3.337 \(\int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx\)

Optimal. Leaf size=310 \[ \frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{6 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}-\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {\left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {B \tanh ^{-1}(\sin (c+d x))}{b^4 d} \]

[Out]

B*arctanh(sin(d*x+c))/b^4/d-(3*A*a^2*b^5+2*A*b^7+2*B*a^7-7*B*a^5*b^2+8*B*a^3*b^4-8*B*a*b^6)*arctanh((a-b)^(1/2
)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^4/(a+b)^(7/2)/d+1/3*a*(A*b-B*a)*sec(d*x+c)^2*tan(d*x+c)/b/(a^2
-b^2)/d/(a+b*sec(d*x+c))^3+1/6*a^2*(5*A*b^3+3*B*a^3-8*B*a*b^2)*tan(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2
-1/6*a*(A*a^2*b^3-16*A*b^5+9*B*a^5-28*B*a^3*b^2+34*B*a*b^4)*tan(d*x+c)/b^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))

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Rubi [A]  time = 1.37, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4029, 4090, 4080, 3998, 3770, 3831, 2659, 208} \[ -\frac {\left (3 a^2 A b^5-7 a^5 b^2 B+8 a^3 b^4 B+2 a^7 B-8 a b^6 B+2 A b^7\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \tan (c+d x)}{6 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}-\frac {a \left (a^2 A b^3-28 a^3 b^2 B+9 a^5 B+34 a b^4 B-16 A b^5\right ) \tan (c+d x)}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac {B \tanh ^{-1}(\sin (c+d x))}{b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)
*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) + (a*(A*b - a*B)*Sec
[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a^2*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*Ta
n[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (a*(a^2*A*b^3 - 16*A*b^5 + 9*a^5*B - 28*a^3*b^2*B
+ 34*a*b^4*B)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 208

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

Rule 2659

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 3770

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

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3998

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x
_Symbol] :> Dist[B/b, Int[Csc[e + f*x], x], x] + Dist[(A*b - a*B)/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]
, x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]

Rule 4029

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

Rule 4080

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_
.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f
*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*Csc[e +
f*x])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Csc[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 4090

