3.245 \(\int \frac{(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx\)

Optimal. Leaf size=98 \[ \frac{a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac{2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6} \]

[Out]

(a^2*c^2*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + (2*a^2*c*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^6)
+ (2*a^2*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^5)

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Rubi [A]  time = 0.182284, antiderivative size = 98, 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} \[ \frac{a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac{2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5,x]

[Out]

(a^2*c^2*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + (2*a^2*c*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^6)
+ (2*a^2*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^5)

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))^2}{(c-c \sin (e+f x))^5} \, dx &=\left (a^2 c^2\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac{a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{1}{9} \left (2 a^2 c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac{a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}+\frac{1}{63} \left (2 a^2\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac{a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}+\frac{2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}\\ \end{align*}

Mathematica [A]  time = 0.536023, size = 121, normalized size = 1.23 \[ \frac{a^2 \left (441 \sin \left (\frac{1}{2} (e+f x)\right )+210 \sin \left (\frac{3}{2} (e+f x)\right )-36 \sin \left (\frac{5}{2} (e+f x)\right )+\sin \left (\frac{9}{2} (e+f x)\right )+315 \cos \left (\frac{1}{2} (e+f x)\right )-126 \cos \left (\frac{3}{2} (e+f x)\right )-9 \cos \left (\frac{7}{2} (e+f x)\right )\right )}{1260 c^5 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5,x]

[Out]

(a^2*(315*Cos[(e + f*x)/2] - 126*Cos[(3*(e + f*x))/2] - 9*Cos[(7*(e + f*x))/2] + 441*Sin[(e + f*x)/2] + 210*Si
n[(3*(e + f*x))/2] - 36*Sin[(5*(e + f*x))/2] + Sin[(9*(e + f*x))/2]))/(1260*c^5*f*(Cos[(e + f*x)/2] - Sin[(e +
f*x)/2])^9)

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Maple [A]  time = 0.104, size = 148, normalized size = 1.5 \begin{align*} 2\,{\frac{{a}^{2}}{f{c}^{5}} \left ( -{\frac{404}{5\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}}}-{\frac{272}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}}}-50\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-4}-{\frac{64}{3\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-32\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-8}-6\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2}- \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-1}-{\frac{64}{9\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-{\frac{480}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x)

[Out]

2/f*a^2/c^5*(-404/5/(tan(1/2*f*x+1/2*e)-1)^5-272/3/(tan(1/2*f*x+1/2*e)-1)^6-50/(tan(1/2*f*x+1/2*e)-1)^4-64/3/(
tan(1/2*f*x+1/2*e)-1)^3-32/(tan(1/2*f*x+1/2*e)-1)^8-6/(tan(1/2*f*x+1/2*e)-1)^2-1/(tan(1/2*f*x+1/2*e)-1)-64/9/(
tan(1/2*f*x+1/2*e)-1)^9-480/7/(tan(1/2*f*x+1/2*e)-1)^7)

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Maxima [B]  time = 1.40376, size = 1449, normalized size = 14.79 \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))^2/(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

-2/315*(a^2*(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) - 10*a^2*(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) + 14*a^2*(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 = 1.31428, size = 684, normalized size = 6.98 \begin{align*} \frac{2 \, a^{2} \cos \left (f x + e\right )^{5} - 8 \, a^{2} \cos \left (f x + e\right )^{4} - 25 \, a^{2} \cos \left (f x + e\right )^{3} - 85 \, a^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{2} \cos \left (f x + e\right ) + 140 \, a^{2} +{\left (2 \, a^{2} \cos \left (f x + e\right )^{4} + 10 \, a^{2} \cos \left (f x + e\right )^{3} - 15 \, a^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{2} \cos \left (f x + e\right ) + 140 \, a^{2}\right )} \sin \left (f x + e\right )}{315 \,{\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))^2/(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

1/315*(2*a^2*cos(f*x + e)^5 - 8*a^2*cos(f*x + e)^4 - 25*a^2*cos(f*x + e)^3 - 85*a^2*cos(f*x + e)^2 + 70*a^2*co
s(f*x + e) + 140*a^2 + (2*a^2*cos(f*x + e)^4 + 10*a^2*cos(f*x + e)^3 - 15*a^2*cos(f*x + e)^2 + 70*a^2*cos(f*x
+ e) + 140*a^2)*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))**2/(c-c*sin(f*x+e))**5,x)

[Out]

Timed out

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Giac [A]  time = 2.20265, size = 219, normalized size = 2.23 \begin{align*} -\frac{2 \,{\left (315 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 630 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 2310 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 2520 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3402 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1638 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 1062 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 108 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 47 \, a^{2}\right )}}{315 \, 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))^2/(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

-2/315*(315*a^2*tan(1/2*f*x + 1/2*e)^8 - 630*a^2*tan(1/2*f*x + 1/2*e)^7 + 2310*a^2*tan(1/2*f*x + 1/2*e)^6 - 25
20*a^2*tan(1/2*f*x + 1/2*e)^5 + 3402*a^2*tan(1/2*f*x + 1/2*e)^4 - 1638*a^2*tan(1/2*f*x + 1/2*e)^3 + 1062*a^2*t
an(1/2*f*x + 1/2*e)^2 - 108*a^2*tan(1/2*f*x + 1/2*e) + 47*a^2)/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)