\(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx\) [90]

Optimal result

Integrand size = 38, antiderivative size = 167 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {64 a^2 (11 A-B) c^5 \cos ^5(e+f x)}{3465 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 (11 A-B) c^4 \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \]

[Out]

64/3465*a^2*(11*A-B)*c^5*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(5/2)+16/693*a^2*(11*A-B)*c^4*cos(f*x+e)^5/f/(c-c*sin
(f*x+e))^(3/2)+2/99*a^2*(11*A-B)*c^3*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^(1/2)-2/11*a^2*B*c^2*cos(f*x+e)^5*(c-c*si
n(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2935, 2753, 2752} \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {64 a^2 c^5 (11 A-B) \cos ^5(e+f x)}{3465 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 c^4 (11 A-B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 (11 A-B) \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \]

[In]

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

[Out]

(64*a^2*(11*A - B)*c^5*Cos[e + f*x]^5)/(3465*f*(c - c*Sin[e + f*x])^(5/2)) + (16*a^2*(11*A - B)*c^4*Cos[e + f*
x]^5)/(693*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^2*(11*A - B)*c^3*Cos[e + f*x]^5)/(99*f*Sqrt[c - c*Sin[e + f*x]
]) - (2*a^2*B*c^2*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(11*f)

Rule 2752

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rule 2753

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 - 1)/(f*g*(m + p))), x] + Dist[a*((2*m + p - 1)/(m + p)), 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] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

Rule 2935

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x
] + Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; F
reeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p +
1, 0]

Rule 3046

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]))

Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx \\ & = -\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{11} \left (a^2 (11 A-B) c^2\right ) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{99} \left (8 a^2 (11 A-B) c^3\right ) \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {16 a^2 (11 A-B) c^4 \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{693} \left (32 a^2 (11 A-B) c^4\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {64 a^2 (11 A-B) c^5 \cos ^5(e+f x)}{3465 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 (11 A-B) c^4 \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1173\) vs. \(2(167)=334\).

Time = 12.62 (sec) , antiderivative size = 1173, normalized size of antiderivative = 7.02 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {(6 A-B) \cos \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(4 A+B) \cos \left (\frac {3}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{24 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(8 A-3 B) \cos \left (\frac {5}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{80 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(2 A+3 B) \cos \left (\frac {7}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(2 A-B) \cos \left (\frac {9}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{144 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {B \cos \left (\frac {11}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{176 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(6 A-B) \sin \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(4 A+B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {3}{2} (e+f x)\right )}{24 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(8 A-3 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {5}{2} (e+f x)\right )}{80 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(2 A+3 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {7}{2} (e+f x)\right )}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(2 A-B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {9}{2} (e+f x)\right )}{144 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {B (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {11}{2} (e+f x)\right )}{176 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \]

[In]

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

[Out]

((6*A - B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e
+ f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((4*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x])^
2*(c - c*Sin[e + f*x])^(5/2))/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/
2])^4) + ((8*A - 3*B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(80*f*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((2*A + 3*B)*Cos[(7*(e + f*x))/2]*(a
+ a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/
2] + Sin[(e + f*x)/2])^4) + ((2*A - B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))
/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - (B*Cos[(11*(e + f*x
))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2))/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(
e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((6*A - B)*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(
5/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((4*A + B)*(a +
a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*Sin[(3*(e + f*x))/2])/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])
^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((8*A - 3*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*S
in[(5*(e + f*x))/2])/(80*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) +
((2*A + 3*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*Sin[(7*(e + f*x))/2])/(112*f*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2*A - B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[
e + f*x])^(5/2)*Sin[(9*(e + f*x))/2])/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(
e + f*x)/2])^4) + (B*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)*Sin[(11*(e + f*x))/2])/(176*f*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)

Maple [A] (verified)

Time = 36.99 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.63

method	result	size
default	\(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (-315 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-385 A +980 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (-1210 A +1370 B \right ) \sin \left (f x +e \right )+1562 A -1402 B \right )}{3465 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(105\)
parts	\(-\frac {2 A \,a^{2} \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )-14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 B \,a^{2} \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (63 \left (\sin ^{5}\left (f x +e \right )\right )-224 \left (\sin ^{4}\left (f x +e \right )\right )+355 \left (\sin ^{3}\left (f x +e \right )\right )-426 \left (\sin ^{2}\left (f x +e \right )\right )+568 \sin \left (f x +e \right )-1136\right )}{693 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{2} \left (A +2 B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )-130 \left (\sin ^{3}\left (f x +e \right )\right )+219 \left (\sin ^{2}\left (f x +e \right )\right )-292 \sin \left (f x +e \right )+584\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{2} \left (2 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right ) \left (3 \left (\sin ^{3}\left (f x +e \right )\right )-12 \left (\sin ^{2}\left (f x +e \right )\right )+23 \sin \left (f x +e \right )-46\right )}{21 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(346\)

