3.257 \(\int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx\)
Optimal. Leaf size=69 \[ \frac{a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7}+\frac{a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8} \]
[Out]
(a^3*c^3*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^8) + (a^3*c^2*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^7)
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Rubi [A] time = 0.139355, antiderivative size = 69, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used =
{2736, 2672, 2671} \[ \frac{a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7}+\frac{a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^5,x]
[Out]
(a^3*c^3*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^8) + (a^3*c^2*Cos[e + f*x]^7)/(63*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))^5} \, dx &=\left (a^3 c^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)}{9 f (c-c \sin (e+f x))^8}+\frac{1}{9} \left (a^3 c^2\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)}{9 f (c-c \sin (e+f x))^8}+\frac{a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7}\\ \end{align*}
Mathematica [A] time = 0.720795, size = 135, normalized size = 1.96 \[ \frac{a^3 \left (189 \sin \left (\frac{1}{2} (e+f x)\right )+105 \sin \left (\frac{3}{2} (e+f x)\right )-27 \sin \left (\frac{5}{2} (e+f x)\right )-\sin \left (\frac{9}{2} (e+f x)\right )+315 \cos \left (\frac{1}{2} (e+f x)\right )-189 \cos \left (\frac{3}{2} (e+f x)\right )-63 \cos \left (\frac{5}{2} (e+f x)\right )+9 \cos \left (\frac{7}{2} (e+f x)\right )\right )}{504 c^5 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^5,x]
[Out]
(a^3*(315*Cos[(e + f*x)/2] - 189*Cos[(3*(e + f*x))/2] - 63*Cos[(5*(e + f*x))/2] + 9*Cos[(7*(e + f*x))/2] + 189
*Sin[(e + f*x)/2] + 105*Sin[(3*(e + f*x))/2] - 27*Sin[(5*(e + f*x))/2] - Sin[(9*(e + f*x))/2]))/(504*c^5*f*(Co
s[(e + f*x)/2] - Sin[(e + f*x)/2])^9)
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Maple [B] time = 0.109, size = 148, normalized size = 2.1 \begin{align*} 2\,{\frac{{a}^{3}}{f{c}^{5}} \left ( -{\frac{928}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-76\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-4}-{\frac{496}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}}}-{\frac{86}{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}-64\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-8}-136\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-5}-{\frac{128}{9\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2} \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))^5,x)
[Out]
2/f*a^3/c^5*(-928/7/(tan(1/2*f*x+1/2*e)-1)^7-76/(tan(1/2*f*x+1/2*e)-1)^4-496/3/(tan(1/2*f*x+1/2*e)-1)^6-86/3/(
tan(1/2*f*x+1/2*e)-1)^3-1/(tan(1/2*f*x+1/2*e)-1)-64/(tan(1/2*f*x+1/2*e)-1)^8-136/(tan(1/2*f*x+1/2*e)-1)^5-128/
9/(tan(1/2*f*x+1/2*e)-1)^9-7/(tan(1/2*f*x+1/2*e)-1)^2)
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Maxima [B] time = 2.19098, size = 1875, normalized size = 27.17 \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))^5,x, algorithm="maxima")
[Out]
-2/315*(a^3*(432*sin(f*x + e)/(cos(f*x + e) + 1) - 1728*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3612*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 - 5418*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5040*sin(f*x + e)^5/(cos(f*x + e) + 1)^5
- 3360*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1260*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 315*sin(f*x + e)^8/(c
os(f*x + e) + 1)^8 - 83)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) - 15*a^3*(4
5*sin(f*x + e)/(cos(f*x + e) + 1) - 117*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 273*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 - 315*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 315*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 147*sin(f*x + e)
^6/(cos(f*x + e) + 1)^6 + 63*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 5)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e)
+ 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f
*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x
+ e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin
(f*x + e)^9/(cos(f*x + e) + 1)^9) - 10*a^3*(9*sin(f*x + e)/(cos(f*x + e) + 1) - 36*sin(f*x + e)^2/(cos(f*x + e
) + 1)^2 + 84*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 63*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 63*sin(f*x + e)^5
/(cos(f*x + e) + 1)^5 - 1)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e)
+ 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*si
n(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*
x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 42*a^3*
(9*sin(f*x + e)/(cos(f*x + e) + 1) - 36*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 54*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 - 81*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 45*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 30*sin(f*x + e)^6/(
cos(f*x + e) + 1)^6 - 1)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9))/f
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Fricas [B] time = 1.38983, size = 675, normalized size = 9.78 \begin{align*} -\frac{a^{3} \cos \left (f x + e\right )^{5} - 4 \, a^{3} \cos \left (f x + e\right )^{4} + 19 \, a^{3} \cos \left (f x + e\right )^{3} + 52 \, a^{3} \cos \left (f x + e\right )^{2} - 28 \, a^{3} \cos \left (f x + e\right ) - 56 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{4} + 5 \, a^{3} \cos \left (f x + e\right )^{3} + 24 \, a^{3} \cos \left (f x + e\right )^{2} - 28 \, a^{3} \cos \left (f x + e\right ) - 56 \, a^{3}\right )} \sin \left (f x + e\right )}{63 \,{\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f -{\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} 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))^5,x, algorithm="fricas")
[Out]
-1/63*(a^3*cos(f*x + e)^5 - 4*a^3*cos(f*x + e)^4 + 19*a^3*cos(f*x + e)^3 + 52*a^3*cos(f*x + e)^2 - 28*a^3*cos(
f*x + e) - 56*a^3 + (a^3*cos(f*x + e)^4 + 5*a^3*cos(f*x + e)^3 + 24*a^3*cos(f*x + e)^2 - 28*a^3*cos(f*x + e) -
56*a^3)*sin(f*x + e))/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(
f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f
*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*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))**5,x)
[Out]
Timed out
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Giac [B] time = 2.08094, size = 219, normalized size = 3.17 \begin{align*} -\frac{2 \,{\left (63 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 63 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 483 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 315 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 693 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 189 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 225 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 9 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 8 \, a^{3}\right )}}{63 \, c^{5} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{9}} \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))^5,x, algorithm="giac")
[Out]
-2/63*(63*a^3*tan(1/2*f*x + 1/2*e)^8 - 63*a^3*tan(1/2*f*x + 1/2*e)^7 + 483*a^3*tan(1/2*f*x + 1/2*e)^6 - 315*a^
3*tan(1/2*f*x + 1/2*e)^5 + 693*a^3*tan(1/2*f*x + 1/2*e)^4 - 189*a^3*tan(1/2*f*x + 1/2*e)^3 + 225*a^3*tan(1/2*f
*x + 1/2*e)^2 - 9*a^3*tan(1/2*f*x + 1/2*e) + 8*a^3)/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)