3.1.21 \(\int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^4} \, dx\) [21]

Optimal. Leaf size=260 \[ \frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} e}+\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))} \]

[Out]

(2*A*a^3+3*A*a*b^2-4*B*a^2*b-B*b^3)*arctan((a-b)^(1/2)*tan(1/2*e*x+1/2*d)/(a+b)^(1/2))/(a-b)^(7/2)/(a+b)^(7/2)
/e+1/3*C/b/e/(a+b*cos(e*x+d))^3-1/3*(A*b-B*a)*sin(e*x+d)/(a^2-b^2)/e/(a+b*cos(e*x+d))^3-1/6*(5*A*a*b-2*B*a^2-3
*B*b^2)*sin(e*x+d)/(a^2-b^2)^2/e/(a+b*cos(e*x+d))^2-1/6*(11*A*a^2*b+4*A*b^3-2*B*a^3-13*B*a*b^2)*sin(e*x+d)/(a^
2-b^2)^3/e/(a+b*cos(e*x+d))

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Rubi [A]
time = 0.35, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4462, 2833, 12, 2738, 211, 2747, 32} \begin {gather*} -\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sin (d+e x)}{6 e \left (a^2-b^2\right )^2 (a+b \cos (d+e x))^2}-\frac {(A b-a B) \sin (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cos (d+e x))^3}+\frac {\left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{e (a-b)^{7/2} (a+b)^{7/2}}-\frac {\left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right ) \sin (d+e x)}{6 e \left (a^2-b^2\right )^3 (a+b \cos (d+e x))}+\frac {C}{3 b e (a+b \cos (d+e x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^4,x]

[Out]

((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*
(a + b)^(7/2)*e) + C/(3*b*e*(a + b*Cos[d + e*x])^3) - ((A*b - a*B)*Sin[d + e*x])/(3*(a^2 - b^2)*e*(a + b*Cos[d
+ e*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sin[d + e*x])/(6*(a^2 - b^2)^2*e*(a + b*Cos[d + e*x])^2) - ((11*a
^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sin[d + e*x])/(6*(a^2 - b^2)^3*e*(a + b*Cos[d + e*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 211

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 2833

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

Rule 4462

Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] :> With[{e = FreeFactors[Cos[c*(a +
b*x)], x]}, Int[ActivateTrig[u*v], x] + Dist[d, Int[ActivateTrig[u]*Sin[c*(a + b*x)]^n, x], x] /; FunctionOfQ[
Cos[c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] &&  !FreeQ[v, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Sin] || EqQ[F, sin])

Rubi steps

\begin {align*} \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^4} \, dx &=C \int \frac {\sin (d+e x)}{(a+b \cos (d+e x))^4} \, dx+\int \frac {A+B \cos (d+e x)}{(a+b \cos (d+e x))^4} \, dx\\ &=-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\int \frac {-3 (a A-b B)+2 (A b-a B) \cos (d+e x)}{(a+b \cos (d+e x))^3} \, dx}{3 \left (a^2-b^2\right )}-\frac {C \text {Subst}\left (\int \frac {1}{(a+x)^4} \, dx,x,b \cos (d+e x)\right )}{b e}\\ &=\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}+\frac {\int \frac {2 \left (3 a^2 A+2 A b^2-5 a b B\right )-\left (5 a A b-2 a^2 B-3 b^2 B\right ) \cos (d+e x)}{(a+b \cos (d+e x))^2} \, dx}{6 \left (a^2-b^2\right )^2}\\ &=\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))}-\frac {\int -\frac {3 \left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right )}{a+b \cos (d+e x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))}+\frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \int \frac {1}{a+b \cos (d+e x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))}+\frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{\left (a^2-b^2\right )^3 e}\\ &=\frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} e}+\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))}\\ \end {align*}

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Mathematica [A]
time = 1.24, size = 302, normalized size = 1.16 \begin {gather*} \frac {\frac {24 \left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+\frac {-8 a^6 C+24 a^4 b^2 C-24 a^2 b^4 C+8 b^6 C-3 b \left (-24 a^4 A b+3 a^2 A b^3-4 A b^5+8 a^5 B+14 a^3 b^2 B+3 a b^4 B\right ) \sin (d+e x)+6 b^2 \left (9 a^3 A b+a A b^3-2 a^4 B-9 a^2 b^2 B+b^4 B\right ) \sin (2 (d+e x))+11 a^2 A b^4 \sin (3 (d+e x))+4 A b^6 \sin (3 (d+e x))-2 a^3 b^3 B \sin (3 (d+e x))-13 a b^5 B \sin (3 (d+e x))}{b \left (-a^2+b^2\right )^3 (a+b \cos (d+e x))^3}}{24 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^4,x]

