2} - 252 \, a^{3} \cos \left (f x + e\right ) - 504 \, a^{3} -{\left (2 \, a^{3} \cos \left (f x + e\right )^{5} - 10 \, a^{3} \cos \left (f x + e\right )^{4} - 35 \, a^{3} \cos \left (f x + e\right )^{3} - 196 \, a^{3} \cos \left (f x + e\right )^{2} + 252 \, a^{3} \cos \left (f x + e\right ) + 504 \, a^{3}\right )} \sin \left (f x + e\right )}{693 \,{\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f +{\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} 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))^6,x, algorithm="fricas")
[Out]
1/693*(2*a^3*cos(f*x + e)^6 + 12*a^3*cos(f*x + e)^5 - 25*a^3*cos(f*x + e)^4 + 161*a^3*cos(f*x + e)^3 + 448*a^3
*cos(f*x + e)^2 - 252*a^3*cos(f*x + e) - 504*a^3 - (2*a^3*cos(f*x + e)^5 - 10*a^3*cos(f*x + e)^4 - 35*a^3*cos(
f*x + e)^3 - 196*a^3*cos(f*x + e)^2 + 252*a^3*cos(f*x + e) + 504*a^3)*sin(f*x + e))/(c^6*f*cos(f*x + e)^6 - 5*
c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*
cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 - 12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*
cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*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))**6,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 2.22028, size = 265, normalized size = 2.62 \begin{align*} -\frac{2 \,{\left (693 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 1386 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 8085 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 10626 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 21252 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 15246 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 15444 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 4950 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 2959 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 176 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 79 \, a^{3}\right )}}{693 \, c^{6} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{11}} \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))^6,x, algorithm="giac")
[Out]
-2/693*(693*a^3*tan(1/2*f*x + 1/2*e)^10 - 1386*a^3*tan(1/2*f*x + 1/2*e)^9 + 8085*a^3*tan(1/2*f*x + 1/2*e)^8 -
10626*a^3*tan(1/2*f*x + 1/2*e)^7 + 21252*a^3*tan(1/2*f*x + 1/2*e)^6 - 15246*a^3*tan(1/2*f*x + 1/2*e)^5 + 15444
*a^3*tan(1/2*f*x + 1/2*e)^4 - 4950*a^3*tan(1/2*f*x + 1/2*e)^3 + 2959*a^3*tan(1/2*f*x + 1/2*e)^2 - 176*a^3*tan(
1/2*f*x + 1/2*e) + 79*a^3)/(c^6*f*(tan(1/2*f*x + 1/2*e) - 1)^11)