3.3.78 \(\int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx\) [278]

Optimal. Leaf size=301 \[ -\frac {\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \]

[Out]

-(8*A*a^6*b-8*A*a^4*b^3+7*A*a^2*b^5-2*A*b^7-2*B*a^7-3*B*a^5*b^2)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(
1/2))/a^4/(a-b)^(7/2)/(a+b)^(7/2)/d+A*arctanh(sin(d*x+c))/a^4/d+1/3*b*(A*b-B*a)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*
cos(d*x+c))^3+1/6*b*(8*A*a^2*b-3*A*b^3-5*B*a^3)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2+1/6*b*(26*A*a^
4*b-17*A*a^2*b^3+6*A*b^5-11*B*a^5-4*B*a^3*b^2)*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))

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Rubi [A]
time = 0.95, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3079, 3134, 3080, 3855, 2738, 211} \begin {gather*} \frac {A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \sin (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {b \left (-11 a^5 B+26 a^4 A b-4 a^3 b^2 B-17 a^2 A b^3+6 A b^5\right ) \sin (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac {\left (-2 a^7 B+8 a^6 A b-3 a^5 b^2 B-8 a^4 A b^3+7 a^2 A b^5-2 A b^7\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]

[Out]

-(((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)
/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^4*d) + (b*(A*b - a*B)*S
in[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*Sin[c + d*x])/(6*
a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B
)*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

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 3079

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^2 - a*b*B))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c +
d*Sin[e + f*x])^(1 + n)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), I
nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*
(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 -
d^2, 0] && RationalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n
, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3080

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*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]

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\int \frac {\left (3 A \left (a^2-b^2\right )-3 a (A b-a B) \cos (c+d x)+2 b (A b-a B) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\int \frac {\left (6 A \left (a^2-b^2\right )^2-2 a \left (6 a^2 A b-A b^3-3 a^3 B-2 a b^2 B\right ) \cos (c+d x)+b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (6 A \left (a^2-b^2\right )^3-3 a \left (6 a^4 A b-2 a^2 A b^3+A b^5-2 a^5 B-3 a^3 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {A \int \sec (c+d x) \, dx}{a^4}-\frac {\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right )^3 d}\\ &=-\frac {\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 1.74, size = 368, normalized size = 1.22 \begin {gather*} \frac {\cos (c+d x) (B+A \sec (c+d x)) \left (\frac {24 \left (-8 a^6 A b+8 a^4 A b^3-7 a^2 A b^5+2 A b^7+2 a^7 B+3 a^5 b^2 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}-24 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2 a b \left (-72 a^6 A b+38 a^4 A b^3-5 a^2 A b^5-6 A b^7+36 a^7 B+a^5 b^2 B+8 a^3 b^4 B+6 a b \left (-20 a^4 A b+15 a^2 A b^3-5 A b^5+9 a^5 B+a^3 b^2 B\right ) \cos (c+d x)+b^2 \left (-26 a^4 A b+17 a^2 A b^3-6 A b^5+11 a^5 B+4 a^3 b^2 B\right ) \cos (2 (c+d x))\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}\right )}{24 a^4 d (A+B \cos (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]

[Out]

(Cos[c + d*x]*(B + A*Sec[c + d*x])*((24*(-8*a^6*A*b + 8*a^4*A*b^3 - 7*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B + 3*a^5*b^
2*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - 24*A*Log[Cos[(c + d*x)/2] - Si
n[(c + d*x)/2]] + 24*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (2*a*b*(-72*a^6*A*b + 38*a^4*A*b^3 - 5*a^2*A
*b^5 - 6*A*b^7 + 36*a^7*B + a^5*b^2*B + 8*a^3*b^4*B + 6*a*b*(-20*a^4*A*b + 15*a^2*A*b^3 - 5*A*b^5 + 9*a^5*B +
a^3*b^2*B)*Cos[c + d*x] + b^2*(-26*a^4*A*b + 17*a^2*A*b^3 - 6*A*b^5 + 11*a^5*B + 4*a^3*b^2*B)*Cos[2*(c + d*x)]
)*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(24*a^4*d*(A + B*Cos[c + d*x]))

