3.4.52 \(\int \frac {(a+b \sin (e+f x))^2 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx\) [352]

Optimal. Leaf size=199 \[ -\frac {b (2 b B c-A b d-2 a B d) x}{d^3}-\frac {2 (b c-a d) \left (a d^2 (A c-B d)-b \left (2 B c^3-A c^2 d-3 B c d^2+2 A d^3\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 \left (c^2-d^2\right )^{3/2} f}-\frac {b^2 B \cos (e+f x)}{d^2 f}-\frac {(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))} \]

[Out]

-b*(-A*b*d-2*B*a*d+2*B*b*c)*x/d^3-2*(-a*d+b*c)*(a*d^2*(A*c-B*d)-b*(-A*c^2*d+2*A*d^3+2*B*c^3-3*B*c*d^2))*arctan
((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/d^3/(c^2-d^2)^(3/2)/f-b^2*B*cos(f*x+e)/d^2/f-(-a*d+b*c)^2*(-A*d+B*c
)*cos(f*x+e)/d^2/(c^2-d^2)/f/(c+d*sin(f*x+e))

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Rubi [A]
time = 0.40, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3067, 3102, 2814, 2739, 632, 210} \begin {gather*} -\frac {2 (b c-a d) \left (a d^2 (A c-B d)-b \left (-A c^2 d+2 A d^3+2 B c^3-3 B c d^2\right )\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{d^3 f \left (c^2-d^2\right )^{3/2}}-\frac {(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 f \left (c^2-d^2\right ) (c+d \sin (e+f x))}-\frac {b x (-2 a B d-A b d+2 b B c)}{d^3}-\frac {b^2 B \cos (e+f x)}{d^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(2*b*B*c - A*b*d - 2*a*B*d)*x)/d^3) - (2*(b*c - a*d)*(a*d^2*(A*c - B*d) - b*(2*B*c^3 - A*c^2*d - 3*B*c*d^
2 + 2*A*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(3/2)*f) - (b^2*B*Cos[e + f*x
])/(d^2*f) - ((b*c - a*d)^2*(B*c - A*d)*Cos[e + f*x])/(d^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x]))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2814

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

Rule 3067

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

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {(a+b \sin (e+f x))^2 (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx &=-\frac {(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))}+\frac {\int \frac {-d \left (B (b c-a d)^2-A d \left (a^2 c+b^2 c-2 a b d\right )\right )-b (b B c-A b d-2 a B d) \left (c^2-d^2\right ) \sin (e+f x)+b^2 B d \left (c^2-d^2\right ) \sin ^2(e+f x)}{c+d \sin (e+f x)} \, dx}{d^2 \left (c^2-d^2\right )}\\ &=-\frac {b^2 B \cos (e+f x)}{d^2 f}-\frac {(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))}+\frac {\int \frac {-d^2 \left (B (b c-a d)^2-A d \left (a^2 c+b^2 c-2 a b d\right )\right )-b d (2 b B c-A b d-2 a B d) \left (c^2-d^2\right ) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{d^3 \left (c^2-d^2\right )}\\ &=-\frac {b (2 b B c-A b d-2 a B d) x}{d^3}-\frac {b^2 B \cos (e+f x)}{d^2 f}-\frac {(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))}-\frac {\left ((b c-a d) \left (a d^2 (A c-B d)-b \left (2 B c^3-A c^2 d-3 B c d^2+2 A d^3\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{d^3 \left (c^2-d^2\right )}\\ &=-\frac {b (2 b B c-A b d-2 a B d) x}{d^3}-\frac {b^2 B \cos (e+f x)}{d^2 f}-\frac {(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))}-\frac {\left (2 (b c-a d) \left (a d^2 (A c-B d)-b \left (2 B c^3-A c^2 d-3 B c d^2+2 A d^3\right )\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 \left (c^2-d^2\right ) f}\\ &=-\frac {b (2 b B c-A b d-2 a B d) x}{d^3}-\frac {b^2 B \cos (e+f x)}{d^2 f}-\frac {(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))}+\frac {\left (4 (b c-a d) \left (a d^2 (A c-B d)-b \left (2 B c^3-A c^2 d-3 B c d^2+2 A d^3\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^3 \left (c^2-d^2\right ) f}\\ &=-\frac {b (2 b B c-A b d-2 a B d) x}{d^3}-\frac {2 (b c-a d) \left (a d^2 (A c-B d)-b \left (2 B c^3-A c^2 d-3 B c d^2+2 A d^3\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{d^3 \left (c^2-d^2\right )^{3/2} f}-\frac {b^2 B \cos (e+f x)}{d^2 f}-\frac {(b c-a d)^2 (B c-A d) \cos (e+f x)}{d^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 1.10, size = 188, normalized size = 0.94 \begin {gather*} \frac {b (-2 b B c+A b d+2 a B d) (e+f x)+\frac {2 (b c-a d) \left (a d^2 (-A c+B d)+b \left (2 B c^3-A c^2 d-3 B c d^2+2 A d^3\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{3/2}}-b^2 B d \cos (e+f x)+\frac {d (b c-a d)^2 (-B c+A d) \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))}}{d^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(-2*b*B*c + A*b*d + 2*a*B*d)*(e + f*x) + (2*(b*c - a*d)*(a*d^2*(-(A*c) + B*d) + b*(2*B*c^3 - A*c^2*d - 3*B*
c*d^2 + 2*A*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(3/2) - b^2*B*d*Cos[e + f*x] +
(d*(b*c - a*d)^2*(-(B*c) + A*d)*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])))/(d^3*f)

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Maple [A]
time = 0.72, size = 374, normalized size = 1.88

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\frac {d^{2} \left (a^{2} A \,d^{3}-2 A a b c \,d^{2}+A \,b^{2} c^{2} d -B \,a^{2} c \,d^{2}+2 B a b \,c^{2} d -c^{3} B \,b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) c}+\frac {d \left (a^{2} A \,d^{3}-2 A a b c \,d^{2}+A \,b^{2} c^{2} d -B \,a^{2} c \,d^{2}+2 B a b \,c^{2} d -c^{3} B \,b^{2}\right )}{c^{2}-d^{2}}\right )}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {2 \left (A \,a^{2} c \,d^{3}-2 A a b \,d^{4}-A \,b^{2} c^{3} d +2 A \,b^{2} c \,d^{3}-B \,a^{2} d^{4}-2 B a b \,c^{3} d +4 B a b c \,d^{3}+2 B \,b^{2} c^{4}-3 B \,b^{2} c^{2} d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{2}-d^{2}\right )^{\frac {3}{2}}}}{d^{3}}+\frac {2 b \left (-\frac {B b d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (A b d +2 B a d -2 B b c \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{d^{3}}}{f}\)	\(374\)
default	\(\frac {\frac {\frac {2 \left (\frac {d^{2} \left (a^{2} A \,d^{3}-2 A a b c \,d^{2}+A \,b^{2} c^{2} d -B \,a^{2} c \,d^{2}+2 B a b \,c^{2} d -c^{3} B \,b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-d^{2}\right ) c}+\frac {d \left (a^{2} A \,d^{3}-2 A a b c \,d^{2}+A \,b^{2} c^{2} d -B \,a^{2} c \,d^{2}+2 B a b \,c^{2} d -c^{3} B \,b^{2}\right )}{c^{2}-d^{2}}\right )}{c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c}+\frac {2 \left (A \,a^{2} c \,d^{3}-2 A a b \,d^{4}-A \,b^{2} c^{3} d +2 A \,b^{2} c \,d^{3}-B \,a^{2} d^{4}-2 B a b \,c^{3} d +4 B a b c \,d^{3}+2 B \,b^{2} c^{4}-3 B \,b^{2} c^{2} d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{2}-d^{2}\right )^{\frac {3}{2}}}}{d^{3}}+\frac {2 b \left (-\frac {B b d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (A b d +2 B a d -2 B b c \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{d^{3}}}{f}\)	\(374\)
risch	\(\text {Expression too large to display}\)	\(1721\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d^2-4*c^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (198) = 396\).
