3.265 \(\int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx\)
Optimal. Leaf size=220 \[ \frac{2 d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{3 a f}+\frac{x \left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (-6 c^2 d+2 c^3+9 c d^2-3 d^3\right )\right )}{2 a}+\frac{d^2 (6 A c-9 A d-11 B c+9 B d) \sin (e+f x) \cos (e+f x)}{6 a f}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac{d (3 A-4 B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f} \]
[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]))
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Rubi [A] time = 0.361171, antiderivative size = 220, normalized size of antiderivative = 1.,
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 =
{2977, 2753, 2734} \[ \frac{2 d \left (3 A \left (c^2-3 c d+d^2\right )-B \left (7 c^2-9 c d+4 d^2\right )\right ) \cos (e+f x)}{3 a f}+\frac{x \left (3 A d \left (2 c^2-2 c d+d^2\right )+B \left (-6 c^2 d+2 c^3+9 c d^2-3 d^3\right )\right )}{2 a}+\frac{d^2 (6 A c-9 A d-11 B c+9 B d) \sin (e+f x) \cos (e+f x)}{6 a f}-\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^3}{f (a \sin (e+f x)+a)}+\frac{d (3 A-4 B) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a f} \]
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 2977
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])
Rule 2753
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)*Simp[
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 2734
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]
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*}
Mathematica [B] time = 1.30916, size = 788, normalized size = 3.58 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (3 \cos \left (\frac{1}{2} (e+f x)\right ) \left (4 A d \left (6 c^2 (e+f x)-3 c d (2 e+2 f x+1)+d^2 (3 e+3 f x+1)\right )+B \left (-12 c^2 d (2 e+2 f x+1)+8 c^3 (e+f x)+12 c d^2 (3 e+3 f x+1)-d^3 (12 e+12 f x+7)\right )\right )+9 d \left (A d (d-4 c)+B \left (-4 c^2+3 c d-2 d^2\right )\right ) \cos \left (\frac{3}{2} (e+f x)\right )-144 A c^2 d \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 A c^2 d f x \sin \left (\frac{1}{2} (e+f x)\right )+48 A c^3 \sin \left (\frac{1}{2} (e+f x)\right )+180 A c d^2 \sin \left (\frac{1}{2} (e+f x)\right )-72 A c d^2 e \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 )-36 A c d^2 \sin \left (\frac{3}{2} (e+f x)\right )-60 A d^3 \sin \left (\frac{1}{2} (e+f x)\right )+36 A d^3 e \sin \left (\frac{1}{2} (e+f x)\right )+36 A d^3 f x \sin \left (\frac{1}{2} (e+f x)\right )+9 A d^3 \sin \left (\frac{3}{2} (e+f x)\right )-3 A d^3 \sin \left (\frac{5}{2} (e+f x)\right )+3 A d^3 \cos \left (\frac{5}{2} (e+f x)\right )+180 B c^2 d \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 B c^2 d 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 )-48 B c^3 \sin \left (\frac{1}{2} (e+f x)\right )+24 B c^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 )-180 B c d^2 \sin \left (\frac{1}{2} (e+f x)\right )+108 B c d^2 e \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 )+27 B c d^2 \sin \left (\frac{3}{2} (e+f x)\right )-9 B c d^2 \sin \left (\frac{5}{2} (e+f x)\right )+9 B c d^2 \cos \left (\frac{5}{2} (e+f x)\right )+69 B d^3 \sin \left (\frac{1}{2} (e+f x)\right )-36 B d^3 e \sin \left (\frac{1}{2} (e+f x)\right )-36 B d^3 f x \sin \left (\frac{1}{2} (e+f x)\right )-18 B d^3 \sin \left (\frac{3}{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 )-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 )\right )}{24 a f (\sin (e+f x)+1)} \]
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 [B] time = 0.102, size = 1110, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}
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)
[Out]
-2/a/f/(tan(1/2*f*x+1/2*e)+1)*A*c^3+2/a/f/(tan(1/2*f*x+1/2*e)+1)*A*d^3+2/a/f/(tan(1/2*f*x+1/2*e)+1)*B*c^3-2/a/
f/(tan(1/2*f*x+1/2*e)+1)*B*d^3+2/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*A*d^3-10/3/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*B*d^
3+3/a/f*arctan(tan(1/2*f*x+1/2*e))*A*d^3+2/a/f*arctan(tan(1/2*f*x+1/2*e))*B*c^3-3/a/f*arctan(tan(1/2*f*x+1/2*e
))*B*d^3-3/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)*B*c*d^2-12/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*A*tan(1
/2*f*x+1/2*e)^2*c*d^2-12/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+1/2*e)^2*c^2*d+12/a/f/(1+tan(1/2*f*x+1/2
*e)^2)^3*B*tan(1/2*f*x+1/2*e)^2*c*d^2+3/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*B*c*d^2-6/a/f/(1+t
an(1/2*f*x+1/2*e)^2)^3*A*tan(1/2*f*x+1/2*e)^4*c*d^2-6/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+1/2*e)^4*c^
2*d+6/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+1/2*e)^4*c*d^2+9/a/f*arctan(tan(1/2*f*x+1/2*e))*B*c*d^2+6/a
/f/(tan(1/2*f*x+1/2*e)+1)*A*c^2*d-6/a/f/(tan(1/2*f*x+1/2*e)+1)*A*c*d^2-6/a/f/(tan(1/2*f*x+1/2*e)+1)*B*c^2*d+6/
a/f/(tan(1/2*f*x+1/2*e)+1)*B*c*d^2+1/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*A*d^3-1/a/f/(1+tan(1/
2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)*A*d^3-1/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5*B*d^3+2/a/f/(
1+tan(1/2*f*x+1/2*e)^2)^3*A*tan(1/2*f*x+1/2*e)^4*d^3-2/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+1/2*e)^4*d
^3+4/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*A*tan(1/2*f*x+1/2*e)^2*d^3-8/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*B*tan(1/2*f*x+
1/2*e)^2*d^3+1/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)*B*d^3-6/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*A*c*d^
2-6/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*B*c^2*d+6/a/f/(1+tan(1/2*f*x+1/2*e)^2)^3*B*c*d^2+6/a/f*arctan(tan(1/2*f*x+1
/2*e))*A*c^2*d-6/a/f*arctan(tan(1/2*f*x+1/2*e))*A*c*d^2-6/a/f*arctan(tan(1/2*f*x+1/2*e))*B*c^2*d
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Maxima [B] time = 1.5744, size = 1517, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}
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
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Fricas [B] time = 2.45702, size = 1062, normalized size = 4.83 \begin{align*} \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{align*}
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)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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]
Timed out
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Giac [B] time = 1.32047, size = 647, normalized size = 2.94 \begin{align*} \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{align*}
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