3.329 \(\int \frac {\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=289 \[ \frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\left (-3 a^2 B+a A b+2 b^2 B\right ) \tan (c+d x)}{2 b^3 d \left (a^2-b^2\right )}-\frac {a^2 \left (-3 a^3 B+a^2 A b+6 a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {a \left (-6 a^5 B+2 a^4 A b+15 a^3 b^2 B-5 a^2 A b^3-12 a b^4 B+6 A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {(A b-3 a B) \tanh ^{-1}(\sin (c+d x))}{b^4 d} \]

[Out]

(A*b-3*B*a)*arctanh(sin(d*x+c))/b^4/d-a*(2*A*a^4*b-5*A*a^2*b^3+6*A*b^5-6*B*a^5+15*B*a^3*b^2-12*B*a*b^4)*arctan
h((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^4/(a+b)^(5/2)/d-1/2*(A*a*b-3*B*a^2+2*B*b^2)*tan(d*
x+c)/b^3/(a^2-b^2)/d+1/2*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))^2-1/2*a^2*(A*a^2*b
-4*A*b^3-3*B*a^3+6*B*a*b^2)*tan(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*sec(d*x+c))

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Rubi [A]  time = 1.42, antiderivative size = 289, 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, 4082, 3998, 3770, 3831, 2659, 208} \[ -\frac {\left (-3 a^2 B+a A b+2 b^2 B\right ) \tan (c+d x)}{2 b^3 d \left (a^2-b^2\right )}-\frac {a \left (-5 a^2 A b^3+2 a^4 A b+15 a^3 b^2 B-6 a^5 B-12 a b^4 B+6 A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {a^2 \left (a^2 A b-3 a^3 B+6 a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {(A b-3 a B) \tanh ^{-1}(\sin (c+d x))}{b^4 d} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]

[Out]

((A*b - 3*a*B)*ArcTanh[Sin[c + d*x]])/(b^4*d) - (a*(2*a^4*A*b - 5*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 15*a^3*b^2*B
- 12*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*
A*b - 3*a^2*B + 2*b^2*B)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*
b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Tan[c + d*x])/(2*b^3*
(a^2 - b^2)^2*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 4082

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[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m
+ 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) + (b*B*
(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

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))^3} \, dx &=\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\int \frac {\sec ^2(c+d x) \left (2 a (A b-a B)-2 b (A b-a B) \sec (c+d x)-\left (a A b-3 a^2 B+2 b^2 B\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (a b \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right )+\left (a^2-b^2\right ) \left (a^2 A b-2 A b^3-3 a^3 B+4 a b^2 B\right ) \sec (c+d x)-b \left (a^2-b^2\right ) \left (a A b-3 a^2 B+2 b^2 B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (a b^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right )+2 b \left (a^2-b^2\right )^2 (A b-3 a B) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {(A b-3 a B) \int \sec (c+d x) \, dx}{b^4}-\frac {\left (a \left (2 a^4 A b-5 a^2 A b^3+6 A b^5-6 a^5 B+15 a^3 b^2 B-12 a b^4 B\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {(A b-3 a B) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (a \left (2 a^4 A b-5 a^2 A b^3+6 A b^5-6 a^5 B+15 a^3 b^2 B-12 a b^4 B\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=\frac {(A b-3 a B) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (a \left (2 a^4 A b-5 a^2 A b^3+6 A b^5-6 a^5 B+15 a^3 b^2 B-12 a b^4 B\right )\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 )^2 d}\\ &=\frac {(A b-3 a B) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {a \left (2 a^4 A b-5 a^2 A b^3+6 A b^5-6 a^5 B+15 a^3 b^2 B-12 a b^4 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} d}-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 6.52, size = 418, normalized size = 1.45 \[ \frac {a^2 A b \sin (c+d x)-a^3 B \sin (c+d x)}{2 b^2 d (b-a) (a+b) (a \cos (c+d x)+b)^2}+\frac {4 a^5 B \sin (c+d x)-2 a^4 A b \sin (c+d x)-7 a^3 b^2 B \sin (c+d x)+5 a^2 A b^3 \sin (c+d x)}{2 b^3 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac {a \left (-6 a^5 B+2 a^4 A b+15 a^3 b^2 B-5 a^2 A b^3-12 a b^4 B+6 A b^5\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 d \sqrt {a^2-b^2} \left (b^2-a^2\right )^2}+\frac {(3 a B-A b) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{b^4 d}+\frac {(A b-3 a B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{b^4 d}+\frac {B \sin \left (\frac {1}{2} (c+d x)\right )}{b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {B \sin \left (\frac {1}{2} (c+d x)\right )}{b^3 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]

[Out]

(a*(2*a^4*A*b - 5*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 15*a^3*b^2*B - 12*a*b^4*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2
])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*(-a^2 + b^2)^2*d) + ((-(A*b) + 3*a*B)*Log[Cos[(c + d*x)/2] - Sin[(c
+ d*x)/2]])/(b^4*d) + ((A*b - 3*a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(b^4*d) + (B*Sin[(c + d*x)/2])/
(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (B*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/
2])) + (a^2*A*b*Sin[c + d*x] - a^3*B*Sin[c + d*x])/(2*b^2*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x])^2) + (-2*a^4
*A*b*Sin[c + d*x] + 5*a^2*A*b^3*Sin[c + d*x] + 4*a^5*B*Sin[c + d*x] - 7*a^3*b^2*B*Sin[c + d*x])/(2*b^3*(-a + b
)^2*(a + b)^2*d*(b + a*Cos[c + d*x]))