time = 0.53, size = 1327, normalized size = 6.67 \begin {gather*} \left [-\frac {2 \, {\left (2 \, B b^{2} c^{6} - 4 \, B b^{2} c^{4} d^{2} + 2 \, B b^{2} c^{2} d^{4} - {\left (2 \, B a b + A b^{2}\right )} c^{5} d + 2 \, {\left (2 \, B a b + A b^{2}\right )} c^{3} d^{3} - {\left (2 \, B a b + A b^{2}\right )} c d^{5}\right )} f x + {\left (2 \, B b^{2} c^{5} - 3 \, B b^{2} c^{3} d^{2} - {\left (2 \, B a b + A b^{2}\right )} c^{4} d + {\left (A a^{2} + 4 \, B a b + 2 \, A b^{2}\right )} c^{2} d^{3} - {\left (B a^{2} + 2 \, A a b\right )} c d^{4} + {\left (2 \, B b^{2} c^{4} d - 3 \, B b^{2} c^{2} d^{3} - {\left (2 \, B a b + A b^{2}\right )} c^{3} d^{2} + {\left (A a^{2} + 4 \, B a b + 2 \, A b^{2}\right )} c d^{4} - {\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) + 2 \, {\left (2 \, B b^{2} c^{5} d + A a^{2} d^{6} - {\left (2 \, B a b + A b^{2}\right )} c^{4} d^{2} + {\left (B a^{2} + 2 \, A a b - 3 \, B b^{2}\right )} c^{3} d^{3} - {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} c^{2} d^{4} - {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} c d^{5}\right )} \cos \left (f x + e\right ) + 2 \, {\left ({\left (2 \, B b^{2} c^{5} d - 4 \, B b^{2} c^{3} d^{3} + 2 \, B b^{2} c d^{5} - {\left (2 \, B a b + A b^{2}\right )} c^{4} d^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} c^{2} d^{4} - {\left (2 \, B a b + A b^{2}\right )} d^{6}\right )} f x + {\left (B b^{2} c^{4} d^{2} - 2 \, B b^{2} c^{2} d^{4} + B b^{2} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{4} d^{4} - 2 \, c^{2} d^{6} + d^{8}\right )} f \sin \left (f x + e\right ) + {\left (c^{5} d^{3} - 2 \, c^{3} d^{5} + c d^{7}\right )} f\right )}}, -\frac {{\left (2 \, B b^{2} c^{6} - 4 \, B b^{2} c^{4} d^{2} + 2 \, B b^{2} c^{2} d^{4} - {\left (2 \, B a b + A b^{2}\right )} c^{5} d + 2 \, {\left (2 \, B a b + A b^{2}\right )} c^{3} d^{3} - {\left (2 \, B a b + A b^{2}\right )} c d^{5}\right )} f x + {\left (2 \, B b^{2} c^{5} - 3 \, B b^{2} c^{3} d^{2} - {\left (2 \, B a b + A b^{2}\right )} c^{4} d + {\left (A a^{2} + 4 \, B a b + 2 \, A b^{2}\right )} c^{2} d^{3} - {\left (B a^{2} + 2 \, A a b\right )} c d^{4} + {\left (2 \, B b^{2} c^{4} d - 3 \, B b^{2} c^{2} d^{3} - {\left (2 \, B a b + A b^{2}\right )} c^{3} d^{2} + {\left (A a^{2} + 4 \, B a b + 2 \, A b^{2}\right )} c d^{4} - {\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) + {\left (2 \, B b^{2} c^{5} d + A a^{2} d^{6} - {\left (2 \, B a b + A b^{2}\right )} c^{4} d^{2} + {\left (B a^{2} + 2 \, A a b - 3 \, B b^{2}\right )} c^{3} d^{3} - {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} c^{2} d^{4} - {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} c