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(a*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e
+ f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*C
sc[e + f*x])^(m + 1)*Simp[b*(m + 1)*(-(a*(b*B - a*C)) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) +
C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx &=\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\int \frac {\sec ^2(c+d x) \left (2 a (A b-a B)-3 b (A b-a B) \sec (c+d x)+3 \left (a^2-b^2\right ) B \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {a^2 \left (5 A b^3+3 a^3 B-8 a b^2 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\sec (c+d x) \left (-2 a b \left (5 A b^3+3 a^3 B-8 a b^2 B\right )-\left (a^2 A b^3-6 A b^5+3 a^5 B-10 a^3 b^2 B+12 a b^4 B\right ) \sec (c+d x)+6 b \left (a^2-b^2\right )^2 B \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {a^2 \left (5 A b^3+3 a^3 B-8 a b^2 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {a \left (a^2 A b^3-16 A b^5+9 a^5 B-28 a^3 b^2 B+34 a b^4 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (3 b \left (3 a^2 A b^4+2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B\right )-6 b \left (a^2-b^2\right )^3 B \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3}\\ &=\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {a^2 \left (5 A b^3+3 a^3 B-8 a b^2 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {a \left (a^2 A b^3-16 A b^5+9 a^5 B-28 a^3 b^2 B+34 a b^4 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {B \int \sec (c+d x) \, dx}{b^4}-\frac {\left (3 a^2 A b^5+2 A b^7+2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B-8 a b^6 B\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^3}\\ &=\frac {B \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {a^2 \left (5 A b^3+3 a^3 B-8 a b^2 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {a \left (a^2 A b^3-16 A b^5+9 a^5 B-28 a^3 b^2 B+34 a b^4 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (3 a^2 A b^5+2 A b^7+2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B-8 a b^6 B\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^5 \left (a^2-b^2\right )^3}\\ &=\frac {B \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {a^2 \left (5 A b^3+3 a^3 B-8 a b^2 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {a \left (a^2 A b^3-16 A b^5+9 a^5 B-28 a^3 b^2 B+34 a b^4 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (3 a^2 A b^5+2 A b^7+2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B-8 a b^6 B\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right )^3 d}\\ &=\frac {B \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {\left (3 a^2 A b^5+2 A b^7+2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B-8 a b^6 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {a^2 \left (5 A b^3+3 a^3 B-8 a b^2 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {a \left (a^2 A b^3-16 A b^5+9 a^5 B-28 a^3 b^2 B+34 a b^4 B\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 1.99, size = 369, normalized size = 1.19 \[ \frac {\cos (c+d x) (A+B \sec (c+d x)) \left (\frac {24 \left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {2 a b \sin (c+d x) \left (-6 a^7 B-5 a^5 b^2 B+8 a^4 A b^3+38 a^3 b^4 B+a^2 A b^5+a^2 \left (-6 a^5 B+17 a^3 b^2 B+4 a^2 A b^3-26 a b^4 B+11 A b^5\right ) \cos (2 (c+d x))-6 a b \left (5 a^5 B-15 a^3 b^2 B-a^2 A b^3+20 a b^4 B-9 A b^5\right ) \cos (c+d x)-72 a b^6 B+36 A b^7\right )}{\left (b^2-a^2\right )^3 (a \cos (c+d x)+b)^3}-24 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{24 b^4 d (A \cos (c+d x)+B)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 48.76, size = 2278, normalized size = 7.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.46, size = 844, normalized size = 2.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*(2*B*a^7 - 7*B*a^5*b^2 + 8*B*a^3*b^4 + 3*A*a^2*b^5 - 8*B*a*b^6 + 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^6*b^4
- 3*a^4*b^6 + 3*a^2*b^8 - b^10)*sqrt(-a^2 + b^2)) - 3*B*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^4 + 3*B*log(abs(
tan(1/2*d*x + 1/2*c) - 1))/b^4 - (6*B*a^8*tan(1/2*d*x + 1/2*c)^5 - 15*B*a^7*b*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^6
*b^2*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 45*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 3*A*a
^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 60*B
*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 27*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 -
18*A*a*b^7*tan(1/2*d*x + 1/2*c)^5 - 12*B*a^8*tan(1/2*d*x + 1/2*c)^3 + 56*B*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 4*
A*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 - 116*B*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 32*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^3
