3.3.65 \(\int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx\) [265]

Optimal. Leaf size=220 \[ \frac {\left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (2 c^3-6 c^2 d+9 c d^2-3 d^3\right )\right ) x}{2 a}+\frac {2 d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{3 a f}+\frac {d^2 (6 A c-11 B c-9 A d+9 B d) \cos (e+f x) \sin (e+f x)}{6 a f}+\frac {(3 A-4 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))} \]

[Out]

1/2*(3*A*d*(2*c^2-2*c*d+d^2)+B*(2*c^3-6*c^2*d+9*c*d^2-3*d^3))*x/a+2/3*d*(3*A*(c^2-3*c*d+d^2)-B*(7*c^2-9*c*d+4*
d^2))*cos(f*x+e)/a/f+1/6*d^2*(6*A*c-9*A*d-11*B*c+9*B*d)*cos(f*x+e)*sin(f*x+e)/a/f+1/3*(3*A-4*B)*d*cos(f*x+e)*(
c+d*sin(f*x+e))^2/a/f-(A-B)*cos(f*x+e)*(c+d*sin(f*x+e))^3/f/(a+a*sin(f*x+e))

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Rubi [A]
time = 0.24, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3056, 2832, 2813} \begin {gather*} \frac {2 d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{3 a f}+\frac {x \left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (2 c^3-6 c^2 d+9 c d^2-3 d^3\right )\right )}{2 a}+\frac {d^2 (6 A c-9 A d-11 B c+9 B d) \sin (e+f x) \cos (e+f x)}{6 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac {d (3 A-4 B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*A*d*(2*c^2 - 2*c*d + d^2) + B*(2*c^3 - 6*c^2*d + 9*c*d^2 - 3*d^3))*x)/(2*a) + (2*d*(3*A*(c^2 - 3*c*d + d^2
) - B*(7*c^2 - 9*c*d + 4*d^2))*Cos[e + f*x])/(3*a*f) + (d^2*(6*A*c - 11*B*c - 9*A*d + 9*B*d)*Cos[e + f*x]*Sin[
e + f*x])/(6*a*f) + ((3*A - 4*B)*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a*f) - ((A - B)*Cos[e + f*x]*(c + d
*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]))

Rule 2813

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*a*c +
b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2832