d^{5}\right )} \cos \left (f x + e\right ) + {\left ({\left (2 \, B b^{2} c^{5} d - 4 \, B b^{2} c^{3} d^{3} + 2 \, B b^{2} c d^{5} - {\left (2 \, B a b + A b^{2}\right )} c^{4} d^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} c^{2} d^{4} - {\left (2 \, B a b + A b^{2}\right )} d^{6}\right )} f x + {\left (B b^{2} c^{4} d^{2} - 2 \, B b^{2} c^{2} d^{4} + B b^{2} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{{\left (c^{4} d^{4} - 2 \, c^{2} d^{6} + d^{8}\right )} f \sin \left (f x + e\right ) + {\left (c^{5} d^{3} - 2 \, c^{3} d^{5} + c d^{7}\right )} f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*(2*B*b^2*c^6 - 4*B*b^2*c^4*d^2 + 2*B*b^2*c^2*d^4 - (2*B*a*b + A*b^2)*c^5*d + 2*(2*B*a*b + A*b^2)*c^3*
d^3 - (2*B*a*b + A*b^2)*c*d^5)*f*x + (2*B*b^2*c^5 - 3*B*b^2*c^3*d^2 - (2*B*a*b + A*b^2)*c^4*d + (A*a^2 + 4*B*a
*b + 2*A*b^2)*c^2*d^3 - (B*a^2 + 2*A*a*b)*c*d^4 + (2*B*b^2*c^4*d - 3*B*b^2*c^2*d^3 - (2*B*a*b + A*b^2)*c^3*d^2
+ (A*a^2 + 4*B*a*b + 2*A*b^2)*c*d^4 - (B*a^2 + 2*A*a*b)*d^5)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2
)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2
+ d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*B*b^2*c^5*d + A*a^2*d^6 - (2*B*a*b + A*
b^2)*c^4*d^2 + (B*a^2 + 2*A*a*b - 3*B*b^2)*c^3*d^3 - (A*a^2 - 2*B*a*b - A*b^2)*c^2*d^4 - (B*a^2 + 2*A*a*b - B*
b^2)*c*d^5)*cos(f*x + e) + 2*((2*B*b^2*c^5*d - 4*B*b^2*c^3*d^3 + 2*B*b^2*c*d^5 - (2*B*a*b + A*b^2)*c^4*d^2 + 2
*(2*B*a*b + A*b^2)*c^2*d^4 - (2*B*a*b + A*b^2)*d^6)*f*x + (B*b^2*c^4*d^2 - 2*B*b^2*c^2*d^4 + B*b^2*d^6)*cos(f*
x + e))*sin(f*x + e))/((c^4*d^4 - 2*c^2*d^6 + d^8)*f*sin(f*x + e) + (c^5*d^3 - 2*c^3*d^5 + c*d^7)*f), -((2*B*b
^2*c^6 - 4*B*b^2*c^4*d^2 + 2*B*b^2*c^2*d^4 - (2*B*a*b + A*b^2)*c^5*d + 2*(2*B*a*b + A*b^2)*c^3*d^3 - (2*B*a*b
+ A*b^2)*c*d^5)*f*x + (2*B*b^2*c^5 - 3*B*b^2*c^3*d^2 - (2*B*a*b + A*b^2)*c^4*d + (A*a^2 + 4*B*a*b + 2*A*b^2)*c
^2*d^3 - (B*a^2 + 2*A*a*b)*c*d^4 + (2*B*b^2*c^4*d - 3*B*b^2*c^2*d^3 - (2*B*a*b + A*b^2)*c^3*d^2 + (A*a^2 + 4*B
*a*b + 2*A*b^2)*c*d^4 - (B*a^2 + 2*A*a*b)*d^5)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqr
t(c^2 - d^2)*cos(f*x + e))) + (2*B*b^2*c^5*d + A*a^2*d^6 - (2*B*a*b + A*b^2)*c^4*d^2 + (B*a^2 + 2*A*a*b - 3*B*
b^2)*c^3*d^3 - (A*a^2 - 2*B*a*b - A*b^2)*c^2*d^4 - (B*a^2 + 2*A*a*b - B*b^2)*c*d^5)*cos(f*x + e) + ((2*B*b^2*c
^5*d - 4*B*b^2*c^3*d^3 + 2*B*b^2*c*d^5 - (2*B*a*b + A*b^2)*c^4*d^2 + 2*(2*B*a*b + A*b^2)*c^2*d^4 - (2*B*a*b +
A*b^2)*d^6)*f*x + (B*b^2*c^4*d^2 - 2*B*b^2*c^2*d^4 + B*b^2*d^6)*cos(f*x + e))*sin(f*x + e))/((c^4*d^4 - 2*c^2*
d^6 + d^8)*f*sin(f*x + e) + (c^5*d^3 - 2*c^3*d^5 + c*d^7)*f)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (198) = 396\).