[Out]

((24*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[((a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b
^2)^(7/2) + (-8*a^6*C + 24*a^4*b^2*C - 24*a^2*b^4*C + 8*b^6*C - 3*b*(-24*a^4*A*b + 3*a^2*A*b^3 - 4*A*b^5 + 8*a
^5*B + 14*a^3*b^2*B + 3*a*b^4*B)*Sin[d + e*x] + 6*b^2*(9*a^3*A*b + a*A*b^3 - 2*a^4*B - 9*a^2*b^2*B + b^4*B)*Si
n[2*(d + e*x)] + 11*a^2*A*b^4*Sin[3*(d + e*x)] + 4*A*b^6*Sin[3*(d + e*x)] - 2*a^3*b^3*B*Sin[3*(d + e*x)] - 13*
a*b^5*B*Sin[3*(d + e*x)])/(b*(-a^2 + b^2)^3*(a + b*Cos[d + e*x])^3))/(24*e)

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Maple [A]
time = 0.53, size = 456, normalized size = 1.75

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\left (6 A \,a^{2} b +3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -2 a^{2} b B -6 B a \,b^{2}-b^{3} B \right ) \left (\tan ^{5}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{\left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 C \left (\tan ^{4}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a -b}-\frac {4 \left (9 A \,a^{2} b +A \,b^{3}-3 a^{3} B -7 B a \,b^{2}\right ) \left (\tan ^{3}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 a C \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a^{2}-2 a b +b^{2}}-\frac {\left (6 A \,a^{2} b -3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B +2 a^{2} b B -6 B a \,b^{2}+b^{3} B \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}-\frac {2 \left (3 a^{2}+b^{2}\right ) C}{3 \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,b^{2}-4 a^{2} b B -b^{3} B \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{e}\)	\(456\)
default	\(\frac {\frac {-\frac {\left (6 A \,a^{2} b +3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -2 a^{2} b B -6 B a \,b^{2}-b^{3} B \right ) \left (\tan ^{5}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{\left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 C \left (\tan ^{4}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a -b}-\frac {4 \left (9 A \,a^{2} b +A \,b^{3}-3 a^{3} B -7 B a \,b^{2}\right ) \left (\tan ^{3}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 a C \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a^{2}-2 a b +b^{2}}-\frac {\left (6 A \,a^{2} b -3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B +2 a^{2} b B -6 B a \,b^{2}+b^{3} B \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}-\frac {2 \left (3 a^{2}+b^{2}\right ) C}{3 \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,b^{2}-4 a^{2} b B -b^{3} B \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{e}\)	\(456\)
risch	\(\text {Expression too large to display}\)	\(1282\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,x,method=_RETURNVERBOSE)

[Out]

1/e*(2*(-1/2*(6*A*a^2*b+3*A*a*b^2+2*A*b^3-2*B*a^3-2*B*a^2*b-6*B*a*b^2-B*b^3)/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*t
an(1/2*e*x+1/2*d)^5-C/(a-b)*tan(1/2*e*x+1/2*d)^4-2/3*(9*A*a^2*b+A*b^3-3*B*a^3-7*B*a*b^2)/(a^2-2*a*b+b^2)/(a^2+
2*a*b+b^2)*tan(1/2*e*x+1/2*d)^3-2*a*C/(a^2-2*a*b+b^2)*tan(1/2*e*x+1/2*d)^2-1/2*(6*A*a^2*b-3*A*a*b^2+2*A*b^3-2*
B*a^3+2*B*a^2*b-6*B*a*b^2+B*b^3)/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*e*x+1/2*d)-1/3*(3*a^2+b^2)*C/(a^3-3*a
^2*b+3*a*b^2-b^3))/(a*tan(1/2*e*x+1/2*d)^2-b*tan(1/2*e*x+1/2*d)^2+a+b)^3+(2*A*a^3+3*A*a*b^2-4*B*a^2*b-B*b^3)/(
a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*e*x+1/2*d)/((a-b)*(a+b))^(1/2)))