+ 72*B*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*b^7*tan(1/2*d*x + 1/2*c)^3 + 6*B*a^8*tan(1/2*d*x + 1/2*c) + 15*
B*a^7*b*tan(1/2*d*x + 1/2*c) - 6*B*a^6*b^2*tan(1/2*d*x + 1/2*c) - 6*A*a^5*b^3*tan(1/2*d*x + 1/2*c) - 45*B*a^5*
b^3*tan(1/2*d*x + 1/2*c) - 3*A*a^4*b^4*tan(1/2*d*x + 1/2*c) - 6*B*a^4*b^4*tan(1/2*d*x + 1/2*c) - 6*A*a^3*b^5*t
an(1/2*d*x + 1/2*c) + 60*B*a^3*b^5*tan(1/2*d*x + 1/2*c) - 27*A*a^2*b^6*tan(1/2*d*x + 1/2*c) + 36*B*a^2*b^6*tan
(1/2*d*x + 1/2*c) - 18*A*a*b^7*tan(1/2*d*x + 1/2*c))/((a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*(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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maple [B]  time = 0.74, size = 2264, normalized size = 7.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1
/2*c)^3*A+12/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan
(1/2*d*x+1/2*c)*B-6/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a/(a-b)/(a^3+3*a^2*b+3*a*b^2+b
^3)*tan(1/2*d*x+1/2*c)^5*A+12/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a-b)/(a^3+3*a^2*b
+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-6/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a/(a+b)/(a^
3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-2/d*a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b
)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A-4/d*a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)
^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+4/d*a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b
-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-2/d*a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2
*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+44/3/d/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d
*x+1/2*c)^2*b-a-b)^3*a^4/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-4/d/b^3/(a*tan(1/2*d*x+1/2*c)^
2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^6/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-24/d*b/(a*tan(1/2*d
*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^2/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+1/d*a^5/b
^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-
1/d*a^5/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+
1/2*c)^5*B-6/d*a^4/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan
(1/2*d*x+1/2*c)*B+2/d*a^6/b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2
-b^3)*tan(1/2*d*x+1/2*c)*B+4/3/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a^2-2*a*b+b^2)/(a^
2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-2/d/b^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*
d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*a^7*B-3/d*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh(ta
n(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*a^2*A+7/d/b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*ar
ctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*a^5*B-3/d*a^2*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)
^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+3/d*a^2*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d
*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+2/d*a^6/b^3/(a*tan(1/2*d*x+1/2*c)^2-
tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-6/d*a^4/b/(a*tan(1/2*d*x+
1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+1/d*B/b^4*ln(tan
(1/2*d*x+1/2*c)+1)-1/d*B/b^4*ln(tan(1/2*d*x+1/2*c)-1)+8/d*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2
)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*a*B-8/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1
/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*a^3*B-2/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(
a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(A+B*sec(d*x+c))/(a+b*sec(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*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 14.14, size = 9713, normalized size = 31.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((tan(c/2 + (d*x)/2)*(2*B*a^6 + 3*A*a^2*b^4 - 2*A*a^3*b^3 + 12*B*a^2*b^4 - 4*B*a^3*b^3 - 6*B*a^4*b^2 - 6*A*a
*b^5 + B*a^5*b))/((a + b)*(3*a*b^5 - b^6 - 3*a^2*b^4 + a^3*b^3)) - (tan(c/2 + (d*x)/2)^5*(3*A*a^2*b^4 - 2*B*a^
6 + 2*A*a^3*b^3 - 12*B*a^2*b^4 - 4*B*a^3*b^3 + 6*B*a^4*b^2 + 6*A*a*b^5 + B*a^5*b))/((a*b^3 - b^4)*(a + b)^3) +
(4*tan(c/2 + (d*x)/2)^3*(A*a^3*b^3 - 3*B*a^6 - 18*B*a^2*b^4 + 11*B*a^4*b^2 + 9*A*a*b^5))/(3*(a + b)^2*(b^5 -
2*a*b^4 + a^2*b^3)))/(d*(tan(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3) - tan(c/2 + (d*x)/2)^4*(3*a*