time = 0.46, size = 776, normalized size = 3.90 \begin {gather*} \frac {\frac {2 \, {\left (2 \, B b^{2} c^{4} - 2 \, B a b c^{3} d - A b^{2} c^{3} d - 3 \, B b^{2} c^{2} d^{2} + A a^{2} c d^{3} + 4 \, B a b c d^{3} + 2 \, A b^{2} c d^{3} - B a^{2} d^{4} - 2 \, A a b d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (c^{2} d^{3} - d^{5}\right )} \sqrt {c^{2} - d^{2}}} - \frac {2 \, {\left (B b^{2} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, B a b c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - A b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + B a^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, A a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - A a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, B b^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, B a b c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - A b^{2} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + B a^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, A a b c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - B b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - A a^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B b^{2} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, B a b c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + B a^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, A a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, B b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - A a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, B b^{2} c^{4} - 2 \, B a b c^{3} d - A b^{2} c^{3} d + B a^{2} c^{2} d^{2} + 2 \, A a b c^{2} d^{2} - B b^{2} c^{2} d^{2} - A a^{2} c d^{3}\right )}}{{\left (c^{3} d^{2} - c d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}} - \frac {{\left (2 \, B b^{2} c - 2 \, B a b d - A b^{2} d\right )} {\left (f x + e\right )}}{d^{3}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(2*B*b^2*c^4 - 2*B*a*b*c^3*d - A*b^2*c^3*d - 3*B*b^2*c^2*d^2 + A*a^2*c*d^3 + 4*B*a*b*c*d^3 + 2*A*b^2*c*d^3
- B*a^2*d^4 - 2*A*a*b*d^4)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt
