3.1.26 \(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx\) [26]

Optimal. Leaf size=229 \[ \frac {9}{128} a^2 (8 A-3 B) c^5 x+\frac {3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac {9 a^2 (8 A-3 B) c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {3 a^2 (8 A-3 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f} \]

[Out]

9/128*a^2*(8*A-3*B)*c^5*x+3/80*a^2*(8*A-3*B)*c^5*cos(f*x+e)^5/f+9/128*a^2*(8*A-3*B)*c^5*cos(f*x+e)*sin(f*x+e)/
f+3/64*a^2*(8*A-3*B)*c^5*cos(f*x+e)^3*sin(f*x+e)/f+1/56*a^2*(8*A-3*B)*c^3*cos(f*x+e)^5*(c-c*sin(f*x+e))^2/f-1/
8*a^2*B*c^2*cos(f*x+e)^5*(c-c*sin(f*x+e))^3/f+3/112*a^2*(8*A-3*B)*cos(f*x+e)^5*(c^5-c^5*sin(f*x+e))/f

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Rubi [A]
time = 0.25, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3046, 2939, 2757, 2748, 2715, 8} \begin {gather*} \frac {3 a^2 c^5 (8 A-3 B) \cos ^5(e+f x)}{80 f}+\frac {3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac {3 a^2 c^5 (8 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac {9 a^2 c^5 (8 A-3 B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac {9}{128} a^2 c^5 x (8 A-3 B)+\frac {a^2 c^3 (8 A-3 B) \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f} \end {gather*}

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]

(9*a^2*(8*A - 3*B)*c^5*x)/128 + (3*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]^5)/(80*f) + (9*a^2*(8*A - 3*B)*c^5*Cos[e +
f*x]*Sin[e + f*x])/(128*f) + (3*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(64*f) + (a^2*(8*A - 3*B)*c^
3*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^2)/(56*f) - (a^2*B*c^2*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^3)/(8*f) + (3
*a^2*(8*A - 3*B)*Cos[e + f*x]^5*(c^5 - c^5*Sin[e + f*x]))/(112*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2757

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] && GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2939

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] && 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*} \int (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 \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx\\ &=-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {1}{8} \left (a^2 (8 A-3 B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^3 \, dx\\ &=\frac {a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {1}{56} \left (9 a^2 (8 A-3 B) c^3\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=\frac {a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac {1}{16} \left (3 a^2 (8 A-3 B) c^4\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac {a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac {1}{16} \left (3 a^2 (8 A-3 B) c^5\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac {3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac {3 a^2 (8 A-3 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac {1}{64} \left (9 a^2 (8 A-3 B) c^5\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac {9 a^2 (8 A-3 B) c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {3 a^2 (8 A-3 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac {1}{128} \left (9 a^2 (8 A-3 B) c^5\right ) \int 1 \, dx\\ &=\frac {9}{128} a^2 (8 A-3 B) c^5 x+\frac {3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac {9 a^2 (8 A-3 B) c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {3 a^2 (8 A-3 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}\\ \end {align*}

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Mathematica [A]
time = 1.24, size = 219, normalized size = 0.96 \begin {gather*} \frac {(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5 (2520 (8 A-3 B) (e+f x)+560 (27 A-17 B) \cos (e+f x)+560 (13 A-7 B) \cos (3 (e+f x))+112 (11 A-B) \cos (5 (e+f x))-80 (A-3 B) \cos (7 (e+f x))+560 (19 A-3 B) \sin (2 (e+f x))-280 (2 A-7 B) \sin (4 (e+f x))-560 (A-B) \sin (6 (e+f x))-35 B \sin (8 (e+f x)))}{35840 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \end {gather*}