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 3056

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rubi steps

\begin {align*} \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx &=-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))}+\frac {\int (c+d \sin (e+f x))^2 (a (B (c-3 d)+3 A d)-a (3 A-4 B) d \sin (e+f x)) \, dx}{a^2}\\ &=\frac {(3 A-4 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))}+\frac {\int (c+d \sin (e+f x)) \left (a \left (3 A (3 c-2 d) d+B \left (3 c^2-9 c d+8 d^2\right )\right )-a d (6 A c-11 B c-9 A d+9 B d) \sin (e+f x)\right ) \, dx}{3 a^2}\\ &=\frac {\left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (2 c^3-6 c^2 d+9 c d^2-3 d^3\right )\right ) x}{2 a}+\frac {2 d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{3 a f}+\frac {d^2 (6 A c-11 B c-9 A d+9 B d) \cos (e+f x) \sin (e+f x)}{6 a f}+\frac {(3 A-4 B) d \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f}-\frac {(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a+a \sin (e+f x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(788\) vs. \(2(220)=440\).
time = 0.90, size = 788, normalized size = 3.58 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (3 \left (4 A d \left (6 c^2 (e+f x)-3 c d (1+2 e+2 f x)+d^2 (1+3 e+3 f x)\right )+B \left (8 c^3 (e+f x)-12 c^2 d (1+2 e+2 f x)+12 c d^2 (1+3 e+3 f x)-d^3 (7+12 e+12 f x)\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right )+9 d \left (A d (-4 c+d)+B \left (-4 c^2+3 c d-2 d^2\right )\right ) \cos \left (\frac {3}{2} (e+f x)\right )+9 B c d^2 \cos \left (\frac {5}{2} (e+f x)\right )+3 A d^3 \cos \left (\frac {5}{2} (e+f x)\right )-2 B d^3 \cos \left (\frac {5}{2} (e+f x)\right )+B d^3 \cos \left (\frac {7}{2} (e+f x)\right )+48 A c^3 \sin \left (\frac {1}{2} (e+f x)\right )-48 B c^3 \sin \left (\frac {1}{2} (e+f x)\right )-144 A c^2 d \sin \left (\frac {1}{2} (e+f x)\right )+180 B c^2 d \sin \left (\frac {1}{2} (e+f x)\right )+180 A c d^2 \sin \left (\frac {1}{2} (e+f x)\right )-180 B c d^2 \sin \left (\frac {1}{2} (e+f x)\right )-60 A d^3 \sin \left (\frac {1}{2} (e+f x)\right )+69 B d^3 \sin \left (\frac {1}{2} (e+f x)\right )+24 B c^3 e \sin \left (\frac {1}{2} (e+f x)\right )+72 A c^2 d e \sin \left (\frac {1}{2} (e+f x)\right )-72 B c^2 d e \sin \left (\frac {1}{2} (e+f x)\right )-72 A c d^2 e \sin \left (\frac {1}{2} (e+f x)\right )+108 B c d^2 e \sin \left (\frac {1}{2} (e+f x)\right )+36 A d^3 