3.35 \(\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx\)
Optimal. Leaf size=115 \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{a^2 (2 A-7 B) \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac{a^2 c (2 A-7 B) \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6} \]
[Out]
(a^2*(A + B)*c^2*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + (a^2*(2*A - 7*B)*c*Cos[e + f*x]^5)/(63*f*(c -
c*Sin[e + f*x])^6) + (a^2*(2*A - 7*B)*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^5)
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Rubi [A] time = 0.285661, antiderivative size = 115, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{2967, 2859, 2672, 2671} \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{a^2 (2 A-7 B) \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac{a^2 c (2 A-7 B) \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6} \]
Antiderivative was successfully verified.
[In]
Int[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x]
[Out]
(a^2*(A + B)*c^2*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + (a^2*(2*A - 7*B)*c*Cos[e + f*x]^5)/(63*f*(c -
c*Sin[e + f*x])^6) + (a^2*(2*A - 7*B)*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^5)
Rule 2967
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Rule 2859
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m +
p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 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 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx &=\left (a^2 c^2\right ) \int \frac{\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{1}{9} \left (a^2 (2 A-7 B) c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{a^2 (2 A-7 B) c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}+\frac{1}{63} \left (a^2 (2 A-7 B)\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac{a^2 (2 A-7 B) c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}+\frac{a^2 (2 A-7 B) \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}\\ \end{align*}
Mathematica [B] time = 1.1964, size = 261, normalized size = 2.27 \[ -\frac{a^2 (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (315 (2 A+3 B) \cos \left (\frac{1}{2} (e+f x)\right )-63 (4 A+11 B) \cos \left (\frac{3}{2} (e+f x)\right )+882 A \sin \left (\frac{1}{2} (e+f x)\right )+420 A \sin \left (\frac{3}{2} (e+f x)\right )-72 A \sin \left (\frac{5}{2} (e+f x)\right )+2 A \sin \left (\frac{9}{2} (e+f x)\right )-18 A \cos \left (\frac{7}{2} (e+f x)\right )+63 B \sin \left (\frac{1}{2} (e+f x)\right )+105 B \sin \left (\frac{3}{2} (e+f x)\right )-63 B \sin \left (\frac{5}{2} (e+f x)\right )-7 B \sin \left (\frac{9}{2} (e+f x)\right )-315 B \cos \left (\frac{5}{2} (e+f x)\right )+63 B \cos \left (\frac{7}{2} (e+f x)\right )\right )}{2520 c^5 f (\sin (e+f x)-1)^5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x]
[Out]
-(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(315*(2*A + 3*B)*Cos[(e + f*x)/2] - 63*(4*A +
11*B)*Cos[(3*(e + f*x))/2] - 315*B*Cos[(5*(e + f*x))/2] - 18*A*Cos[(7*(e + f*x))/2] + 63*B*Cos[(7*(e + f*x))/
2] + 882*A*Sin[(e + f*x)/2] + 63*B*Sin[(e + f*x)/2] + 420*A*Sin[(3*(e + f*x))/2] + 105*B*Sin[(3*(e + f*x))/2]
- 72*A*Sin[(5*(e + f*x))/2] - 63*B*Sin[(5*(e + f*x))/2] + 2*A*Sin[(9*(e + f*x))/2] - 7*B*Sin[(9*(e + f*x))/2])
)/(2520*c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x])^5)
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Maple [A] time = 0.153, size = 205, normalized size = 1.8 \begin{align*} 2\,{\frac{{a}^{2}}{f{c}^{5}} \left ( -1/3\,{\frac{64\,A+22\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-1/9\,{\frac{64\,A+64\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-1/8\,{\frac{256\,A+256\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{8}}}-1/6\,{\frac{544\,A+448\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}}}-1/4\,{\frac{200\,A+104\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-1/7\,{\frac{480\,A+448\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-1/2\,{\frac{12\,A+2\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-{\frac{A}{\tan \left ( 1/2\,fx+e/2 \right ) -1}}-1/5\,{\frac{404\,A+276\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x)
[Out]
2/f*a^2/c^5*(-1/3*(64*A+22*B)/(tan(1/2*f*x+1/2*e)-1)^3-1/9*(64*A+64*B)/(tan(1/2*f*x+1/2*e)-1)^9-1/8*(256*A+256
*B)/(tan(1/2*f*x+1/2*e)-1)^8-1/6*(544*A+448*B)/(tan(1/2*f*x+1/2*e)-1)^6-1/4*(200*A+104*B)/(tan(1/2*f*x+1/2*e)-