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 + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5*(2520*(8*A - 3*B)*(e + f*x) + 560*(27*A - 17*B)*Cos[e + f*x] +
560*(13*A - 7*B)*Cos[3*(e + f*x)] + 112*(11*A - B)*Cos[5*(e + f*x)] - 80*(A - 3*B)*Cos[7*(e + f*x)] + 560*(19*
A - 3*B)*Sin[2*(e + f*x)] - 280*(2*A - 7*B)*Sin[4*(e + f*x)] - 560*(A - B)*Sin[6*(e + f*x)] - 35*B*Sin[8*(e +
f*x)]))/(35840*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(568\) vs. \(2(215)=430\).
time = 0.36, size = 569, normalized size = 2.48 Too large to display

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,method=_RETURNVERBOSE)

[Out]

1/f*(a^2*A*c^5*(f*x+e)-B*a^2*c^5*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/1
6*e)+B*a^2*c^5*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)+5*B*a^2*c^5*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*
cos(f*x+e)+3/8*f*x+3/8*e)+3*a^2*A*c^5*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*
x+5/16*e)-3/7*B*a^2*c^5*(16/5+sin(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(f*x+e)^2)*cos(f*x+e)+1/7*a^2*A*c^5*(16/5+s
in(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(f*x+e)^2)*cos(f*x+e)-B*a^2*c^5*(-1/8*(sin(f*x+e)^7+7/6*sin(f*x+e)^5+35/24
*sin(f*x+e)^3+35/16*sin(f*x+e))*cos(f*x+e)+35/128*f*x+35/128*e)-B*a^2*c^5*cos(f*x+e)+3*a^2*A*c^5*cos(f*x+e)-3*
B*a^2*c^5*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)+a^2*A*c^5*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-1/3*
B*a^2*c^5*(2+sin(f*x+e)^2)*cos(f*x+e)+1/5*a^2*A*c^5*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-5*a^2*A*c^5
*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-5/3*a^2*A*c^5*(2+sin(f*x+e)^2)*cos(f*x+e))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (229) = 458\).
time = 0.31, size = 615, normalized size = 2.69 \begin {gather*} -\frac {3072 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} A a^{2} c^{5} - 7168 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} A a^{2} c^{5} - 179200 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c^{5} - 1680 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{5} + 16800 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{5} - 26880 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{5} - 107520 \, {\left (f x + e\right )} A a^{2} c^{5} - 9216 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} B a^{2} c^{5} - 35840 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{2} c^{5} - 35840 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c^{5} + 35 \, {\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{5} + 560 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{5} - 16800 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{5} + 80640 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{5} - 322560 \, A a^{2} c^{5} \cos \left (f x + e\right ) + 107520 \, B a^{2} c^{5} \cos \left (f x + e\right )}{107520 \, f} \end {gather*}

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]

-1/107520*(3072*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*A*a^2*c^5 - 7168*
(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^2*c^5 - 179200*(cos(f*x + e)^3 - 3*cos(f*x + e))*
A*a^2*c^5 - 1680*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*A*a^2*c^5 +
16800*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^2*c^5 - 26880*(2*f*x + 2*e - sin(2*f*x + 2*
e))*A*a^2*c^5 - 107520*(f*x + e)*A*a^2*c^5 - 9216*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 -
35*cos(f*x + e))*B*a^2*c^5 - 35840*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^2*c^5 - 35840*
(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^2*c^5 + 35*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e
) + 168*sin(4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*B*a^2*c^5 + 560*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin
(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*B*a^2*c^5 - 16800*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))
*B*a^2*c^5 + 80640*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^2*c^5 - 322560*A*a^2*c^5*cos(f*x + e) + 107520*B*a^2*c
^5*cos(f*x + e))/f

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Fricas [A]
time = 0.38, size = 165, normalized size = 0.72 \begin {gather*} -\frac {640 \, {\left (A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{7} - 3584 \, {\left (A - B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{5} - 315 \, {\left (8 \, A - 3 \, B\right )} a^{2} c^{5} f x + 35 \, {\left (16 \, B a^{2} c^{5} \cos \left (f x + e\right )^{7} + 8 \, {\left (8 \, A - 11 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{5} - 6 \, {\left (8 \, A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{3} - 9 \, {\left (8 \, A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4480 \, f} \end {gather*}