e \sin \left (\frac {1}{2} (e+f x)\right )-36 B d^3 e \sin \left (\frac {1}{2} (e+f x)\right )+24 B c^3 f x \sin \left (\frac {1}{2} (e+f x)\right )+72 A c^2 d f x \sin \left (\frac {1}{2} (e+f x)\right )-72 B c^2 d f x \sin \left (\frac {1}{2} (e+f x)\right )-72 A c d^2 f x \sin \left (\frac {1}{2} (e+f x)\right )+108 B c d^2 f x \sin \left (\frac {1}{2} (e+f x)\right )+36 A d^3 f x \sin \left (\frac {1}{2} (e+f x)\right )-36 B d^3 f x \sin \left (\frac {1}{2} (e+f x)\right )-36 B c^2 d \sin \left (\frac {3}{2} (e+f x)\right )-36 A c d^2 \sin \left (\frac {3}{2} (e+f x)\right )+27 B c d^2 \sin \left (\frac {3}{2} (e+f x)\right )+9 A d^3 \sin \left (\frac {3}{2} (e+f x)\right )-18 B d^3 \sin \left (\frac {3}{2} (e+f x)\right )-9 B c d^2 \sin \left (\frac {5}{2} (e+f x)\right )-3 A d^3 \sin \left (\frac {5}{2} (e+f x)\right )+2 B d^3 \sin \left (\frac {5}{2} (e+f x)\right )+B d^3 \sin \left (\frac {7}{2} (e+f x)\right )\right )}{24 a f (1+\sin (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*(4*A*d*(6*c^2*(e + f*x) - 3*c*d*(1 + 2*e + 2*f*x) + d^2*(1 + 3*e + 3
*f*x)) + B*(8*c^3*(e + f*x) - 12*c^2*d*(1 + 2*e + 2*f*x) + 12*c*d^2*(1 + 3*e + 3*f*x) - d^3*(7 + 12*e + 12*f*x
)))*Cos[(e + f*x)/2] + 9*d*(A*d*(-4*c + d) + B*(-4*c^2 + 3*c*d - 2*d^2))*Cos[(3*(e + f*x))/2] + 9*B*c*d^2*Cos[
(5*(e + f*x))/2] + 3*A*d^3*Cos[(5*(e + f*x))/2] - 2*B*d^3*Cos[(5*(e + f*x))/2] + B*d^3*Cos[(7*(e + f*x))/2] +
48*A*c^3*Sin[(e + f*x)/2] - 48*B*c^3*Sin[(e + f*x)/2] - 144*A*c^2*d*Sin[(e + f*x)/2] + 180*B*c^2*d*Sin[(e + f*
x)/2] + 180*A*c*d^2*Sin[(e + f*x)/2] - 180*B*c*d^2*Sin[(e + f*x)/2] - 60*A*d^3*Sin[(e + f*x)/2] + 69*B*d^3*Sin
[(e + f*x)/2] + 24*B*c^3*e*Sin[(e + f*x)/2] + 72*A*c^2*d*e*Sin[(e + f*x)/2] - 72*B*c^2*d*e*Sin[(e + f*x)/2] -
72*A*c*d^2*e*Sin[(e + f*x)/2] + 108*B*c*d^2*e*Sin[(e + f*x)/2] + 36*A*d^3*e*Sin[(e + f*x)/2] - 36*B*d^3*e*Sin[
(e + f*x)/2] + 24*B*c^3*f*x*Sin[(e + f*x)/2] + 72*A*c^2*d*f*x*Sin[(e + f*x)/2] - 72*B*c^2*d*f*x*Sin[(e + f*x)/
2] - 72*A*c*d^2*f*x*Sin[(e + f*x)/2] + 108*B*c*d^2*f*x*Sin[(e + f*x)/2] + 36*A*d^3*f*x*Sin[(e + f*x)/2] - 36*B
*d^3*f*x*Sin[(e + f*x)/2] - 36*B*c^2*d*Sin[(3*(e + f*x))/2] - 36*A*c*d^2*Sin[(3*(e + f*x))/2] + 27*B*c*d^2*Sin
[(3*(e + f*x))/2] + 9*A*d^3*Sin[(3*(e + f*x))/2] - 18*B*d^3*Sin[(3*(e + f*x))/2] - 9*B*c*d^2*Sin[(5*(e + f*x))
/2] - 3*A*d^3*Sin[(5*(e + f*x))/2] + 2*B*d^3*Sin[(5*(e + f*x))/2] + B*d^3*Sin[(7*(e + f*x))/2]))/(24*a*f*(1 +
Sin[e + f*x]))