(c^2 - d^2)))/((c^2*d^3 - d^5)*sqrt(c^2 - d^2)) - 2*(B*b^2*c^3*d*tan(1/2*f*x + 1/2*e)^3 - 2*B*a*b*c^2*d^2*tan(
1/2*f*x + 1/2*e)^3 - A*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 + B*a^2*c*d^3*tan(1/2*f*x + 1/2*e)^3 + 2*A*a*b*c*d^3
*tan(1/2*f*x + 1/2*e)^3 - A*a^2*d^4*tan(1/2*f*x + 1/2*e)^3 + 2*B*b^2*c^4*tan(1/2*f*x + 1/2*e)^2 - 2*B*a*b*c^3*
d*tan(1/2*f*x + 1/2*e)^2 - A*b^2*c^3*d*tan(1/2*f*x + 1/2*e)^2 + B*a^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 + 2*A*a*b
*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 - B*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 - A*a^2*c*d^3*tan(1/2*f*x + 1/2*e)^2 +
3*B*b^2*c^3*d*tan(1/2*f*x + 1/2*e) - 2*B*a*b*c^2*d^2*tan(1/2*f*x + 1/2*e) - A*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e)
+ B*a^2*c*d^3*tan(1/2*f*x + 1/2*e) + 2*A*a*b*c*d^3*tan(1/2*f*x + 1/2*e) - 2*B*b^2*c*d^3*tan(1/2*f*x + 1/2*e)
- A*a^2*d^4*tan(1/2*f*x + 1/2*e) + 2*B*b^2*c^4 - 2*B*a*b*c^3*d - A*b^2*c^3*d + B*a^2*c^2*d^2 + 2*A*a*b*c^2*d^2
- B*b^2*c^2*d^2 - A*a^2*c*d^3)/((c^3*d^2 - c*d^4)*(c*tan(1/2*f*x + 1/2*e)^4 + 2*d*tan(1/2*f*x + 1/2*e)^3 + 2*
c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)) - (2*B*b^2*c - 2*B*a*b*d - A*b^2*d)*(f*x + e)/d^3)/f

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Mupad [B]
time = 27.62, size = 2500, normalized size = 12.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*(A*a^2*d^3 - 2*B*b^2*c^3 + A*b^2*c^2*d - B*a^2*c*d^2 + B*b^2*c*d^2 - 2*A*a*b*c*d^2 + 2*B*a*b*c^2*d))/(d^2*
(c^2 - d^2)) + (2*tan(e/2 + (f*x)/2)^2*(A*a^2*d^3 - 2*B*b^2*c^3 + A*b^2*c^2*d - B*a^2*c*d^2 + B*b^2*c*d^2 - 2*
A*a*b*c*d^2 + 2*B*a*b*c^2*d))/(d^2*(c^2 - d^2)) + (2*tan(e/2 + (f*x)/2)^3*(A*a^2*d^3 - B*b^2*c^3 + A*b^2*c^2*d
- B*a^2*c*d^2 - 2*A*a*b*c*d^2 + 2*B*a*b*c^2*d))/(c*d*(c^2 - d^2)) + (2*tan(e/2 + (f*x)/2)*(A*a^2*d^3 - 3*B*b^
2*c^3 + A*b^2*c^2*d - B*a^2*c*d^2 + 2*B*b^2*c*d^2 - 2*A*a*b*c*d^2 + 2*B*a*b*c^2*d))/(c*d*(c^2 - d^2)))/(f*(c +
2*d*tan(e/2 + (f*x)/2) + 2*c*tan(e/2 + (f*x)/2)^2 + c*tan(e/2 + (f*x)/2)^4 + 2*d*tan(e/2 + (f*x)/2)^3)) + (at
an((((b*d*(A*b + 2*B*a)*1i - B*b^2*c*2i)*((32*(A^2*b^4*c^2*d^8 - 2*A^2*b^4*c^4*d^6 + A^2*b^4*c^6*d^4 + 4*B^2*b