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]

-1/4480*(640*(A - 3*B)*a^2*c^5*cos(f*x + e)^7 - 3584*(A - B)*a^2*c^5*cos(f*x + e)^5 - 315*(8*A - 3*B)*a^2*c^5*
f*x + 35*(16*B*a^2*c^5*cos(f*x + e)^7 + 8*(8*A - 11*B)*a^2*c^5*cos(f*x + e)^5 - 6*(8*A - 3*B)*a^2*c^5*cos(f*x
+ e)^3 - 9*(8*A - 3*B)*a^2*c^5*cos(f*x + e))*sin(f*x + e))/f

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1586 vs. \(2 (218) = 436\).
time = 1.17, size = 1586, normalized size = 6.93 \begin {gather*} \text {Too large to display} \end {gather*}

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]

Piecewise((15*A*a**2*c**5*x*sin(e + f*x)**6/16 + 45*A*a**2*c**5*x*sin(e + f*x)**4*cos(e + f*x)**2/16 - 15*A*a*
*2*c**5*x*sin(e + f*x)**4/8 + 45*A*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**4/16 - 15*A*a**2*c**5*x*sin(e + f
*x)**2*cos(e + f*x)**2/4 + A*a**2*c**5*x*sin(e + f*x)**2/2 + 15*A*a**2*c**5*x*cos(e + f*x)**6/16 - 15*A*a**2*c
**5*x*cos(e + f*x)**4/8 + A*a**2*c**5*x*cos(e + f*x)**2/2 + A*a**2*c**5*x + A*a**2*c**5*sin(e + f*x)**6*cos(e
+ f*x)/f - 33*A*a**2*c**5*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 2*A*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)**3/
f + A*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)/f - 5*A*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)**3/(2*f) + 25*A*a*
*2*c**5*sin(e + f*x)**3*cos(e + f*x)/(8*f) + 8*A*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 4*A*a**2*c*
*5*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 5*A*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)/f - 15*A*a**2*c**5*sin(e
+ f*x)*cos(e + f*x)**5/(16*f) + 15*A*a**2*c**5*sin(e + f*x)*cos(e + f*x)**3/(8*f) - A*a**2*c**5*sin(e + f*x)*
cos(e + f*x)/(2*f) + 16*A*a**2*c**5*cos(e + f*x)**7/(35*f) + 8*A*a**2*c**5*cos(e + f*x)**5/(15*f) - 10*A*a**2*
c**5*cos(e + f*x)**3/(3*f) + 3*A*a**2*c**5*cos(e + f*x)/f - 35*B*a**2*c**5*x*sin(e + f*x)**8/128 - 35*B*a**2*c
**5*x*sin(e + f*x)**6*cos(e + f*x)**2/32 - 5*B*a**2*c**5*x*sin(e + f*x)**6/16 - 105*B*a**2*c**5*x*sin(e + f*x)
**4*cos(e + f*x)**4/64 - 15*B*a**2*c**5*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 15*B*a**2*c**5*x*sin(e + f*x)**
4/8 - 35*B*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**6/32 - 15*B*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**4/1
6 + 15*B*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**2/4 - 3*B*a**2*c**5*x*sin(e + f*x)**2/2 - 35*B*a**2*c**5*x*
cos(e + f*x)**8/128 - 5*B*a**2*c**5*x*cos(e + f*x)**6/16 + 15*B*a**2*c**5*x*cos(e + f*x)**4/8 - 3*B*a**2*c**5*
x*cos(e + f*x)**2/2 + 93*B*a**2*c**5*sin(e + f*x)**7*cos(e + f*x)/(128*f) - 3*B*a**2*c**5*sin(e + f*x)**6*cos(
e + f*x)/f + 511*B*a**2*c**5*sin(e + f*x)**5*cos(e + f*x)**3/(384*f) + 11*B*a**2*c**5*sin(e + f*x)**5*cos(e +
f*x)/(16*f) - 6*B*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)**3/f + 5*B*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)/f +
385*B*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)**5/(384*f) + 5*B*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)**3/(6*f)
- 25*B*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 24*B*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) +
20*B*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - B*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)/f + 35*B*a**2*
c**5*sin(e + f*x)*cos(e + f*x)**7/(128*f) + 5*B*a**2*c**5*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 15*B*a**2*c**5
*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 3*B*a**2*c**5*sin(e + f*x)*cos(e + f*x)/(2*f) - 48*B*a**2*c**5*cos(e + f
*x)**7/(35*f) + 8*B*a**2*c**5*cos(e + f*x)**5/(3*f) - 2*B*a**2*c**5*cos(e + f*x)**3/(3*f) - B*a**2*c**5*cos(e
+ f*x)/f, Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)**2*(-c*sin(e) + c)**5, True))