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Maple [A]
time = 1.40, size = 479, normalized size = 1.59

method	result	size
derivativedivides	\(\frac {\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4}}-\frac {2 \left (\frac {-\frac {\left (12 A \,a^{4} b +4 A \,a^{3} b^{2}-6 A \,a^{2} b^{3}-A a \,b^{4}+2 A \,b^{5}-6 B \,a^{5}-3 B \,a^{4} b -2 B \,a^{3} b^{2}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (18 A \,a^{4} b -11 A \,a^{2} b^{3}+3 A \,b^{5}-9 B \,a^{5}-B \,a^{3} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (12 A \,a^{4} b -4 A \,a^{3} b^{2}-6 A \,a^{2} b^{3}+A a \,b^{4}+2 A \,b^{5}-6 B \,a^{5}+3 B \,a^{4} b -2 B \,a^{3} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (8 A \,a^{6} b -8 A \,a^{4} b^{3}+7 A \,a^{2} b^{5}-2 A \,b^{7}-2 B \,a^{7}-3 B \,a^{5} b^{2}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}}{d}\)	\(479\)
default	\(\frac {\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{4}}-\frac {2 \left (\frac {-\frac {\left (12 A \,a^{4} b +4 A \,a^{3} b^{2}-6 A \,a^{2} b^{3}-A a \,b^{4}+2 A \,b^{5}-6 B \,a^{5}-3 B \,a^{4} b -2 B \,a^{3} b^{2}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (18 A \,a^{4} b -11 A \,a^{2} b^{3}+3 A \,b^{5}-9 B \,a^{5}-B \,a^{3} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (12 A \,a^{4} b -4 A \,a^{3} b^{2}-6 A \,a^{2} b^{3}+A a \,b^{4}+2 A \,b^{5}-6 B \,a^{5}+3 B \,a^{4} b -2 B \,a^{3} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (8 A \,a^{6} b -8 A \,a^{4} b^{3}+7 A \,a^{2} b^{5}-2 A \,b^{7}-2 B \,a^{7}-3 B \,a^{5} b^{2}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}}{d}\)	\(479\)
risch	\(\text {Expression too large to display}\)	\(1676\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c))*sec(d*x+c)/(a+b*cos(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(A/a^4*ln(tan(1/2*d*x+1/2*c)+1)-2/a^4*((-1/2*(12*A*a^4*b+4*A*a^3*b^2-6*A*a^2*b^3-A*a*b^4+2*A*b^5-6*B*a^5-3
*B*a^4*b-2*B*a^3*b^2)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5-2/3*(18*A*a^4*b-11*A*a^2*b^3+3*
A*b^5-9*B*a^5-B*a^3*b^2)*a*b/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(12*A*a^4*b-4*A*a^3*b^2-
6*A*a^2*b^3+A*a*b^4+2*A*b^5-6*B*a^5+3*B*a^4*b-2*B*a^3*b^2)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2
*c))/(a*tan(1/2*d*x+1/2*c)^2-b*tan(1/2*d*x+1/2*c)^2+a+b)^3+1/2*(8*A*a^6*b-8*A*a^4*b^3+7*A*a^2*b^5-2*A*b^7-2*B*
a^7-3*B*a^5*b^2)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b
))^(1/2)))-A/a^4*ln(tan(1/2*d*x+1/2*c)-1))