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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*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,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*b^2-4*a^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. 647 vs. \(2 (246) = 492\).
time = 0.45, size = 1368, normalized size = 5.26 \begin {gather*} \left [\frac {4 \, C a^{8} - 16 \, C a^{6} b^{2} + 24 \, C a^{4} b^{4} - 16 \, C a^{2} b^{6} + 4 \, C b^{8} - 3 \, {\left (2 \, A a^{6} b - 4 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} - B a^{3} b^{4} + {\left (2 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 3 \, A a b^{6} - B b^{7}\right )} \cos \left (x e + d\right )^{3} + 3 \, {\left (2 \, A a^{4} b^{3} - 4 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5} - B a b^{6}\right )} \cos \left (x e + d\right )^{2} + 3 \, {\left (2 \, A a^{5} b^{2} - 4 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4} - B a^{2} b^{5}\right )} \cos \left (x e + d\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (x e + d\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (x e + d\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (x e + d\right ) + b\right )} \sin \left (x e + d\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x e + d\right )^{2} + 2 \, a b \cos \left (x e + d\right ) + a^{2}}\right ) + 2 \, {\left (6 \, B a^{7} b - 18 \, A a^{6} b^{2} + 4 \, B a^{5} b^{3} + 23 \, A a^{4} b^{4} - 11 \, B a^{3} b^{5} - 7 \, A a^{2} b^{6} + B a b^{7} + 2 \, A b^{8} + {\left (2 \, B a^{5} b^{3} - 11 \, A a^{4} b^{4} + 11 \, B a^{3} b^{5} + 7 \, A a^{2} b^{6} - 13 \, B a b^{7} + 4 \, A b^{8}\right )} \cos \left (x e + d\right )^{2} + 3 \, {\left (2 \, B a^{6} b^{2} - 9 \, A a^{5} b^{3} + 7 \, B a^{4} b^{4} + 8 \, A a^{3} b^{5} - 10 \, B a^{2} b^{6} + A a b^{7} + B b^{8}\right )} \cos \left (x e + d\right )\right )} \sin \left (x e + d\right )}{12 \, {\left ({\left (a^{8} b^{4} - 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} - 4 \, a^{2} b^{10} + b^{12}\right )} \cos \left (x e + d\right )^{3} e + 3 \, {\left (a^{9} b^{3} - 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} - 4 \, a^{3} b^{9} + a b^{11}\right )} \cos \left (x e + d\right )^{2} e + 3 \, {\left (a^{10} b^{2} - 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} - 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \cos \left (x e + d\right ) e + {\left (a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}\right )} e\right )}}, \frac {2 \, C a^{8} - 8 \, C a^{6} b^{2} + 12 \, C a^{4} b^{4} - 8 \, C a^{2} b^{6} + 2 \, C b^{8} + 3 \, {\left (2 \, A a^{6} b - 4 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} - B a^{3} b^{4} + {\left (2 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 3 \, A a b^{6} - B b^{7}\right )} \cos \left (x e + d\right )^{3} + 3 \, {\left (2 \, A a^{4} b^{3} - 4 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5} - B a b^{6}\right )} \cos \left (x e + d\right )^{2} + 3 \, {\left (2 \, A a^{5} b^{2} - 4 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4} - B a^{2} b^{5}\right )} \cos \left (x e + d\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (x e + d\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (x e + d\right )}\right ) + {\left (6 \, B a^{7} b - 18 \, A a^{6} b^{2} + 4 \, B a^{5} b^{3} + 23 \, A a^{4} b^{4} - 11 \, B a^{3} b^{5} - 7 \, A a^{2} b^{6} + B a b^{7} + 2 \, A b^{8} + {\left (2 \, B a^{5} b^{3} - 11 \, A a^{4} b^{4} + 11 \, B a^{3} b^{5} + 7 \, A a^{2} b^{6} - 13 \, B a b^{7} + 4 \, A b^{8}\right )} \cos \left (x e + d\right )^{2} + 3 \, {\left (2 \, B a^{6} b^{2} - 9 \, A a^{5} b^{3} + 7 \, B a^{4} b^{4} + 8 \, A a^{3} b^{5} - 10 \, B a^{2} b^{6} + A a b^{7} + B b^{8}\right )} \cos \left (x e + d\right )\right )} \sin \left (x e + d\right )}{6 \, {\left ({\left (a^{8} b^{4} - 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} - 4 \, a^{2} b^{10} + b^{12}\right )} \cos \left (x e + d\right )^{3} e + 3 \, {\left (a^{9} b^{3} - 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} - 4 \, a^{3} b^{9} + a b^{11}\right )} \cos \left (x e + d\right )^{2} e + 3 \, {\left (a^{10} b^{2} - 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} - 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \cos \left (x e + d\right ) e + {\left (a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,x, algorithm="fricas")