^4*c^4*d^6 - 8*B^2*b^4*c^6*d^4 + 4*B^2*b^4*c^8*d^2 + 4*B^2*a^2*b^2*c^2*d^8 - 8*B^2*a^2*b^2*c^4*d^6 + 4*B^2*a^2
*b^2*c^6*d^4 - 4*A*B*b^4*c^3*d^7 + 8*A*B*b^4*c^5*d^5 - 4*A*B*b^4*c^7*d^3 - 8*B^2*a*b^3*c^3*d^7 + 16*B^2*a*b^3*
c^5*d^5 - 8*B^2*a*b^3*c^7*d^3 + 4*A*B*a*b^3*c^2*d^8 - 8*A*B*a*b^3*c^4*d^6 + 4*A*B*a*b^3*c^6*d^4))/(d^9 - 2*c^2
*d^7 + c^4*d^5) + ((b*d*(A*b + 2*B*a)*1i - B*b^2*c*2i)*((((32*(c^2*d^12 - 2*c^4*d^10 + c^6*d^8))/(d^9 - 2*c^2*
d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^10 - 2*c^7*d^8))/(d^10 - 2*c^2*d^8 +
c^4*d^6))*(b*d*(A*b + 2*B*a)*1i - B*b^2*c*2i))/d^3 - (32*(A*a^2*c^5*d^7 - A*a^2*c^3*d^9 - A*b^2*c*d^11 + A*b^2
*c^3*d^9 + B*a^2*c^2*d^10 - B*a^2*c^4*d^8 + 2*B*b^2*c^2*d^10 - 3*B*b^2*c^4*d^8 + B*b^2*c^6*d^6 - 2*B*a*b*c*d^1
1 + 2*A*a*b*c^2*d^10 - 2*A*a*b*c^4*d^8 + 2*B*a*b*c^3*d^9))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (32*tan(e/2 + (f*x)/2
)*(2*A*a^2*c^2*d^11 - 2*B*a^2*c*d^12 - 2*A*a^2*c^4*d^9 + 4*A*b^2*c^2*d^11 - 6*A*b^2*c^4*d^9 + 2*A*b^2*c^6*d^7
+ 2*B*a^2*c^3*d^10 - 6*B*b^2*c^3*d^10 + 10*B*b^2*c^5*d^8 - 4*B*b^2*c^7*d^6 - 4*A*a*b*c*d^12 + 4*A*a*b*c^3*d^10
+ 8*B*a*b*c^2*d^11 - 12*B*a*b*c^4*d^9 + 4*B*a*b*c^6*d^7))/(d^10 - 2*c^2*d^8 + c^4*d^6)))/d^3 - (32*tan(e/2 +
(f*x)/2)*(A^2*a^4*c^3*d^8 + 9*A^2*b^4*c^3*d^8 - 8*A^2*b^4*c^5*d^6 + 2*A^2*b^4*c^7*d^4 - 8*B^2*b^4*c^3*d^8 + 29
*B^2*b^4*c^5*d^6 - 28*B^2*b^4*c^7*d^4 + 8*B^2*b^4*c^9*d^2 - 2*A^2*b^4*c*d^10 + B^2*a^4*c*d^10 + 4*A^2*a^2*b^2*
c^3*d^8 - 2*A^2*a^2*b^2*c^5*d^6 + 42*B^2*a^2*b^2*c^3*d^8 - 36*B^2*a^2*b^2*c^5*d^6 + 8*B^2*a^2*b^2*c^7*d^4 - 2*
A*B*a^4*c^2*d^9 + 8*A*B*b^4*c^2*d^9 - 32*A*B*b^4*c^4*d^7 + 30*A*B*b^4*c^6*d^5 - 8*A*B*b^4*c^8*d^3 - 8*A^2*a*b^
3*c^2*d^9 + 4*A^2*a*b^3*c^4*d^7 + 4*A^2*a^2*b^2*c*d^10 - 4*A^2*a^3*b*c^2*d^9 + 16*B^2*a*b^3*c^2*d^9 - 64*B^2*a
*b^3*c^4*d^7 + 60*B^2*a*b^3*c^6*d^5 - 16*B^2*a*b^3*c^8*d^3 - 8*B^2*a^2*b^2*c*d^10 - 8*B^2*a^3*b*c^2*d^9 + 4*B^
2*a^3*b*c^4*d^7 - 8*A*B*a*b^3*c*d^10 + 4*A*B*a^3*b*c*d^10 + 48*A*B*a*b^3*c^3*d^8 - 40*A*B*a*b^3*c^5*d^6 + 8*A*
B*a*b^3*c^7*d^4 + 8*A*B*a^3*b*c^3*d^8 - 4*A*B*a^3*b*c^5*d^6 - 20*A*B*a^2*b^2*c^2*d^9 + 4*A*B*a^2*b^2*c^4*d^7 +
4*A*B*a^2*b^2*c^6*d^5))/(d^10 - 2*c^2*d^8 + c^4*d^6))*1i)/d^3 + ((b*d*(A*b + 2*B*a)*1i - B*b^2*c*2i)*((32*(A^