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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(d*x+c))*sec(d*x+c)/(a+b*cos(d*x+c))^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. 1100 vs. \(2 (285) = 570\).
time = 35.53, size = 2269, normalized size = 7.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*(2*B*a^10 - 8*A*a^9*b + 3*B*a^8*b^2 + 8*A*a^7*b^3 - 7*A*a^5*b^5 + 2*A*a^3*b^7 + (2*B*a^7*b^3 - 8*A*a^
6*b^4 + 3*B*a^5*b^5 + 8*A*a^4*b^6 - 7*A*a^2*b^8 + 2*A*b^10)*cos(d*x + c)^3 + 3*(2*B*a^8*b^2 - 8*A*a^7*b^3 + 3*
B*a^6*b^4 + 8*A*a^5*b^5 - 7*A*a^3*b^7 + 2*A*a*b^9)*cos(d*x + c)^2 + 3*(2*B*a^9*b - 8*A*a^8*b^2 + 3*B*a^7*b^3 +
8*A*a^6*b^4 - 7*A*a^4*b^6 + 2*A*a^2*b^8)*cos(d*x + c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^
2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 +
2*a*b*cos(d*x + c) + a^2)) + 6*(A*a^11 - 4*A*a^9*b^2 + 6*A*a^7*b^4 - 4*A*a^5*b^6 + A*a^3*b^8 + (A*a^8*b^3 - 4*
A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*b^11)*cos(d*x + c)^3 + 3*(A*a^9*b^2 - 4*A*a^7*b^4 + 6*A*a^5*b^6 - 4*
A*a^3*b^8 + A*a*b^10)*cos(d*x + c)^2 + 3*(A*a^10*b - 4*A*a^8*b^3 + 6*A*a^6*b^5 - 4*A*a^4*b^7 + A*a^2*b^9)*cos(
d*x + c))*log(sin(d*x + c) + 1) - 6*(A*a^11 - 4*A*a^9*b^2 + 6*A*a^7*b^4 - 4*A*a^5*b^6 + A*a^3*b^8 + (A*a^8*b^3
- 4*A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*b^11)*cos(d*x + c)^3 + 3*(A*a^9*b^2 - 4*A*a^7*b^4 + 6*A*a^5*b^6
- 4*A*a^3*b^8 + A*a*b^10)*cos(d*x + c)^2 + 3*(A*a^10*b - 4*A*a^8*b^3 + 6*A*a^6*b^5 - 4*A*a^4*b^7 + A*a^2*b^9)
*cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(18*B*a^10*b - 36*A*a^9*b^2 - 23*B*a^8*b^3 + 68*A*a^7*b^4 + 7*B*a^6*
b^5 - 43*A*a^5*b^6 - 2*B*a^4*b^7 + 11*A*a^3*b^8 + (11*B*a^8*b^3 - 26*A*a^7*b^4 - 7*B*a^6*b^5 + 43*A*a^5*b^6 -
4*B*a^4*b^7 - 23*A*a^3*b^8 + 6*A*a*b^10)*cos(d*x + c)^2 + 3*(9*B*a^9*b^2 - 20*A*a^8*b^3 - 8*B*a^7*b^4 + 35*A*a
^6*b^5 - B*a^5*b^6 - 20*A*a^4*b^7 + 5*A*a^2*b^9)*cos(d*x + c))*sin(d*x + c))/((a^12*b^3 - 4*a^10*b^5 + 6*a^8*b
^7 - 4*a^6*b^9 + a^4*b^11)*d*cos(d*x + c)^3 + 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*c
os(d*x + c)^2 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c) + (a^15 - 4*a^13*b^2
+ 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d), 1/6*(3*(2*B*a^10 - 8*A*a^9*b + 3*B*a^8*b^2 + 8*A*a^7*b^3 - 7*A*a^5*b^
5 + 2*A*a^3*b^7 + (2*B*a^7*b^3 - 8*A*a^6*b^4 + 3*B*a^5*b^5 + 8*A*a^4*b^6 - 7*A*a^2*b^8 + 2*A*b^10)*cos(d*x + c
)^3 + 3*(2*B*a^8*b^2 - 8*A*a^7*b^3 + 3*B*a^6*b^4 + 8*A*a^5*b^5 - 7*A*a^3*b^7 + 2*A*a*b^9)*cos(d*x + c)^2 + 3*(
2*B*a^9*b - 8*A*a^8*b^2 + 3*B*a^7*b^3 + 8*A*a^6*b^4 - 7*A*a^4*b^6 + 2*A*a^2*b^8)*cos(d*x + c))*sqrt(a^2 - b^2)
*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) + 3*(A*a^11 - 4*A*a^9*b^2 + 6*A*a^7*b^4 - 4*A*a^
5*b^6 + A*a^3*b^8 + (A*a^8*b^3 - 4*A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*b^11)*cos(d*x + c)^3 + 3*(A*a^9*b
^2 - 4*A*a^7*b^4 + 6*A*a^5*b^6 - 4*A*a^3*b^8 + A*a*b^10)*cos(d*x + c)^2 + 3*(A*a^10*b - 4*A*a^8*b^3 + 6*A*a^6*
b^5 - 4*A*a^4*b^7 + A*a^2*b^9)*cos(d*x + c))*log(sin(d*x + c) + 1) - 3*(A*a^11 - 4*A*a^9*b^2 + 6*A*a^7*b^4 - 4
*A*a^5*b^6 + A*a^3*b^8 + (A*a^8*b^3 - 4*A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*b^11)*cos(d*x + c)^3 + 3*(A*
a^9*b^2 - 4*A*a^7*b^4 + 6*A*a^5*b^6 - 4*A*a^3*b^8 + A*a*b^10)*cos(d*x + c)^2 + 3*(A*a^10*b - 4*A*a^8*b^3 + 6*A
*a^6*b^5 - 4*A*a^4*b^7 + A*a^2*b^9)*cos(d*x + c))*log(-sin(d*x + c) + 1) - (18*B*a^10*b - 36*A*a^9*b^2 - 23*B*
a^8*b^3 + 68*A*a^7*b^4 + 7*B*a^6*b^5 - 43*A*a^5*b^6 - 2*B*a^4*b^7 + 11*A*a^3*b^8 + (11*B*a^8*b^3 - 26*A*a^7*b^
4 - 7*B*a^6*b^5 + 43*A*a^5*b^6 - 4*B*a^4*b^7 - 23*A*a^3*b^8 + 6*A*a*b^10)*cos(d*x + c)^2 + 3*(9*B*a^9*b^2 - 20
*A*a^8*b^3 - 8*B*a^7*b^4 + 35*A*a^6*b^5 - B*a^5*b^6 - 20*A*a^4*b^7 + 5*A*a^2*b^9)*cos(d*x + c))*sin(d*x + c))/
((a^12*b^3 - 4*a^10*b^5 + 6*a^8*b^7 - 4*a^6*b^9 + a^4*b^11)*d*cos(d*x + c)^3 + 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^
9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c)^2 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*
cos(d*x + c) + (a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \cos {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)/(a+b*cos(d*x+c))**4,x)