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Maple [A]
time = 0.30, size = 337, normalized size = 1.53 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/a*(-(A*c^3-3*A*c^2*d+3*A*c*d^2-A*d^3-B*c^3+3*B*c^2*d-3*B*c*d^2+B*d^3)/(tan(1/2*f*x+1/2*e)+1)+((1/2*A*d^3+3
/2*B*c*d^2-1/2*B*d^3)*tan(1/2*f*x+1/2*e)^5+(-3*A*c*d^2+A*d^3-3*B*c^2*d+3*B*c*d^2-B*d^3)*tan(1/2*f*x+1/2*e)^4+(
-6*A*c*d^2+2*A*d^3-6*B*c^2*d+6*B*c*d^2-4*B*d^3)*tan(1/2*f*x+1/2*e)^2+(-1/2*A*d^3-3/2*B*c*d^2+1/2*B*d^3)*tan(1/
2*f*x+1/2*e)-3*A*c*d^2+A*d^3-3*B*c^2*d+3*B*c*d^2-5/3*B*d^3)/(1+tan(1/2*f*x+1/2*e)^2)^3+1/2*(6*A*c^2*d-6*A*c*d^
2+3*A*d^3+2*B*c^3-6*B*c^2*d+9*B*c*d^2-3*B*d^3)*arctan(tan(1/2*f*x+1/2*e)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1226 vs. \(2 (220) = 440\).
time = 0.59, size = 1226, normalized size = 5.57 \begin {gather*} -\frac {B d^{3} {\left (\frac {\frac {7 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {39 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {24 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {24 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {9 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {9 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + 16}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {3 \, a \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {a \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {a \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} + \frac {9 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} - 9 \, B c d^{2} {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} - 3 \, A d^{3} {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} + 18 \, B c^{2} d {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} + 18 \, A c d^{2} {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} - 6 \, B c^{3} {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - 18 \, A c^{2} d {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} + \frac {6 \, A c^{3}}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(B*d^3*((7*sin(f*x + e)/(cos(f*x + e) + 1) + 39*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 24*sin(f*x + e)^3/(
cos(f*x + e) + 1)^3 + 24*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 9*sin(f
*x + e)^6/(cos(f*x + e) + 1)^6 + 16)/(a + a*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a*sin(f*x + e)^2/(cos(f*x + e)
+ 1)^2 + 3*a*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3*a*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + a*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 9*a
rctan(sin(f*x + e)/(cos(f*x + e) + 1))/a) - 9*B*c*d^2*((sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(co
s(f*x + e) + 1)^2 + 3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 4)/(a + a*
sin(f*x + e)/(cos(f*x + e) + 1) + 2*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2*a*sin(f*x + e)^3/(cos(f*x + e) +
1)^3 + a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)
/(cos(f*x + e) + 1))/a) - 3*A*d^3*((sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 +
3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 4)/(a + a*sin(f*x + e)/(cos(f*
x + e) + 1) + 2*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2*a*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + a*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 + a*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))
/a) + 18*B*c^2*d*((sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a + a*sin(f*x +
e)/(cos(f*x + e) + 1) + a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + arct
an(sin(f*x + e)/(cos(f*x + e) + 1))/a) + 18*A*c*d^2*((sin(f*x + e)/(cos(f*x + e) + 1) + sin(f*x + e)^2/(cos(f*
x + e) + 1)^2 + 2)/(a + a*sin(f*x + e)/(cos(f*x + e) + 1) + a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3) + arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a) - 6*B*c^3*(arctan(sin(f*x + e)/(cos(
f*x + e) + 1))/a + 1/(a + a*sin(f*x + e)/(cos(f*x + e) + 1))) - 18*A*c^2*d*(arctan(sin(f*x + e)/(cos(f*x + e)
+ 1))/a + 1/(a + a*sin(f*x + e)/(cos(f*x + e) + 1))) + 6*A*c^3/(a + a*sin(f*x + e)/(cos(f*x + e) + 1)))/f