2*b^4*c^2*d^8 - 2*A^2*b^4*c^4*d^6 + A^2*b^4*c^6*d^4 + 4*B^2*b^4*c^4*d^6 - 8*B^2*b^4*c^6*d^4 + 4*B^2*b^4*c^8*d^
2 + 4*B^2*a^2*b^2*c^2*d^8 - 8*B^2*a^2*b^2*c^4*d^6 + 4*B^2*a^2*b^2*c^6*d^4 - 4*A*B*b^4*c^3*d^7 + 8*A*B*b^4*c^5*
d^5 - 4*A*B*b^4*c^7*d^3 - 8*B^2*a*b^3*c^3*d^7 + 16*B^2*a*b^3*c^5*d^5 - 8*B^2*a*b^3*c^7*d^3 + 4*A*B*a*b^3*c^2*d
^8 - 8*A*B*a*b^3*c^4*d^6 + 4*A*B*a*b^3*c^6*d^4))/(d^9 - 2*c^2*d^7 + c^4*d^5) + ((b*d*(A*b + 2*B*a)*1i - B*b^2*
c*2i)*((32*(A*a^2*c^5*d^7 - A*a^2*c^3*d^9 - A*b^2*c*d^11 + A*b^2*c^3*d^9 + B*a^2*c^2*d^10 - B*a^2*c^4*d^8 + 2*
B*b^2*c^2*d^10 - 3*B*b^2*c^4*d^8 + B*b^2*c^6*d^6 - 2*B*a*b*c*d^11 + 2*A*a*b*c^2*d^10 - 2*A*a*b*c^4*d^8 + 2*B*a
*b*c^3*d^9))/(d^9 - 2*c^2*d^7 + c^4*d^5) + (((32*(c^2*d^12 - 2*c^4*d^10 + c^6*d^8))/(d^9 - 2*c^2*d^7 + c^4*d^5
) + (32*tan(e/2 + (f*x)/2)*(3*c*d^14 - 8*c^3*d^12 + 7*c^5*d^10 - 2*c^7*d^8))/(d^10 - 2*c^2*d^8 + c^4*d^6))*(b*
d*(A*b + 2*B*a)*1i - B*b^2*c*2i))/d^3 - (32*tan(e/2 + (f*x)/2)*(2*A*a^2*c^2*d^11 - 2*B*a^2*c*d^12 - 2*A*a^2*c^
4*d^9 + 4*A*b^2*c^2*d^11 - 6*A*b^2*c^4*d^9 + 2*A*b^2*c^6*d^7 + 2*B*a^2*c^3*d^10 - 6*B*b^2*c^3*d^10 + 10*B*b^2*
c^5*d^8 - 4*B*b^2*c^7*d^6 - 4*A*a*b*c*d^12 + 4*A*a*b*c^3*d^10 + 8*B*a*b*c^2*d^11 - 12*B*a*b*c^4*d^9 + 4*B*a*b*
c^6*d^7))/(d^10 - 2*c^2*d^8 + c^4*d^6)))/d^3 - (32*tan(e/2 + (f*x)/2)*(A^2*a^4*c^3*d^8 + 9*A^2*b^4*c^3*d^8 - 8
*A^2*b^4*c^5*d^6 + 2*A^2*b^4*c^7*d^4 - 8*B^2*b^4*c^3*d^8 + 29*B^2*b^4*c^5*d^6 - 28*B^2*b^4*c^7*d^4 + 8*B^2*b^4
*c^9*d^2 - 2*A^2*b^4*c*d^10 + B^2*a^4*c*d^10 + 4*A^2*a^2*b^2*c^3*d^8 - 2*A^2*a^2*b^2*c^5*d^6 + 42*B^2*a^2*b^2*
c^3*d^8 - 36*B^2*a^2*b^2*c^5*d^6 + 8*B^2*a^2*b^2*c^7*d^4 - 2*A*B*a^4*c^2*d^9 + 8*A*B*b^4*c^2*d^9 - 32*A*B*b^4*
c^4*d^7 + 30*A*B*b^4*c^6*d^5 - 8*A*B*b^4*c^8*d^3 - 8*A^2*a*b^3*c^2*d^9 + 4*A^2*a*b^3*c^4*d^7 + 4*A^2*a^2*b^2*c
*d^10 - 4*A^2*a^3*b*c^2*d^9 + 16*B^2*a*b^3*c^2*d^9 - 64*B^2*a*b^3*c^4*d^7 + 60*B^2*a*b^3*c^6*d^5 - 16*B^2*a*b^
3*c^8*d^3 - 8*B^2*a^2*b^2*c*d^10 - 8*B^2*a^3*b*c^2*d^9 + 4*B^2*a^3*b*c^4*d^7 - 8*A*B*a*b^3*c*d^10 + 4*A*B*a^3*
b*c*d^10 + 48*A*B*a*b^3*c^3*d^8 - 40*A*B*a*b^3*...

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