[Out]

Integral((A + B*cos(c + d*x))*sec(c + d*x)/(a + b*cos(c + d*x))**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (285) = 570\).
time = 0.52, size = 837, normalized size = 2.78 \begin {gather*} \frac {\frac {3 \, {\left (2 \, B a^{7} - 8 \, A a^{6} b + 3 \, B a^{5} b^{2} + 8 \, A a^{4} b^{3} - 7 \, A a^{2} b^{5} + 2 \, A b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {3 \, A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {3 \, A \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {18 \, B a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, A a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, B a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, A a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, A a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, A a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, B a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 116 \, A a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 56 \, A a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, B a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, A a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, B a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 60 \, A a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, A a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, A a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{3}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(2*B*a^7 - 8*A*a^6*b + 3*B*a^5*b^2 + 8*A*a^4*b^3 - 7*A*a^2*b^5 + 2*A*b^7)*(pi*floor(1/2*(d*x + c)/pi +
1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^10 - 3*a^
8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(a^2 - b^2)) + 3*A*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 3*A*log(abs(tan(1
/2*d*x + 1/2*c) - 1))/a^4 - (18*B*a^7*b*tan(1/2*d*x + 1/2*c)^5 - 36*A*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 - 27*B*a^
6*b^2*tan(1/2*d*x + 1/2*c)^5 + 60*A*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*
a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*B*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 45*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*
B*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 15*A*a*b^7*tan(1/2*d*x + 1/2*c)^5 - 6*
A*b^8*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^7*b*tan(1/2*d*x + 1/2*c)^3 - 72*A*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 - 32*B*
a^5*b^3*tan(1/2*d*x + 1/2*c)^3 + 116*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 - 4*B*a^3*b^5*tan(1/2*d*x + 1/2*c)^3 - 5
6*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 + 12*A*b^8*tan(1/2*d*x + 1/2*c)^3 + 18*B*a^7*b*tan(1/2*d*x + 1/2*c) - 36*A*
a^6*b^2*tan(1/2*d*x + 1/2*c) + 27*B*a^6*b^2*tan(1/2*d*x + 1/2*c) - 60*A*a^5*b^3*tan(1/2*d*x + 1/2*c) + 6*B*a^5
*b^3*tan(1/2*d*x + 1/2*c) + 6*A*a^4*b^4*tan(1/2*d*x + 1/2*c) + 3*B*a^4*b^4*tan(1/2*d*x + 1/2*c) + 45*A*a^3*b^5
*tan(1/2*d*x + 1/2*c) + 6*B*a^3*b^5*tan(1/2*d*x + 1/2*c) + 6*A*a^2*b^6*tan(1/2*d*x + 1/2*c) - 15*A*a*b^7*tan(1
/2*d*x + 1/2*c) - 6*A*b^8*tan(1/2*d*x + 1/2*c))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(a*tan(1/2*d*x + 1/2*
c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^3))/d