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (220) = 440\).
time = 0.47, size = 480, normalized size = 2.18 \begin {gather*} \frac {2 \, B d^{3} \cos \left (f x + e\right )^{4} - 6 \, {\left (A - B\right )} c^{3} + 18 \, {\left (A - B\right )} c^{2} d - 18 \, {\left (A - B\right )} c d^{2} + 6 \, {\left (A - B\right )} d^{3} + {\left (9 \, B c d^{2} + {\left (3 \, A - B\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x - 6 \, {\left (3 \, B c^{2} d + 3 \, {\left (A - B\right )} c d^{2} - {\left (A - 2 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (2 \, {\left (A - B\right )} c^{3} - 6 \, {\left (A - 2 \, B\right )} c^{2} d + 3 \, {\left (4 \, A - 3 \, B\right )} c d^{2} - {\left (3 \, A - 5 \, B\right )} d^{3} - {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x\right )} \cos \left (f x + e\right ) + {\left (2 \, B d^{3} \cos \left (f x + e\right )^{3} + 6 \, {\left (A - B\right )} c^{3} - 18 \, {\left (A - B\right )} c^{2} d + 18 \, {\left (A - B\right )} c d^{2} - 6 \, {\left (A - B\right )} d^{3} + 3 \, {\left (2 \, B c^{3} + 6 \, {\left (A - B\right )} c^{2} d - 3 \, {\left (2 \, A - 3 \, B\right )} c d^{2} + 3 \, {\left (A - B\right )} d^{3}\right )} f x - 3 \, {\left (3 \, B c d^{2} + {\left (A - B\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (6 \, B c^{2} d + 3 \, {\left (2 \, A - B\right )} c d^{2} - {\left (A - 3 \, B\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*B*d^3*cos(f*x + e)^4 - 6*(A - B)*c^3 + 18*(A - B)*c^2*d - 18*(A - B)*c*d^2 + 6*(A - B)*d^3 + (9*B*c*d^2
+ (3*A - B)*d^3)*cos(f*x + e)^3 + 3*(2*B*c^3 + 6*(A - B)*c^2*d - 3*(2*A - 3*B)*c*d^2 + 3*(A - B)*d^3)*f*x - 6
*(3*B*c^2*d + 3*(A - B)*c*d^2 - (A - 2*B)*d^3)*cos(f*x + e)^2 - 3*(2*(A - B)*c^3 - 6*(A - 2*B)*c^2*d + 3*(4*A
- 3*B)*c*d^2 - (3*A - 5*B)*d^3 - (2*B*c^3 + 6*(A - B)*c^2*d - 3*(2*A - 3*B)*c*d^2 + 3*(A - B)*d^3)*f*x)*cos(f*
x + e) + (2*B*d^3*cos(f*x + e)^3 + 6*(A - B)*c^3 - 18*(A - B)*c^2*d + 18*(A - B)*c*d^2 - 6*(A - B)*d^3 + 3*(2*
B*c^3 + 6*(A - B)*c^2*d - 3*(2*A - 3*B)*c*d^2 + 3*(A - B)*d^3)*f*x - 3*(3*B*c*d^2 + (A - B)*d^3)*cos(f*x + e)^
2 - 3*(6*B*c^2*d + 3*(2*A - B)*c*d^2 - (A - 3*B)*d^3)*cos(f*x + e))*sin(f*x + e))/(a*f*cos(f*x + e) + a*f*sin(
f*x + e) + a*f)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 14644 vs. \(2 (204) = 408\).
time = 4.99, size = 14644, normalized size = 66.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-12*A*c**3*tan(e/2 + f*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(
e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f
*tan(e/2 + f*x/2) + 6*a*f) - 36*A*c**3*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)
**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 +
f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 36*A*c**3*tan(e/2 + f*x/2)**2/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f
*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 +
18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 12*A*c**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*ta
n(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*
a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 18*A*c**2*d*f*x*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2
+ f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*ta
n(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 18*A*c**2*d*f*x*tan(e/2 + f
*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2
+ f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 54*A
*c**2*d*f*x*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/
2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 +
f*x/2) + 6*a*f) + 54*A*c**2*d*f*x*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6
+ 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/
2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 54*A*c**2*d*f*x*tan(e/2 + f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a
*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3
+ 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 54*A*c**2*d*f*x*tan(e/2 + f*x/2)**2/(6*a*f*ta
n(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a
*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 18*A*c**2*d*f*x*tan(e/