b^2 + 3*a^2*b - 3*a^3 - 3*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 - tan(c/2 + (d*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3
- b^3))) - (B*atan(((B*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2*a^14 + 4*B^2*b^14 - 8*B^2*a*b^13 - 8*B^2*a^
13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44*B^2*a^2*b^12 + 48*B^2*a^3*b^11 - 92*B^2*a^4*b^10 - 120*B^2*a^5*b^
9 + 156*B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*a^9*b^5 + 117*B^2*a^10*b^4 + 48*B^2*a^11*b^3
- 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^3*b^11 + 20*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 12*A*B*a^9*b^5))/(a*b
^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a
^9*b^8 - a^10*b^7 - a^11*b^6) + (B*((8*(4*A*b^21 + 4*B*b^21 - 6*A*a^2*b^19 + 6*A*a^3*b^18 - 6*A*a^4*b^17 + 6*A
*a^5*b^16 + 14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9*b^12 - 12*B*a^2*b^19 + 64*B*a^3*b^18 + 20*B
*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 110*B*a^7*b^14 + 30*B*a^8*b^13 - 70*B*a^9*b^12 - 14*B*a^10*b^11 +
26*B*a^11*b^10 + 2*B*a^12*b^9 - 4*B*a^13*b^8 - 4*A*a*b^20 - 16*B*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3
*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9
) + (8*B*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 -
160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^1
4*b^8))/(b^4*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10
+ 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6))))/b^4)*1i)/b^4 + (B*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2
*a^14 + 4*B^2*b^14 - 8*B^2*a*b^13 - 8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44*B^2*a^2*b^12 + 48*B^2
*a^3*b^11 - 92*B^2*a^4*b^10 - 120*B^2*a^5*b^9 + 156*B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*
a^9*b^5 + 117*B^2*a^10*b^4 + 48*B^2*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^3*b^11 + 20*A*B*a^5*
b^9 - 34*A*B*a^7*b^7 + 12*A*B*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 -
10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) - (B*((8*(4*A*b^21 + 4*B*b^21 - 6*A*
a^2*b^19 + 6*A*a^3*b^18 - 6*A*a^4*b^17 + 6*A*a^5*b^16 + 14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9
*b^12 - 12*B*a^2*b^19 + 64*B*a^3*b^18 + 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 110*B*a^7*b^14 + 30*B
*a^8*b^13 - 70*B*a^9*b^12 - 14*B*a^10*b^11 + 26*B*a^11*b^10 + 2*B*a^12*b^9 - 4*B*a^13*b^8 - 4*A*a*b^20 - 16*B*
a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*
a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) - (8*B*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 +
48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a
^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8))/(b^4*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^1
3 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6))))/b^4)*1i)/b^4)/((
16*(4*B^3*a^13 - 4*A*B^2*b^13 + 4*A^2*B*b^13 + 16*B^3*a*b^12 - 2*B^3*a^12*b + 48*B^3*a^2*b^11 - 64*B^3*a^3*b^1
0 - 64*B^3*a^4*b^9 + 110*B^3*a^5*b^8 + 66*B^3*a^6*b^7 - 110*B^3*a^7*b^6 - 34*B^3*a^8*b^5 + 70*B^3*a^9*b^4 + 11
*B^3*a^10*b^3 - 26*B^3*a^11*b^2 - 28*A*B^2*a*b^12 + 6*A*B^2*a^2*b^11 - 22*A*B^2*a^3*b^10 + 6*A*B^2*a^4*b^9 + 1
4*A*B^2*a^5*b^8 - 14*A*B^2*a^6*b^7 - 20*A*B^2*a^7*b^6 + 6*A*B^2*a^8*b^5 + 6*A*B^2*a^9*b^4 + 12*A^2*B*a^2*b^11
+ 9*A^2*B*a^4*b^9))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^
7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) + (B*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2*a^14
+ 4*B^2*b^14 - 8*B^2*a*b^13 - 8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44*B^2*a^2*b^12 + 48*B^2*a^3*b
^11 - 92*B^2*a^4*b^10 - 120*B^2*a^5*b^9 + 156*B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*a^9*b^
5 + 117*B^2*a^10*b^4 + 48*B^2*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^3*b^11 + 20*A*B*a^5*b^9 -
34*A*B*a^7*b^7 + 12*A*B*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^
6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) + (B*((8*(4*A*b^21 + 4*B*b^21 - 6*A*a^2*b^
19 + 6*A*a^3*b^18 - 6*A*a^4*b^17 + 6*A*a^5*b^16 + 14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9*b^12
- 12*B*a^2*b^19 + 64*B*a^3*b^18 + 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 110*B*a^7*b^14 + 30*B*a^8*b
^13 - 70*B*a^9*b^12 - 14*B*a^10*b^11 + 26*B*a^11*b^10 + 2*B*a^12*b^9 - 4*B*a^13*b^8 - 4*A*a*b^20 - 16*B*a*b^20
))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^
12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) + (8*B*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^
4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^
11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8))/(b^4*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10
*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6))))/b^4))/b^4 - (B*((8*tan