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Mupad [B]
time = 12.81, size = 2500, normalized size = 8.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*atan(-((A*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A
^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A
^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a
^12*b^2 - 32*A*B*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3))/(a^16*b + a^17
- a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14
*b^3 - 5*a^15*b^2) + (A*((8*(4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*A*a^10*b^11 - 14*A*a^11*b^
10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^5 + 20*A*a^17*b^4 + 64*A*a^
18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^6 + 6*B*a^16*b^5 - 6*B*a^17
*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11
*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) + (8*A*ta
n(c/2 + (d*x)/2)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10 + 120*a^13*
b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^2))/(a^
4*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*
a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2))))/a^4)*1i)/a^4 + (A*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B
^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^5*b^9
- 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 +
44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^2 - 32*A*B*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B
*a^9*b^5 - 16*A*B*a^11*b^3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a
^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) - (A*((8*(4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 +
2*A*a^9*b^12 + 26*A*a^10*b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^
6 - 110*A*a^16*b^5 + 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b
^7 + 14*B*a^15*b^6 + 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b))/(a
^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a
^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) - (8*A*tan(c/2 + (d*x)/2)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^1
2 - 48*a^11*b^11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 +
48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^2))/(a^4*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 1
0*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2))))/a^4)*1i)/a^4)/((16*(4*A^3*b
^13 + 4*A*B^2*a^13 - 4*A^2*B*a^13 - 2*A^3*a*b^12 + 16*A^3*a^12*b - 26*A^3*a^2*b^11 + 11*A^3*a^3*b^10 + 70*A^3*
a^4*b^9 - 34*A^3*a^5*b^8 - 110*A^3*a^6*b^7 + 66*A^3*a^7*b^6 + 110*A^3*a^8*b^5 - 64*A^3*a^9*b^4 - 64*A^3*a^10*b
^3 + 48*A^3*a^11*b^2 - 28*A^2*B*a^12*b + 9*A*B^2*a^9*b^4 + 12*A*B^2*a^11*b^2 + 6*A^2*B*a^4*b^9 + 6*A^2*B*a^5*b
^8 - 20*A^2*B*a^6*b^7 - 14*A^2*B*a^7*b^6 + 14*A^2*B*a^8*b^5 + 6*A^2*B*a^9*b^4 - 22*A^2*B*a^10*b^3 + 6*A^2*B*a^
11*b^2))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15
*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) + (A*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B^2*a^1
4 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^5*b^9 - 164
*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 + 44*A^
2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^2 - 32*A*B*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b
^5 - 16*A*B*a^11*b^3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^
6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) + (A*((8*(4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^
9*b^12 + 26*A*a^10*b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 11
0*A*a^16*b^5 + 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 1
4*B*a^15*b^6 + 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b))/(a^19*b
+ a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^
4 - 5*a^17*b^3 - 5*a^18*b^2) + (8*A*tan(c/2 + (d*x)/2)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^12 - 48
*a^11*b^11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 + 48*a^1
8*b^4 - 48*a^19*b^3 - 8*a^20*b^2))/(a^4*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10
*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 ...

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