\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx\) [98]

Optimal result

Integrand size = 38, antiderivative size = 210 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a^3 (15 A-B) c^7 \cos ^7(e+f x)}{45045 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 (15 A-B) c^6 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac {8 a^3 (15 A-B) c^5 \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 B c^3 \cos ^7(e+f x) \sqrt {c-c \sin (e+f x)}}{15 f} \]

[Out]

256/45045*a^3*(15*A-B)*c^7*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)+64/6435*a^3*(15*A-B)*c^6*cos(f*x+e)^7/f/(c-c*
sin(f*x+e))^(5/2)+8/715*a^3*(15*A-B)*c^5*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(3/2)+2/195*a^3*(15*A-B)*c^4*cos(f*x+
e)^7/f/(c-c*sin(f*x+e))^(1/2)-2/15*a^3*B*c^3*cos(f*x+e)^7*(c-c*sin(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2935, 2753, 2752} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a^3 c^7 (15 A-B) \cos ^7(e+f x)}{45045 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 c^6 (15 A-B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac {8 a^3 c^5 (15 A-B) \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 (15 A-B) \cos ^7(e+f x)}{195 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 B c^3 \cos ^7(e+f x) \sqrt {c-c \sin (e+f x)}}{15 f} \]

[In]

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

(256*a^3*(15*A - B)*c^7*Cos[e + f*x]^7)/(45045*f*(c - c*Sin[e + f*x])^(7/2)) + (64*a^3*(15*A - B)*c^6*Cos[e +
f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^(5/2)) + (8*a^3*(15*A - B)*c^5*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x]
)^(3/2)) + (2*a^3*(15*A - B)*c^4*Cos[e + f*x]^7)/(195*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a^3*B*c^3*Cos[e + f*x]^
7*Sqrt[c - c*Sin[e + f*x]])/(15*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^3 c^3\right ) \int \cos ^6(e+f x) (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx \\ & = -\frac {2 a^3 B c^3 \cos ^7(e+f x) \sqrt {c-c \sin (e+f x)}}{15 f}+\frac {1}{15} \left (a^3 (15 A-B) c^3\right ) \int \cos ^6(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 B c^3 \cos ^7(e+f x) \sqrt {c-c \sin (e+f x)}}{15 f}+\frac {1}{65} \left (4 a^3 (15 A-B) c^4\right ) \int \frac {\cos ^6(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {8 a^3 (15 A-B) c^5 \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 B c^3 \cos ^7(e+f x) \sqrt {c-c \sin (e+f x)}}{15 f}+\frac {1}{715} \left (32 a^3 (15 A-B) c^5\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {64 a^3 (15 A-B) c^6 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac {8 a^3 (15 A-B) c^5 \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 B c^3 \cos ^7(e+f x) \sqrt {c-c \sin (e+f x)}}{15 f}+\frac {\left (128 a^3 (15 A-B) c^6\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx}{6435} \\ & = \frac {256 a^3 (15 A-B) c^7 \cos ^7(e+f x)}{45045 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 (15 A-B) c^6 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac {8 a^3 (15 A-B) c^5 \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 B c^3 \cos ^7(e+f x) \sqrt {c-c \sin (e+f x)}}{15 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1569\) vs. \(2(210)=420\).

Time = 11.61 (sec) , antiderivative size = 1569, normalized size of antiderivative = 7.47 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {5 (8 A-B) \cos \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{64 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {5 (6 A+B) \cos \left (\frac {3}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{192 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {3 (10 A-3 B) \cos \left (\frac {5}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{320 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {3 (4 A+3 B) \cos \left (\frac {7}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{448 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {(12 A-5 B) \cos \left (\frac {9}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{576 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {(2 A+5 B) \cos \left (\frac {11}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{704 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {(2 A-B) \cos \left (\frac {13}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{832 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \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 {15}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{960 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {5 (8 A-B) \sin \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2}}{64 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {5 (6 A+B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {3}{2} (e+f x)\right )}{192 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {3 (10 A-3 B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {5}{2} (e+f x)\right )}{320 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {3 (4 A+3 B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {7}{2} (e+f x)\right )}{448 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {(12 A-5 B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {9}{2} (e+f x)\right )}{576 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {(2 A+5 B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {11}{2} (e+f x)\right )}{704 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {(2 A-B) (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {13}{2} (e+f x)\right )}{832 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \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))^{7/2} \sin \left (\frac {15}{2} (e+f x)\right )}{960 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \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])^(7/2),x]