[Out]

[1/12*(4*C*a^8 - 16*C*a^6*b^2 + 24*C*a^4*b^4 - 16*C*a^2*b^6 + 4*C*b^8 - 3*(2*A*a^6*b - 4*B*a^5*b^2 + 3*A*a^4*b
^3 - B*a^3*b^4 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cos(x*e + d)^3 + 3*(2*A*a^4*b^3 - 4*B*a^3*b^4
+ 3*A*a^2*b^5 - B*a*b^6)*cos(x*e + d)^2 + 3*(2*A*a^5*b^2 - 4*B*a^4*b^3 + 3*A*a^3*b^4 - B*a^2*b^5)*cos(x*e + d
))*sqrt(-a^2 + b^2)*log((2*a*b*cos(x*e + d) + (2*a^2 - b^2)*cos(x*e + d)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(x*e + d
) + b)*sin(x*e + d) - a^2 + 2*b^2)/(b^2*cos(x*e + d)^2 + 2*a*b*cos(x*e + d) + a^2)) + 2*(6*B*a^7*b - 18*A*a^6*
b^2 + 4*B*a^5*b^3 + 23*A*a^4*b^4 - 11*B*a^3*b^5 - 7*A*a^2*b^6 + B*a*b^7 + 2*A*b^8 + (2*B*a^5*b^3 - 11*A*a^4*b^
4 + 11*B*a^3*b^5 + 7*A*a^2*b^6 - 13*B*a*b^7 + 4*A*b^8)*cos(x*e + d)^2 + 3*(2*B*a^6*b^2 - 9*A*a^5*b^3 + 7*B*a^4
*b^4 + 8*A*a^3*b^5 - 10*B*a^2*b^6 + A*a*b^7 + B*b^8)*cos(x*e + d))*sin(x*e + d))/((a^8*b^4 - 4*a^6*b^6 + 6*a^4
*b^8 - 4*a^2*b^10 + b^12)*cos(x*e + d)^3*e + 3*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*cos(x*e
+ d)^2*e + 3*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*cos(x*e + d)*e + (a^11*b - 4*a^9*b^3 +
6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9)*e), 1/6*(2*C*a^8 - 8*C*a^6*b^2 + 12*C*a^4*b^4 - 8*C*a^2*b^6 + 2*C*b^8 + 3*(2*
A*a^6*b - 4*B*a^5*b^2 + 3*A*a^4*b^3 - B*a^3*b^4 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cos(x*e + d)
^3 + 3*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cos(x*e + d)^2 + 3*(2*A*a^5*b^2 - 4*B*a^4*b^3 + 3*A
*a^3*b^4 - B*a^2*b^5)*cos(x*e + d))*sqrt(a^2 - b^2)*arctan(-(a*cos(x*e + d) + b)/(sqrt(a^2 - b^2)*sin(x*e + d)
)) + (6*B*a^7*b - 18*A*a^6*b^2 + 4*B*a^5*b^3 + 23*A*a^4*b^4 - 11*B*a^3*b^5 - 7*A*a^2*b^6 + B*a*b^7 + 2*A*b^8 +
(2*B*a^5*b^3 - 11*A*a^4*b^4 + 11*B*a^3*b^5 + 7*A*a^2*b^6 - 13*B*a*b^7 + 4*A*b^8)*cos(x*e + d)^2 + 3*(2*B*a^6*
b^2 - 9*A*a^5*b^3 + 7*B*a^4*b^4 + 8*A*a^3*b^5 - 10*B*a^2*b^6 + A*a*b^7 + B*b^8)*cos(x*e + d))*sin(x*e + d))/((
a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*cos(x*e + d)^3*e + 3*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4
*a^3*b^9 + a*b^11)*cos(x*e + d)^2*e + 3*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*cos(x*e + d)
*e + (a^11*b - 4*a^9*b^3 + 6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9)*e)]