2 + f*x/2)/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/
2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 18
*A*c**2*d*f*x/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan
(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) +
36*A*c**2*d*tan(e/2 + f*x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x
/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2
+ f*x/2) + 6*a*f) + 108*A*c**2*d*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 +
18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)
**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 108*A*c**2*d*tan(e/2 + f*x/2)**2/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*ta
n(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*
a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) + 36*A*c**2*d/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan
(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a
*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 18*A*c*d**2*f*x*tan(e/2 + f*x/2)**7/(6*a*f*tan(e/2
+ f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan
(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 18*A*c*d**2*f*x*tan(e/2 + f*
x/2)**6/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 +
f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 54*A*
c*d**2*f*x*tan(e/2 + f*x/2)**5/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2
)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 +
f*x/2) + 6*a*f) - 54*A*c*d**2*f*x*tan(e/2 + f*x/2)**4/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*f*tan(e/2 + f*x/2)**6 +
18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 + 18*a*f*tan(e/2 + f*x/2
)**2 + 6*a*f*tan(e/2 + f*x/2) + 6*a*f) - 54*A*c*d**2*f*x*tan(e/2 + f*x/2)**3/(6*a*f*tan(e/2 + f*x/2)**7 + 6*a*
f*tan(e/2 + f*x/2)**6 + 18*a*f*tan(e/2 + f*x/2)**5 + 18*a*f*tan(e/2 + f*x/2)**4 + 18*a*f*tan(e/2 + f*x/2)**3 +
18*a*f*tan(e/2 + f*x/2)**2 + 6*a*f*tan(e/2 + f...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (220) = 440\).
time = 0.54, size = 479, normalized size = 2.18 \begin {gather*} \frac {\frac {3 \, {\left (2 \, B c^{3} + 6 \, A c^{2} d - 6 \, B c^{2} d - 6 \, A c d^{2} + 9 \, B c d^{2} + 3 \, A d^{3} - 3 \, B d^{3}\right )} {\left (f x + e\right )}}{a} - \frac {12 \, {\left (A c^{3} - B c^{3} - 3 \, A c^{2} d + 3 \, B c^{2} d + 3 \, A c d^{2} - 3 \, B c d^{2} - A d^{3} + B d^{3}\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {2 \, {\left (9 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, B c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 18 \, A c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 18 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 6 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 6 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 36 \, B c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 36 \, A c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 24 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, B c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, A d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, B d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 18 \, B c^{2} d - 18 \, A c d^{2} + 18 \, B c d^{2} + 6 \, A d^{3} - 10 \, B d^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} a}}{6 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(2*B*c^3 + 6*A*c^2*d - 6*B*c^2*d - 6*A*c*d^2 + 9*B*c*d^2 + 3*A*d^3 - 3*B*d^3)*(f*x + e)/a - 12*(A*c^3 -
B*c^3 - 3*A*c^2*d + 3*B*c^2*d + 3*A*c*d^2 - 3*B*c*d^2 - A*d^3 + B*d^3)/(a*(tan(1/2*f*x + 1/2*e) + 1)) + 2*(9*
B*c*d^2*tan(1/2*f*x + 1/2*e)^5 + 3*A*d^3*tan(1/2*f*x + 1/2*e)^5 - 3*B*d^3*tan(1/2*f*x + 1/2*e)^5 - 18*B*c^2*d*
tan(1/2*f*x + 1/2*e)^4 - 18*A*c*d^2*tan(1/2*f*x + 1/2*e)^4 + 18*B*c*d^2*tan(1/2*f*x + 1/2*e)^4 + 6*A*d^3*tan(1
/2*f*x + 1/2*e)^4 - 6*B*d^3*tan(1/2*f*x + 1/2*e)^4 - 36*B*c^2*d*tan(1/2*f*x + 1/2*e)^2 - 36*A*c*d^2*tan(1/2*f*
x + 1/2*e)^2 + 36*B*c*d^2*tan(1/2*f*x + 1/2*e)^2 + 12*A*d^3*tan(1/2*f*x + 1/2*e)^2 - 24*B*d^3*tan(1/2*f*x + 1/
2*e)^2 - 9*B*c*d^2*tan(1/2*f*x + 1/2*e) - 3*A*d^3*tan(1/2*f*x + 1/2*e) + 3*B*d^3*tan(1/2*f*x + 1/2*e) - 18*B*c
^2*d - 18*A*c*d^2 + 18*B*c*d^2 + 6*A*d^3 - 10*B*d^3)/((tan(1/2*f*x + 1/2*e)^2 + 1)^3*a))/f