[Out]

(5*(8*A - B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(64*f*(Cos[(e + f*x)/2] - Sin
[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (5*(6*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f
*x])^3*(c - c*Sin[e + f*x])^(7/2))/(192*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])^6) + (3*(10*A - 3*B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(320*f*
(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (3*(4*A + 3*B)*Cos[(7*(e +
f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(448*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((12*A - 5*B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e
+ f*x])^(7/2))/(576*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((2*
A + 5*B)*Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(704*f*(Cos[(e + f*x)/2] - S
in[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A - B)*Cos[(13*(e + f*x))/2]*(a + a*Sin[e +
f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(832*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^6) - (B*Cos[(15*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(960*f*(Cos[(e + f
*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (5*(8*A - B)*Sin[(e + f*x)/2]*(a + a*S
in[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(64*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + S
in[(e + f*x)/2])^6) + (5*(6*A + B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(3*(e + f*x))/2])/(19
2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (3*(10*A - 3*B)*(a + a*
Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(5*(e + f*x))/2])/(320*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^
7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (3*(4*A + 3*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*
Sin[(7*(e + f*x))/2])/(448*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)
+ ((12*A - 5*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(9*(e + f*x))/2])/(576*f*(Cos[(e + f*x)/
2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A + 5*B)*(a + a*Sin[e + f*x])^3*(c - c
*Sin[e + f*x])^(7/2)*Sin[(11*(e + f*x))/2])/(704*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2])^6) + ((2*A - B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(13*(e + f*x))/2])/(8
32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(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])^(7/2)*Sin[(15*(e + f*x))/2])/(960*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e +
f*x)/2] + Sin[(e + f*x)/2])^6)

Maple [A] (verified)

Time = 167.16 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.58

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right )^{4} a^{3} \left (3003 B \left (\cos ^{4}\left (f x +e \right )\right )+\left (-3465 A +12243 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (14175 A -24969 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (24780 A -25676 B \right ) \sin \left (f x +e \right )-26700 A +25804 B \right )}{45045 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(121\)
parts	\(\frac {2 A \,a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (5 \left (\sin ^{3}\left (f x +e \right )\right )-27 \left (\sin ^{2}\left (f x +e \right )\right )+71 \sin \left (f x +e \right )-177\right )}{35 \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^{4} \left (1+\sin \left (f x +e \right )\right ) \left (1001 \left (\sin ^{7}\left (f x +e \right )\right )-4543 \left (\sin ^{6}\left (f x +e \right )\right )+9051 \left (\sin ^{5}\left (f x +e \right )\right )-11725 \left (\sin ^{4}\left (f x +e \right )\right )+13400 \left (\sin ^{3}\left (f x +e \right )\right )-16080 \left (\sin ^{2}\left (f x +e \right )\right )+21440 \sin \left (f x +e \right )-42880\right )}{15015 \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^{4} \left (1+\sin \left (f x +e \right )\right ) \left (165 \left (\sin ^{6}\left (f x +e \right )\right )-765 \left (\sin ^{5}\left (f x +e \right )\right )+1565 \left (\sin ^{4}\left (f x +e \right )\right )-2095 \left (\sin ^{3}\left (f x +e \right )\right )+2514 \left (\sin ^{2}\left (f x +e \right )\right )-3352 \sin \left (f x +e \right )+6704\right )}{2145 \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^{4} \left (1+\sin \left (f x +e \right )\right ) \left (5 \left (\sin ^{4}\left (f x +e \right )\right )-25 \left (\sin ^{3}\left (f x +e \right )\right )+57 \left (\sin ^{2}\left (f x +e \right )\right )-91 \sin \left (f x +e \right )+182\right )}{45 \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^{4} \left (1+\sin \left (f x +e \right )\right ) \left (315 \left (\sin ^{5}\left (f x +e \right )\right )-1505 \left (\sin ^{4}\left (f x +e \right )\right )+3205 \left (\sin ^{3}\left (f x +e \right )\right )-4539 \left (\sin ^{2}\left (f x +e \right )\right )+6052 \sin \left (f x +e \right )-12104\right )}{1155 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(507\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/45045*(sin(f*x+e)-1)*c^4*(1+sin(f*x+e))^4*a^3*(3003*B*cos(f*x+e)^4+(-3465*A+12243*B)*cos(f*x+e)^2*sin(f*x+e)
+(14175*A-24969*B)*cos(f*x+e)^2+(24780*A-25676*B)*sin(f*x+e)-26700*A+25804*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. 405 vs. \(2 (190) = 380\).