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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*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))**4,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (246) = 492\).
time = 0.55, size = 960, normalized size = 3.69 \begin {gather*} -\frac {1}{3} \, {\left (\frac {3 \, {\left (2 \, A a^{3} - 4 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} {\left (\pi \left \lfloor \frac {x e + d}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} - \frac {6 \, B a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 18 \, A a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 6 \, B a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 27 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 12 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 6 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 27 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 3 \, A a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 12 \, B a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 6 \, A b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 3 \, B b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 6 \, C a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} - 6 \, C a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 12 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 12 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} - 6 \, C a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} - 6 \, C b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 12 \, B a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 36 \, A a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 16 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 32 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 28 \, B a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 4 \, A b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 12 \, C a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 24 \, C a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 12 \, C a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 6 \, B a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 18 \, A a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 6 \, B a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 27 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 12 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 27 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 3 \, A a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 12 \, B a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \, A b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 3 \, B b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \, C a^{5} - 18 \, C a^{4} b - 20 \, C a^{3} b^{2} - 12 \, C a^{2} b^{3} - 6 \, C a b^{4} - 2 \, C b^{5}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + a + b\right )}^{3}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,x, algorithm="giac")

[Out]

-1/3*(3*(2*A*a^3 - 4*B*a^2*b + 3*A*a*b^2 - B*b^3)*(pi*floor(1/2*(x*e + d)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-
(a*tan(1/2*x*e + 1/2*d) - b*tan(1/2*x*e + 1/2*d))/sqrt(a^2 - b^2)))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(
a^2 - b^2)) - (6*B*a^5*tan(1/2*x*e + 1/2*d)^5 - 18*A*a^4*b*tan(1/2*x*e + 1/2*d)^5 - 6*B*a^4*b*tan(1/2*x*e + 1/
2*d)^5 + 27*A*a^3*b^2*tan(1/2*x*e + 1/2*d)^5 + 12*B*a^3*b^2*tan(1/2*x*e + 1/2*d)^5 - 6*A*a^2*b^3*tan(1/2*x*e +
1/2*d)^5 - 27*B*a^2*b^3*tan(1/2*x*e + 1/2*d)^5 + 3*A*a*b^4*tan(1/2*x*e + 1/2*d)^5 + 12*B*a*b^4*tan(1/2*x*e +
1/2*d)^5 - 6*A*b^5*tan(1/2*x*e + 1/2*d)^5 + 3*B*b^5*tan(1/2*x*e + 1/2*d)^5 - 6*C*a^5*tan(1/2*x*e + 1/2*d)^4 -
6*C*a^4*b*tan(1/2*x*e + 1/2*d)^4 + 12*C*a^3*b^2*tan(1/2*x*e + 1/2*d)^4 + 12*C*a^2*b^3*tan(1/2*x*e + 1/2*d)^4 -
6*C*a*b^4*tan(1/2*x*e + 1/2*d)^4 - 6*C*b^5*tan(1/2*x*e + 1/2*d)^4 + 12*B*a^5*tan(1/2*x*e + 1/2*d)^3 - 36*A*a^
4*b*tan(1/2*x*e + 1/2*d)^3 + 16*B*a^3*b^2*tan(1/2*x*e + 1/2*d)^3 + 32*A*a^2*b^3*tan(1/2*x*e + 1/2*d)^3 - 28*B*
a*b^4*tan(1/2*x*e + 1/2*d)^3 + 4*A*b^5*tan(1/2*x*e + 1/2*d)^3 - 12*C*a^5*tan(1/2*x*e + 1/2*d)^2 - 24*C*a^4*b*t
an(1/2*x*e + 1/2*d)^2 + 24*C*a^2*b^3*tan(1/2*x*e + 1/2*d)^2 + 12*C*a*b^4*tan(1/2*x*e + 1/2*d)^2 + 6*B*a^5*tan(
1/2*x*e + 1/2*d) - 18*A*a^4*b*tan(1/2*x*e + 1/2*d) + 6*B*a^4*b*tan(1/2*x*e + 1/2*d) - 27*A*a^3*b^2*tan(1/2*x*e
+ 1/2*d) + 12*B*a^3*b^2*tan(1/2*x*e + 1/2*d) - 6*A*a^2*b^3*tan(1/2*x*e + 1/2*d) + 27*B*a^2*b^3*tan(1/2*x*e +
1/2*d) - 3*A*a*b^4*tan(1/2*x*e + 1/2*d) + 12*B*a*b^4*tan(1/2*x*e + 1/2*d) - 6*A*b^5*tan(1/2*x*e + 1/2*d) - 3*B
*b^5*tan(1/2*x*e + 1/2*d) - 6*C*a^5 - 18*C*a^4*b - 20*C*a^3*b^2 - 12*C*a^2*b^3 - 6*C*a*b^4 - 2*C*b^5)/((a^6 -
3*a^4*b^2 + 3*a^2*b^4 - b^6)*(a*tan(1/2*x*e + 1/2*d)^2 - b*tan(1/2*x*e + 1/2*d)^2 + a + b)^3))*e^(-1)