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Mupad [B]
time = 14.05, size = 839, normalized size = 3.81 \begin {gather*} -\frac {12\,A\,c^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-18\,A\,d^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-12\,B\,c^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+18\,B\,d^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+6\,A\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-12\,A\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-6\,B\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+36\,B\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-16\,B\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-9\,A\,d^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-6\,B\,c^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )+9\,B\,d^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-9\,A\,d^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-6\,B\,c^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )+9\,B\,d^3\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-18\,A\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+12\,A\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+18\,B\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+12\,B\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-16\,B\,d^3\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+36\,A\,c\,d^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-36\,A\,c^2\,d\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-54\,B\,c\,d^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+36\,B\,c^2\,d\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+36\,A\,c\,d^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+18\,B\,c\,d^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+36\,B\,c^2\,d\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-36\,B\,c\,d^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+18\,A\,c\,d^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-18\,A\,c^2\,d\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-27\,B\,c\,d^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )+18\,B\,c^2\,d\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )+18\,A\,c\,d^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-18\,A\,c^2\,d\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-27\,B\,c\,d^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )+18\,B\,c^2\,d\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )+36\,A\,c\,d^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )-54\,B\,c\,d^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+36\,B\,c^2\,d\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+36\,B\,c\,d^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{6\,a\,f\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+6\,a\,f\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(12*A*c^3*cos(e/2 + (f*x)/2) - 18*A*d^3*cos(e/2 + (f*x)/2) - 12*B*c^3*cos(e/2 + (f*x)/2) + 18*B*d^3*cos(e/2 +
(f*x)/2) + 6*A*d^3*cos(e/2 + (f*x)/2)^3 - 12*A*d^3*cos(e/2 + (f*x)/2)^5 - 6*B*d^3*cos(e/2 + (f*x)/2)^3 + 36*B
*d^3*cos(e/2 + (f*x)/2)^5 - 16*B*d^3*cos(e/2 + (f*x)/2)^7 - 9*A*d^3*cos(e/2 + (f*x)/2)*(e + f*x) - 6*B*c^3*cos
(e/2 + (f*x)/2)*(e + f*x) + 9*B*d^3*cos(e/2 + (f*x)/2)*(e + f*x) - 9*A*d^3*sin(e/2 + (f*x)/2)*(e + f*x) - 6*B*
c^3*sin(e/2 + (f*x)/2)*(e + f*x) + 9*B*d^3*sin(e/2 + (f*x)/2)*(e + f*x) - 18*A*d^3*cos(e/2 + (f*x)/2)^2*sin(e/
2 + (f*x)/2) + 12*A*d^3*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2) + 18*B*d^3*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x
)/2) + 12*B*d^3*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2) - 16*B*d^3*cos(e/2 + (f*x)/2)^6*sin(e/2 + (f*x)/2) + 3
6*A*c*d^2*cos(e/2 + (f*x)/2) - 36*A*c^2*d*cos(e/2 + (f*x)/2) - 54*B*c*d^2*cos(e/2 + (f*x)/2) + 36*B*c^2*d*cos(
e/2 + (f*x)/2) + 36*A*c*d^2*cos(e/2 + (f*x)/2)^3 + 18*B*c*d^2*cos(e/2 + (f*x)/2)^3 + 36*B*c^2*d*cos(e/2 + (f*x
)/2)^3 - 36*B*c*d^2*cos(e/2 + (f*x)/2)^5 + 18*A*c*d^2*cos(e/2 + (f*x)/2)*(e + f*x) - 18*A*c^2*d*cos(e/2 + (f*x
)/2)*(e + f*x) - 27*B*c*d^2*cos(e/2 + (f*x)/2)*(e + f*x) + 18*B*c^2*d*cos(e/2 + (f*x)/2)*(e + f*x) + 18*A*c*d^
2*sin(e/2 + (f*x)/2)*(e + f*x) - 18*A*c^2*d*sin(e/2 + (f*x)/2)*(e + f*x) - 27*B*c*d^2*sin(e/2 + (f*x)/2)*(e +
f*x) + 18*B*c^2*d*sin(e/2 + (f*x)/2)*(e + f*x) + 36*A*c*d^2*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2) - 54*B*c*d
^2*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2) + 36*B*c^2*d*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2) + 36*B*c*d^2*c
os(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2))/(6*a*f*cos(e/2 + (f*x)/2) + 6*a*f*sin(e/2 + (f*x)/2))

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