\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx\) [100]

Optimal result

Integrand size = 38, antiderivative size = 124 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 a^3 (11 A+3 B) c^5 \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 (11 A+3 B) c^4 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}} \]

[Out]

8/693*a^3*(11*A+3*B)*c^5*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)+2/99*a^3*(11*A+3*B)*c^4*cos(f*x+e)^7/f/(c-c*sin
(f*x+e))^(5/2)-2/11*a^3*B*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(3/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2935, 2753, 2752} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 a^3 c^5 (11 A+3 B) \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 c^4 (11 A+3 B) \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}} \]

[In]

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]

[Out]

(8*a^3*(11*A + 3*B)*c^5*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^(7/2)) + (2*a^3*(11*A + 3*B)*c^4*Cos[e + f
*x]^7)/(99*f*(c - c*Sin[e + f*x])^(5/2)) - (2*a^3*B*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^(3/2))

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^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = -\frac {2 a^3 B c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}+\frac {1}{11} \left (a^3 (11 A+3 B) c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {2 a^3 (11 A+3 B) c^4 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}}+\frac {1}{99} \left (4 a^3 (11 A+3 B) c^4\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx \\ & = \frac {8 a^3 (11 A+3 B) c^5 \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 (11 A+3 B) c^4 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^{3/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1157\) vs. \(2(124)=248\).

Time = 12.55 (sec) , antiderivative size = 1157, normalized size of antiderivative = 9.33 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {(6 A+B) \cos \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {(8 A+3 B) \cos \left (\frac {3}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}}{24 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {B \cos \left (\frac {5}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}}{16 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {(6 A+B) \cos \left (\frac {7}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {(2 A+3 B) \cos \left (\frac {9}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}}{144 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {B \cos \left (\frac {11}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}}{176 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {(6 A+B) \sin \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {(8 A+3 B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \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 )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {B (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \sin \left (\frac {5}{2} (e+f x)\right )}{16 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {(6 A+B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \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 )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {(2 A+3 B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \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 )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {B (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \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 )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \]

[In]

Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2),x]

[Out]

((6*A + B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e
+ f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((8*A + 3*B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x]
)^3*(c - c*Sin[e + f*x])^(3/2))/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x
)/2])^6) - (B*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(16*f*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((6*A + B)*Cos[(7*(e + f*x))/2]*(a + a*Sin[e
+ f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(
e + f*x)/2])^6) - ((2*A + 3*B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(144*f*
(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (B*Cos[(11*(e + f*x))/2]*(a
+ a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(176*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)
/2] + Sin[(e + f*x)/2])^6) + ((6*A + B)*Sin[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2))/(8
*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((8*A + 3*B)*(a + a*Sin[
e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(3*(e + f*x))/2])/(24*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (B*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(5*(e + f*x))
/2])/(16*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((6*A + B)*(a +
a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(7*(e + f*x))/2])/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]
)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((2*A + 3*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*
Sin[(9*(e + f*x))/2])/(144*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)
- (B*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)*Sin[(11*(e + f*x))/2])/(176*f*(Cos[(e + f*x)/2] - Sin[(
e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.67

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right )^{4} a^{3} \left (-63 B \left (\cos ^{2}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \left (77 A -105 B \right )-121 A +93 B \right )}{693 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(83\)
parts	\(\frac {2 A \,a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-5\right )}{3 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 B \,a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (15 \left (\sin ^{5}\left (f x +e \right )\right )-35 \left (\sin ^{4}\left (f x +e \right )\right )+40 \left (\sin ^{3}\left (f x +e \right )\right )-48 \left (\sin ^{2}\left (f x +e \right )\right )+64 \sin \left (f x +e \right )-128\right )}{165 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (A +3 B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )-85 \left (\sin ^{3}\left (f x +e \right )\right )+102 \left (\sin ^{2}\left (f x +e \right )\right )-136 \sin \left (f x +e \right )+272\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (3 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )-3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (1+\sin \left (f x +e \right )\right ) \left (15 \left (\sin ^{3}\left (f x +e \right )\right )-39 \left (\sin ^{2}\left (f x +e \right )\right )+52 \sin \left (f x +e \right )-104\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(403\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/693*(sin(f*x+e)-1)*c^2*(1+sin(f*x+e))^4*a^3*(-63*B*cos(f*x+e)^2+sin(f*x+e)*(77*A-105*B)-121*A+93*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. 287 vs. \(2 (112) = 224\).

Time = 0.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.31 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {2 \, {\left (63 \, B a^{3} c \cos \left (f x + e\right )^{6} - 7 \, {\left (11 \, A + 12 \, B\right )} a^{3} c \cos \left (f x + e\right )^{5} - {\left (187 \, A + 177 \, B\right )} a^{3} c \cos \left (f x + e\right )^{4} + 2 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right )^{3} - 4 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right )^{2} + 16 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right ) + 32 \, {\left (11 \, A + 3 \, B\right )} a^{3} c - {\left (63 \, B a^{3} c \cos \left (f x + e\right )^{5} + 7 \, {\left (11 \, A + 21 \, B\right )} a^{3} c \cos \left (f x + e\right )^{4} - 10 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right )^{3} - 12 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right )^{2} - 16 \, {\left (11 \, A + 3 \, B\right )} a^{3} c \cos \left (f x + e\right ) - 32 \, {\left (11 \, A + 3 \, B\right )} a^{3} c\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{693 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

2/693*(63*B*a^3*c*cos(f*x + e)^6 - 7*(11*A + 12*B)*a^3*c*cos(f*x + e)^5 - (187*A + 177*B)*a^3*c*cos(f*x + e)^4
+ 2*(11*A + 3*B)*a^3*c*cos(f*x + e)^3 - 4*(11*A + 3*B)*a^3*c*cos(f*x + e)^2 + 16*(11*A + 3*B)*a^3*c*cos(f*x +
e) + 32*(11*A + 3*B)*a^3*c - (63*B*a^3*c*cos(f*x + e)^5 + 7*(11*A + 21*B)*a^3*c*cos(f*x + e)^4 - 10*(11*A + 3
*B)*a^3*c*cos(f*x + e)^3 - 12*(11*A + 3*B)*a^3*c*cos(f*x + e)^2 - 16*(11*A + 3*B)*a^3*c*cos(f*x + e) - 32*(11*
A + 3*B)*a^3*c)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)

Sympy [F]

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

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(3/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*(-c*sin(f*x + e) + c)^(3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (112) = 224\).

Time = 0.49 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.37 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (693 \, B a^{3} c \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 63 \, B a^{3} c \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 ) + 1386 \, {\left (6 \, A a^{3} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{3} c \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 ) + 462 \, {\left (8 \, A a^{3} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{3} c \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 ) - 99 \, {\left (6 \, A a^{3} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{3} c \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 ) - 77 \, {\left (2 \, A a^{3} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{3} c \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}}{11088 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

-1/11088*sqrt(2)*(693*B*a^3*c*cos(-5/4*pi + 5/2*f*x + 5/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 63*B*a^3*c*
cos(-11/4*pi + 11/2*f*x + 11/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 1386*(6*A*a^3*c*sgn(sin(-1/4*pi + 1/2*
f*x + 1/2*e)) + B*a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1/2*f*x + 1/2*e) + 462*(8*A*a^3*c*s
gn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*B*a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-3/4*pi + 3/2*f*x + 3/
2*e) - 99*(6*A*a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + B*a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-7
/4*pi + 7/2*f*x + 7/2*e) - 77*(2*A*a^3*c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*B*a^3*c*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. \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(3/2),x)

[Out]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(3/2), x)