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Giac [A]
time = 0.49, size = 278, normalized size = 1.21 \begin {gather*} -\frac {B a^{2} c^{5} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac {9}{128} \, {\left (8 \, A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} x - \frac {{\left (A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {{\left (11 \, A a^{2} c^{5} - B a^{2} c^{5}\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac {{\left (13 \, A a^{2} c^{5} - 7 \, B a^{2} c^{5}\right )} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {{\left (27 \, A a^{2} c^{5} - 17 \, B a^{2} c^{5}\right )} \cos \left (f x + e\right )}{64 \, f} - \frac {{\left (A a^{2} c^{5} - B a^{2} c^{5}\right )} \sin \left (6 \, f x + 6 \, e\right )}{64 \, f} - \frac {{\left (2 \, A a^{2} c^{5} - 7 \, B a^{2} c^{5}\right )} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {{\left (19 \, A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end {gather*}

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]

-1/1024*B*a^2*c^5*sin(8*f*x + 8*e)/f + 9/128*(8*A*a^2*c^5 - 3*B*a^2*c^5)*x - 1/448*(A*a^2*c^5 - 3*B*a^2*c^5)*c
os(7*f*x + 7*e)/f + 1/320*(11*A*a^2*c^5 - B*a^2*c^5)*cos(5*f*x + 5*e)/f + 1/64*(13*A*a^2*c^5 - 7*B*a^2*c^5)*co
s(3*f*x + 3*e)/f + 1/64*(27*A*a^2*c^5 - 17*B*a^2*c^5)*cos(f*x + e)/f - 1/64*(A*a^2*c^5 - B*a^2*c^5)*sin(6*f*x
+ 6*e)/f - 1/128*(2*A*a^2*c^5 - 7*B*a^2*c^5)*sin(4*f*x + 4*e)/f + 1/64*(19*A*a^2*c^5 - 3*B*a^2*c^5)*sin(2*f*x
+ 2*e)/f