(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2*a^14 + 4*B^2*b^14 - 8*B^2*a*b^13 - 8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*
a^4*b^10 + 44*B^2*a^2*b^12 + 48*B^2*a^3*b^11 - 92*B^2*a^4*b^10 - 120*B^2*a^5*b^9 + 156*B^2*a^6*b^8 + 160*B^2*a
^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*a^9*b^5 + 117*B^2*a^10*b^4 + 48*B^2*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b
^13 - 16*A*B*a^3*b^11 + 20*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 12*A*B*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3
*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) -
(B*((8*(4*A*b^21 + 4*B*b^21 - 6*A*a^2*b^19 + 6*A*a^3*b^18 - 6*A*a^4*b^17 + 6*A*a^5*b^16 + 14*A*a^6*b^15 - 14*
A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9*b^12 - 12*B*a^2*b^19 + 64*B*a^3*b^18 + 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30
*B*a^6*b^15 + 110*B*a^7*b^14 + 30*B*a^8*b^13 - 70*B*a^9*b^12 - 14*B*a^10*b^11 + 26*B*a^11*b^10 + 2*B*a^12*b^9
- 4*B*a^13*b^8 - 4*A*a*b^20 - 16*B*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^
15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) - (8*B*tan(c/2 + (d*x)/2)*(8*
a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 +
120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8))/(b^4*(a*b^16 + b^17 - 5
*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*
b^7 - a^11*b^6))))/b^4))/b^4))*2i)/(b^4*d) - (atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2*a^14 + 4*B^2*b
^14 - 8*B^2*a*b^13 - 8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44*B^2*a^2*b^12 + 48*B^2*a^3*b^11 - 92*
B^2*a^4*b^10 - 120*B^2*a^5*b^9 + 156*B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*a^9*b^5 + 117*B
^2*a^10*b^4 + 48*B^2*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^3*b^11 + 20*A*B*a^5*b^9 - 34*A*B*a^
7*b^7 + 12*A*B*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 -
10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) - (((8*(4*A*b^21 + 4*B*b^21 - 6*A*a^2*b^19 + 6*A*a^
3*b^18 - 6*A*a^4*b^17 + 6*A*a^5*b^16 + 14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9*b^12 - 12*B*a^2*
b^19 + 64*B*a^3*b^18 + 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 110*B*a^7*b^14 + 30*B*a^8*b^13 - 70*B*
a^9*b^12 - 14*B*a^10*b^11 + 26*B*a^11*b^10 + 2*B*a^12*b^9 - 4*B*a^13*b^8 - 4*A*a*b^20 - 16*B*a*b^20))/(a*b^19
+ b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*
b^11 - a^10*b^10 - a^11*b^9) - (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*
b^5 + 8*B*a^3*b^4 - 7*B*a^5*b^2 - 8*B*a*b^6)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17
- 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8
*a^13*b^9 - 8*a^14*b^8))/((b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*
b^6 - a^14*b^4)*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^
10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b
^5 + 8*B*a^3*b^4 - 7*B*a^5*b^2 - 8*B*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 -
21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b^5 + 8*B*a^3
*b^4 - 7*B*a^5*b^2 - 8*B*a*b^6)*1i)/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*
b^8 + 7*a^12*b^6 - a^14*b^4)) + (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2*a^14 + 4*B^2*b^14 - 8*B^2*a*b^13 -
8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44*B^2*a^2*b^12 + 48*B^2*a^3*b^11 - 92*B^2*a^4*b^10 - 120*B
^2*a^5*b^9 + 156*B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*a^9*b^5 + 117*B^2*a^10*b^4 + 48*B^2
*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^3*b^11 + 20*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 12*A*B*a^9*b
^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*
b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) + (((8*(4*A*b^21 + 4*B*b^21 - 6*A*a^2*b^19 + 6*A*a^3*b^18 - 6*A*a^4*b^1
7 + 6*A*a^5*b^16 + 14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9*b^12 - 12*B*a^2*b^19 + 64*B*a^3*b^18
+ 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 110*B*a^7*b^14 + 30*B*a^8*b^13 - 70*B*a^9*b^12 - 14*B*a^10
*b^11 + 26*B*a^11*b^10 + 2*B*a^12*b^9 - 4*B*a^13*b^8 - 4*A*a*b^20 - 16*B*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18
- 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a
^11*b^9) + (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b^5 + 8*B*a^3*b^4 -
7*B*a^5*b^2 - 8*B*a*b^6)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 16
0*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b
^8))/((b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)*(a*b
^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a
^9*b^8 - a^10*b^7 - a^11*b^6)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b^5 + 8*B*a^3*b^4 - 7
*B*a^5*b^2 - 8*B*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^1