1)^4-1/7*(480*A+448*B)/(tan(1/2*f*x+1/2*e)-1)^7-1/2*(12*A+2*B)/(tan(1/2*f*x+1/2*e)-1)^2-A/(tan(1/2*f*x+1/2*e)-
1)-1/5*(404*A+276*B)/(tan(1/2*f*x+1/2*e)-1)^5)
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Maxima [B] time = 1.25865, size = 2817, normalized size = 24.5 \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*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="maxima")
[Out]
-2/315*(A*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/
(cos(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*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) - 10*A*a^
2*(45*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*s
in(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) - 5*B*a^2*(45*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*s
in(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*B*a^2*(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*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*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) + 28*B*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 = 1.45711, size = 833, normalized size = 7.24 \begin{align*} \frac{{\left (2 \, A - 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 4 \,{\left (2 \, A - 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 5 \,{\left (5 \, A + 14 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 5 \,{\left (17 \, A + 35 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 70 \,{\left (A + B\right )} a^{2} \cos \left (f x + e\right ) + 140 \,{\left (A + B\right )} a^{2} +{\left ({\left (2 \, A - 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} + 5 \,{\left (2 \, A - 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 15 \,{\left (A + 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 70 \,{\left (A + B\right )} a^{2} \cos \left (f x + e\right ) + 140 \,{\left (A + B\right )} 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*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="fricas")
[Out]
1/315*((2*A - 7*B)*a^2*cos(f*x + e)^5 - 4*(2*A - 7*B)*a^2*cos(f*x + e)^4 - 5*(5*A + 14*B)*a^2*cos(f*x + e)^3 -
5*(17*A + 35*B)*a^2*cos(f*x + e)^2 + 70*(A + B)*a^2*cos(f*x + e) + 140*(A + B)*a^2 + ((2*A - 7*B)*a^2*cos(f*x
+ e)^4 + 5*(2*A - 7*B)*a^2*cos(f*x + e)^3 - 15*(A + 7*B)*a^2*cos(f*x + e)^2 + 70*(A + B)*a^2*cos(f*x + e) + 1
40*(A + B)*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))
________________________________________________________________________________________
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*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**5,x)
[Out]
Timed out
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Giac [B] time = 1.23672, size = 406, normalized size = 3.53 \begin{align*} -\frac{2 \,{\left (315 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 630 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 315 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 2310 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 105 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 2520 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 945 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3402 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 63 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 1638 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 693 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 1062 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 63 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 108 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 63 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 47 \, A a^{2} - 7 \, B 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*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="giac")
[Out]
-2/315*(315*A*a^2*tan(1/2*f*x + 1/2*e)^8 - 630*A*a^2*tan(1/2*f*x + 1/2*e)^7 + 315*B*a^2*tan(1/2*f*x + 1/2*e)^7
+ 2310*A*a^2*tan(1/2*f*x + 1/2*e)^6 + 105*B*a^2*tan(1/2*f*x + 1/2*e)^6 - 2520*A*a^2*tan(1/2*f*x + 1/2*e)^5 +
945*B*a^2*tan(1/2*f*x + 1/2*e)^5 + 3402*A*a^2*tan(1/2*f*x + 1/2*e)^4 + 63*B*a^2*tan(1/2*f*x + 1/2*e)^4 - 1638*
A*a^2*tan(1/2*f*x + 1/2*e)^3 + 693*B*a^2*tan(1/2*f*x + 1/2*e)^3 + 1062*A*a^2*tan(1/2*f*x + 1/2*e)^2 + 63*B*a^2
*tan(1/2*f*x + 1/2*e)^2 - 108*A*a^2*tan(1/2*f*x + 1/2*e) + 63*B*a^2*tan(1/2*f*x + 1/2*e) + 47*A*a^2 - 7*B*a^2)
/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)