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Mupad [B]
time = 15.13, size = 661, normalized size = 2.89 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,\left (6\,A\,a^2\,c^5-2\,B\,a^2\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (30\,A\,a^2\,c^5-10\,B\,a^2\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (22\,A\,a^2\,c^5-18\,B\,a^2\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (46\,A\,a^2\,c^5-26\,B\,a^2\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {74\,A\,a^2\,c^5}{5}-\frac {14\,B\,a^2\,c^5}{5}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}\,\left (\frac {7\,A\,a^2\,c^5}{8}+\frac {27\,B\,a^2\,c^5}{64}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {158\,A\,a^2\,c^5}{35}-\frac {138\,B\,a^2\,c^5}{35}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {218\,A\,a^2\,c^5}{5}-\frac {158\,B\,a^2\,c^5}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {75\,A\,a^2\,c^5}{8}-\frac {305\,B\,a^2\,c^5}{64}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}\,\left (\frac {75\,A\,a^2\,c^5}{8}-\frac {305\,B\,a^2\,c^5}{64}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {55\,A\,a^2\,c^5}{8}-\frac {437\,B\,a^2\,c^5}{64}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (\frac {55\,A\,a^2\,c^5}{8}-\frac {437\,B\,a^2\,c^5}{64}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {13\,A\,a^2\,c^5}{8}-\frac {919\,B\,a^2\,c^5}{64}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {13\,A\,a^2\,c^5}{8}-\frac {919\,B\,a^2\,c^5}{64}\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {7\,A\,a^2\,c^5}{8}+\frac {27\,B\,a^2\,c^5}{64}\right )+\frac {46\,A\,a^2\,c^5}{35}-\frac {26\,B\,a^2\,c^5}{35}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {9\,a^2\,c^5\,\mathrm {atan}\left (\frac {9\,a^2\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,A-3\,B\right )}{64\,\left (\frac {9\,A\,a^2\,c^5}{8}-\frac {27\,B\,a^2\,c^5}{64}\right )}\right )\,\left (8\,A-3\,B\right )}{64\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(e/2 + (f*x)/2)^14*(6*A*a^2*c^5 - 2*B*a^2*c^5) + tan(e/2 + (f*x)/2)^10*(30*A*a^2*c^5 - 10*B*a^2*c^5) + tan
(e/2 + (f*x)/2)^12*(22*A*a^2*c^5 - 18*B*a^2*c^5) + tan(e/2 + (f*x)/2)^8*(46*A*a^2*c^5 - 26*B*a^2*c^5) + tan(e/
2 + (f*x)/2)^4*((74*A*a^2*c^5)/5 - (14*B*a^2*c^5)/5) - tan(e/2 + (f*x)/2)^15*((7*A*a^2*c^5)/8 + (27*B*a^2*c^5)
/64) + tan(e/2 + (f*x)/2)^2*((158*A*a^2*c^5)/35 - (138*B*a^2*c^5)/35) + tan(e/2 + (f*x)/2)^6*((218*A*a^2*c^5)/
5 - (158*B*a^2*c^5)/5) + tan(e/2 + (f*x)/2)^3*((75*A*a^2*c^5)/8 - (305*B*a^2*c^5)/64) - tan(e/2 + (f*x)/2)^13*
((75*A*a^2*c^5)/8 - (305*B*a^2*c^5)/64) + tan(e/2 + (f*x)/2)^5*((55*A*a^2*c^5)/8 - (437*B*a^2*c^5)/64) - tan(e
/2 + (f*x)/2)^11*((55*A*a^2*c^5)/8 - (437*B*a^2*c^5)/64) - tan(e/2 + (f*x)/2)^7*((13*A*a^2*c^5)/8 - (919*B*a^2
*c^5)/64) + tan(e/2 + (f*x)/2)^9*((13*A*a^2*c^5)/8 - (919*B*a^2*c^5)/64) + tan(e/2 + (f*x)/2)*((7*A*a^2*c^5)/8
+ (27*B*a^2*c^5)/64) + (46*A*a^2*c^5)/35 - (26*B*a^2*c^5)/35)/(f*(8*tan(e/2 + (f*x)/2)^2 + 28*tan(e/2 + (f*x)
/2)^4 + 56*tan(e/2 + (f*x)/2)^6 + 70*tan(e/2 + (f*x)/2)^8 + 56*tan(e/2 + (f*x)/2)^10 + 28*tan(e/2 + (f*x)/2)^1
2 + 8*tan(e/2 + (f*x)/2)^14 + tan(e/2 + (f*x)/2)^16 + 1)) + (9*a^2*c^5*atan((9*a^2*c^5*tan(e/2 + (f*x)/2)*(8*A
- 3*B))/(64*((9*A*a^2*c^5)/8 - (27*B*a^2*c^5)/64)))*(8*A - 3*B))/(64*f)

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