2*b^6 - a^14*b^4)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b^5 + 8*B*a^3*b^4 - 7*B*a^5*b^2 -
8*B*a*b^6)*1i)/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a
^14*b^4)))/((16*(4*B^3*a^13 - 4*A*B^2*b^13 + 4*A^2*B*b^13 + 16*B^3*a*b^12 - 2*B^3*a^12*b + 48*B^3*a^2*b^11 - 6
4*B^3*a^3*b^10 - 64*B^3*a^4*b^9 + 110*B^3*a^5*b^8 + 66*B^3*a^6*b^7 - 110*B^3*a^7*b^6 - 34*B^3*a^8*b^5 + 70*B^3
*a^9*b^4 + 11*B^3*a^10*b^3 - 26*B^3*a^11*b^2 - 28*A*B^2*a*b^12 + 6*A*B^2*a^2*b^11 - 22*A*B^2*a^3*b^10 + 6*A*B^
2*a^4*b^9 + 14*A*B^2*a^5*b^8 - 14*A*B^2*a^6*b^7 - 20*A*B^2*a^7*b^6 + 6*A*B^2*a^8*b^5 + 6*A*B^2*a^9*b^4 + 12*A^
2*B*a^2*b^11 + 9*A^2*B*a^4*b^9))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6
*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) - (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 +
8*B^2*a^14 + 4*B^2*b^14 - 8*B^2*a*b^13 - 8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44*B^2*a^2*b^12 + 4
8*B^2*a^3*b^11 - 92*B^2*a^4*b^10 - 120*B^2*a^5*b^9 + 156*B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120
*B^2*a^9*b^5 + 117*B^2*a^10*b^4 + 48*B^2*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^3*b^11 + 20*A*B
*a^5*b^9 - 34*A*B*a^7*b^7 + 12*A*B*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b
^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) - (((8*(4*A*b^21 + 4*B*b^21 - 6
*A*a^2*b^19 + 6*A*a^3*b^18 - 6*A*a^4*b^17 + 6*A*a^5*b^16 + 14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*
a^9*b^12 - 12*B*a^2*b^19 + 64*B*a^3*b^18 + 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 110*B*a^7*b^14 + 3
0*B*a^8*b^13 - 70*B*a^9*b^12 - 14*B*a^10*b^11 + 26*B*a^11*b^10 + 2*B*a^12*b^9 - 4*B*a^13*b^8 - 4*A*a*b^20 - 16
*B*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 +
5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) - (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7
+ 2*B*a^7 + 3*A*a^2*b^5 + 8*B*a^3*b^4 - 7*B*a^5*b^2 - 8*B*a*b^6)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4
*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^1
1 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^14*b^8))/((b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 2
1*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*
a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 +
2*B*a^7 + 3*A*a^2*b^5 + 8*B*a^3*b^4 - 7*B*a^5*b^2 - 8*B*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*
b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3
*A*a^2*b^5 + 8*B*a^3*b^4 - 7*B*a^5*b^2 - 8*B*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^
8*b^10 - 21*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)) + (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2*a^14 + 4*B^2*b^14
- 8*B^2*a*b^13 - 8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44*B^2*a^2*b^12 + 48*B^2*a^3*b^11 - 92*B^2
*a^4*b^10 - 120*B^2*a^5*b^9 + 156*B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*a^9*b^5 + 117*B^2*
a^10*b^4 + 48*B^2*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^3*b^11 + 20*A*B*a^5*b^9 - 34*A*B*a^7*b
^7 + 12*A*B*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*
a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) + (((8*(4*A*b^21 + 4*B*b^21 - 6*A*a^2*b^19 + 6*A*a^3*b
^18 - 6*A*a^4*b^17 + 6*A*a^5*b^16 + 14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9*b^12 - 12*B*a^2*b^1
9 + 64*B*a^3*b^18 + 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 110*B*a^7*b^14 + 30*B*a^8*b^13 - 70*B*a^9
*b^12 - 14*B*a^10*b^11 + 26*B*a^11*b^10 + 2*B*a^12*b^9 - 4*B*a^13*b^8 - 4*A*a*b^20 - 16*B*a*b^20))/(a*b^19 + b
^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^1
1 - a^10*b^10 - a^11*b^9) + (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b^5
+ 8*B*a^3*b^4 - 7*B*a^5*b^2 - 8*B*a*b^6)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^4*b^18 + 120*a^5*b^17 -
120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^
13*b^9 - 8*a^14*b^8))/((b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b^6
- a^14*b^4)*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10
+ 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b^5
+ 8*B*a^3*b^4 - 7*B*a^5*b^2 - 8*B*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21
*a^10*b^8 + 7*a^12*b^6 - a^14*b^4)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b^5 + 8*B*a^3*b^
4 - 7*B*a^5*b^2 - 8*B*a*b^6))/(2*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 +
7*a^12*b^6 - a^14*b^4))))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 + 3*A*a^2*b^5 + 8*B*a^3*b^4 - 7*B*a^5
*b^2 - 8*B*a*b^6)*1i)/(d*(b^18 - 7*a^2*b^16 + 21*a^4*b^14 - 35*a^6*b^12 + 35*a^8*b^10 - 21*a^10*b^8 + 7*a^12*b
^6 - a^14*b^4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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