Time = 0.28 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.93 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=-\frac {2 \, {\left (3003 \, B a^{3} c^{3} \cos \left (f x + e\right )^{8} - 231 \, {\left (15 \, A - 14 \, B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{7} + 21 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{6} - 28 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{5} + 40 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{4} - 64 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{3} + 128 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{2} - 512 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right ) - 1024 \, {\left (15 \, A - B\right )} a^{3} c^{3} - {\left (3003 \, B a^{3} c^{3} \cos \left (f x + e\right )^{7} + 231 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{6} + 252 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{5} + 280 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{4} + 320 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{3} + 384 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, {\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, {\left (15 \, A - B\right )} a^{3} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{45045 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

-2/45045*(3003*B*a^3*c^3*cos(f*x + e)^8 - 231*(15*A - 14*B)*a^3*c^3*cos(f*x + e)^7 + 21*(15*A - B)*a^3*c^3*cos
(f*x + e)^6 - 28*(15*A - B)*a^3*c^3*cos(f*x + e)^5 + 40*(15*A - B)*a^3*c^3*cos(f*x + e)^4 - 64*(15*A - B)*a^3*
c^3*cos(f*x + e)^3 + 128*(15*A - B)*a^3*c^3*cos(f*x + e)^2 - 512*(15*A - B)*a^3*c^3*cos(f*x + e) - 1024*(15*A
- B)*a^3*c^3 - (3003*B*a^3*c^3*cos(f*x + e)^7 + 231*(15*A - B)*a^3*c^3*cos(f*x + e)^6 + 252*(15*A - B)*a^3*c^3
*cos(f*x + e)^5 + 280*(15*A - B)*a^3*c^3*cos(f*x + e)^4 + 320*(15*A - B)*a^3*c^3*cos(f*x + e)^3 + 384*(15*A -
B)*a^3*c^3*cos(f*x + e)^2 + 512*(15*A - B)*a^3*c^3*cos(f*x + e) + 1024*(15*A - B)*a^3*c^3)*sin(f*x + e))*sqrt(
-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)

Sympy [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*(-c*sin(f*x + e) + c)^(7/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (190) = 380\).

Time = 0.56 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.18 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {\sqrt {2} {\left (3003 \, B a^{3} c^{3} \cos \left (-\frac {15}{4} \, \pi + \frac {15}{2} \, f x + \frac {15}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 225225 \, {\left (8 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{3} c^{3} \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 ) - 75075 \, {\left (6 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{3} c^{3} \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 ) + 27027 \, {\left (10 \, A a^{3} c^{3} \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^{3} \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 ) + 19305 \, {\left (4 \, A a^{3} c^{3} \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^{3} \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 ) - 5005 \, {\left (12 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 5 \, B a^{3} c^{3} \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 ) - 4095 \, {\left (2 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) + 3465 \, {\left (2 \, A a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{3} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, f x + \frac {13}{2} \, e\right )\right )} \sqrt {c}}{2882880 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

1/2882880*sqrt(2)*(3003*B*a^3*c^3*cos(-15/4*pi + 15/2*f*x + 15/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 2252
25*(8*A*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - B*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*
pi + 1/2*f*x + 1/2*e) - 75075*(6*A*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + B*a^3*c^3*sgn(sin(-1/4*pi + 1
/2*f*x + 1/2*e)))*cos(-3/4*pi + 3/2*f*x + 3/2*e) + 27027*(10*A*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 3
*B*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-5/4*pi + 5/2*f*x + 5/2*e) + 19305*(4*A*a^3*c^3*sgn(sin(-1
/4*pi + 1/2*f*x + 1/2*e)) + 3*B*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-7/4*pi + 7/2*f*x + 7/2*e) -
5005*(12*A*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 5*B*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(
-9/4*pi + 9/2*f*x + 9/2*e) - 4095*(2*A*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*B*a^3*c^3*sgn(sin(-1/4*
pi + 1/2*f*x + 1/2*e)))*cos(-11/4*pi + 11/2*f*x + 11/2*e) + 3465*(2*A*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*
e)) - B*a^3*c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-13/4*pi + 13/2*f*x + 13/2*e))*sqrt(c)/f

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(7/2),x)

[Out]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(7/2), x)