[In]

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3465*(sin(f*x+e)-1)*c^3*(1+sin(f*x+e))^3*a^2*(-315*B*cos(f*x+e)^2*sin(f*x+e)+(-385*A+980*B)*cos(f*x+e)^2+(-
1210*A+1370*B)*sin(f*x+e)+1562*A-1402*B)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (151) = 302\).

Time = 0.28 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.87 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (315 \, B a^{2} c^{2} \cos \left (f x + e\right )^{6} - 35 \, {\left (11 \, A - 10 \, B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{5} + 5 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{3} + 16 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{2} - 64 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right ) - 128 \, {\left (11 \, A - B\right )} a^{2} c^{2} - {\left (315 \, B a^{2} c^{2} \cos \left (f x + e\right )^{5} + 35 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{4} + 40 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{3} + 48 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{2} + 64 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right ) + 128 \, {\left (11 \, A - B\right )} a^{2} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3465 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-2/3465*(315*B*a^2*c^2*cos(f*x + e)^6 - 35*(11*A - 10*B)*a^2*c^2*cos(f*x + e)^5 + 5*(11*A - B)*a^2*c^2*cos(f*x
+ e)^4 - 8*(11*A - B)*a^2*c^2*cos(f*x + e)^3 + 16*(11*A - B)*a^2*c^2*cos(f*x + e)^2 - 64*(11*A - B)*a^2*c^2*c
os(f*x + e) - 128*(11*A - B)*a^2*c^2 - (315*B*a^2*c^2*cos(f*x + e)^5 + 35*(11*A - B)*a^2*c^2*cos(f*x + e)^4 +
40*(11*A - B)*a^2*c^2*cos(f*x + e)^3 + 48*(11*A - B)*a^2*c^2*cos(f*x + e)^2 + 64*(11*A - B)*a^2*c^2*cos(f*x +
e) + 128*(11*A - B)*a^2*c^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)

Sympy [F]

\[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=a^{2} \left (\int A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int \left (- 2 A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\, dx + \int B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\right )\, dx + \int B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{5}{\left (e + f x \right )}\, dx\right ) \]

[In]

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(5/2),x)

[Out]

a**2*(Integral(A*c**2*sqrt(-c*sin(e + f*x) + c), x) + Integral(-2*A*c**2*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x
)**2, x) + Integral(A*c**2*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**4, x) + Integral(B*c**2*sqrt(-c*sin(e + f*x
) + c)*sin(e + f*x), x) + Integral(-2*B*c**2*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**3, x) + Integral(B*c**2*s
qrt(-c*sin(e + f*x) + c)*sin(e + f*x)**5, x))

Maxima [F]

\[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (151) = 302\).

Time = 0.62 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.04 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {\sqrt {2} {\left (315 \, B a^{2} c^{2} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6930 \, {\left (6 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2310 \, {\left (4 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) - 693 \, {\left (8 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 3 \, B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) - 495 \, {\left (2 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 385 \, {\left (2 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right )\right )} \sqrt {c}}{55440 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

-1/55440*sqrt(2)*(315*B*a^2*c^2*cos(-11/4*pi + 11/2*f*x + 11/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 6930*(
6*A*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - B*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi +
1/2*f*x + 1/2*e) + 2310*(4*A*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + B*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*
x + 1/2*e)))*cos(-3/4*pi + 3/2*f*x + 3/2*e) - 693*(8*A*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*B*a^2*c
^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-5/4*pi + 5/2*f*x + 5/2*e) - 495*(2*A*a^2*c^2*sgn(sin(-1/4*pi + 1/
2*f*x + 1/2*e)) + 3*B*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-7/4*pi + 7/2*f*x + 7/2*e) + 385*(2*A*a
^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - B*a^2*c^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-9/4*pi + 9/2*
f*x + 9/2*e))*sqrt(c)/f

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

[In]

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

[Out]

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