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Mupad [B]
time = 5.22, size = 502, normalized size = 1.93 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{2\,\sqrt {a+b}\,{\left (a-b\right )}^{7/2}}\right )\,\left (2\,A\,a^3-4\,B\,a^2\,b+3\,A\,a\,b^2-B\,b^3\right )}{e\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}}-\frac {\frac {2\,C\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4}{a-b}+\frac {2\,C\,\left (3\,a^2+b^2\right )}{3\,{\left (a-b\right )}^3}+\frac {4\,C\,a\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2}{{\left (a-b\right )}^2}+\frac {4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (-3\,B\,a^3+9\,A\,a^2\,b-7\,B\,a\,b^2+A\,b^3\right )}{3\,{\left (a+b\right )}^2\,\left (a^2-2\,a\,b+b^2\right )}-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (2\,B\,a^3-2\,A\,b^3+B\,b^3-3\,A\,a\,b^2-6\,A\,a^2\,b+6\,B\,a\,b^2+2\,B\,a^2\,b\right )}{{\left (a+b\right )}^3\,\left (a-b\right )}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,A\,b^3-2\,B\,a^3+B\,b^3-3\,A\,a\,b^2+6\,A\,a^2\,b-6\,B\,a\,b^2+2\,B\,a^2\,b\right )}{\left (a+b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}}{e\,\left (3\,a\,b^2-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (-3\,a^3+3\,a^2\,b+3\,a\,b^2-3\,b^3\right )-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (-3\,a^3-3\,a^2\,b+3\,a\,b^2+3\,b^3\right )+3\,a^2\,b+a^3+b^3+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cos(d + e*x) + C*sin(d + e*x))/(a + b*cos(d + e*x))^4,x)

[Out]

(atan((tan(d/2 + (e*x)/2)*(2*a - 2*b)*(3*a*b^2 - 3*a^2*b + a^3 - b^3))/(2*(a + b)^(1/2)*(a - b)^(7/2)))*(2*A*a
^3 - B*b^3 + 3*A*a*b^2 - 4*B*a^2*b))/(e*(a + b)^(7/2)*(a - b)^(7/2)) - ((2*C*tan(d/2 + (e*x)/2)^4)/(a - b) + (
2*C*(3*a^2 + b^2))/(3*(a - b)^3) + (4*C*a*tan(d/2 + (e*x)/2)^2)/(a - b)^2 + (4*tan(d/2 + (e*x)/2)^3*(A*b^3 - 3
*B*a^3 + 9*A*a^2*b - 7*B*a*b^2))/(3*(a + b)^2*(a^2 - 2*a*b + b^2)) - (tan(d/2 + (e*x)/2)^5*(2*B*a^3 - 2*A*b^3
+ B*b^3 - 3*A*a*b^2 - 6*A*a^2*b + 6*B*a*b^2 + 2*B*a^2*b))/((a + b)^3*(a - b)) + (tan(d/2 + (e*x)/2)*(2*A*b^3 -
2*B*a^3 + B*b^3 - 3*A*a*b^2 + 6*A*a^2*b - 6*B*a*b^2 + 2*B*a^2*b))/((a + b)*(3*a*b^2 - 3*a^2*b + a^3 - b^3)))/
(e*(3*a*b^2 - tan(d/2 + (e*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) - tan(d/2 + (e*x)/2)^2*(3*a*b^2 - 3*a^2
*b - 3*a^3 + 3*b^3) + 3*a^2*b + a^3 + b^3 + tan(d/2 + (e*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3)))

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