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fricas [B]  time = 48.00, size = 2111, normalized size = 7.30 \[ \text {result too large to display} \]

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))^3,x, algorithm="fricas")

[Out]

[-1/4*(((6*B*a^8 - 2*A*a^7*b - 15*B*a^6*b^2 + 5*A*a^5*b^3 + 12*B*a^4*b^4 - 6*A*a^3*b^5)*cos(d*x + c)^3 + 2*(6*
B*a^7*b - 2*A*a^6*b^2 - 15*B*a^5*b^3 + 5*A*a^4*b^4 + 12*B*a^3*b^5 - 6*A*a^2*b^6)*cos(d*x + c)^2 + (6*B*a^6*b^2
- 2*A*a^5*b^3 - 15*B*a^4*b^4 + 5*A*a^3*b^5 + 12*B*a^2*b^6 - 6*A*a*b^7)*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)) + 2*((3*B*a^9 - A*a^8*b - 9*B*a^7*b^2 + 3*A*a^6*b^3 + 9
*B*a^5*b^4 - 3*A*a^4*b^5 - 3*B*a^3*b^6 + A*a^2*b^7)*cos(d*x + c)^3 + 2*(3*B*a^8*b - A*a^7*b^2 - 9*B*a^6*b^3 +
3*A*a^5*b^4 + 9*B*a^4*b^5 - 3*A*a^3*b^6 - 3*B*a^2*b^7 + A*a*b^8)*cos(d*x + c)^2 + (3*B*a^7*b^2 - A*a^6*b^3 - 9
*B*a^5*b^4 + 3*A*a^4*b^5 + 9*B*a^3*b^6 - 3*A*a^2*b^7 - 3*B*a*b^8 + A*b^9)*cos(d*x + c))*log(sin(d*x + c) + 1)
- 2*((3*B*a^9 - A*a^8*b - 9*B*a^7*b^2 + 3*A*a^6*b^3 + 9*B*a^5*b^4 - 3*A*a^4*b^5 - 3*B*a^3*b^6 + A*a^2*b^7)*cos
(d*x + c)^3 + 2*(3*B*a^8*b - A*a^7*b^2 - 9*B*a^6*b^3 + 3*A*a^5*b^4 + 9*B*a^4*b^5 - 3*A*a^3*b^6 - 3*B*a^2*b^7 +
A*a*b^8)*cos(d*x + c)^2 + (3*B*a^7*b^2 - A*a^6*b^3 - 9*B*a^5*b^4 + 3*A*a^4*b^5 + 9*B*a^3*b^6 - 3*A*a^2*b^7 -
3*B*a*b^8 + A*b^9)*cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(2*B*a^6*b^3 - 6*B*a^4*b^5 + 6*B*a^2*b^7 - 2*B*b^9
+ (6*B*a^8*b - 2*A*a^7*b^2 - 17*B*a^6*b^3 + 7*A*a^5*b^4 + 13*B*a^4*b^5 - 5*A*a^3*b^6 - 2*B*a^2*b^7)*cos(d*x +
c)^2 + (9*B*a^7*b^2 - 3*A*a^6*b^3 - 25*B*a^5*b^4 + 9*A*a^4*b^5 + 20*B*a^3*b^6 - 6*A*a^2*b^7 - 4*B*a*b^8)*cos(
d*x + c))*sin(d*x + c))/((a^8*b^4 - 3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10)*d*cos(d*x + c)^3 + 2*(a^7*b^5 - 3*a^5*b^
7 + 3*a^3*b^9 - a*b^11)*d*cos(d*x + c)^2 + (a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*d*cos(d*x + c)), 1/2*(((6
*B*a^8 - 2*A*a^7*b - 15*B*a^6*b^2 + 5*A*a^5*b^3 + 12*B*a^4*b^4 - 6*A*a^3*b^5)*cos(d*x + c)^3 + 2*(6*B*a^7*b -
2*A*a^6*b^2 - 15*B*a^5*b^3 + 5*A*a^4*b^4 + 12*B*a^3*b^5 - 6*A*a^2*b^6)*cos(d*x + c)^2 + (6*B*a^6*b^2 - 2*A*a^5
*b^3 - 15*B*a^4*b^4 + 5*A*a^3*b^5 + 12*B*a^2*b^6 - 6*A*a*b^7)*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^9 - A*a^8*b - 9*B*a^7*b^2 + 3*A*a^6*b^3 + 9
*B*a^5*b^4 - 3*A*a^4*b^5 - 3*B*a^3*b^6 + A*a^2*b^7)*cos(d*x + c)^3 + 2*(3*B*a^8*b - A*a^7*b^2 - 9*B*a^6*b^3 +
3*A*a^5*b^4 + 9*B*a^4*b^5 - 3*A*a^3*b^6 - 3*B*a^2*b^7 + A*a*b^8)*cos(d*x + c)^2 + (3*B*a^7*b^2 - A*a^6*b^3 - 9
*B*a^5*b^4 + 3*A*a^4*b^5 + 9*B*a^3*b^6 - 3*A*a^2*b^7 - 3*B*a*b^8 + A*b^9)*cos(d*x + c))*log(sin(d*x + c) + 1)
+ ((3*B*a^9 - A*a^8*b - 9*B*a^7*b^2 + 3*A*a^6*b^3 + 9*B*a^5*b^4 - 3*A*a^4*b^5 - 3*B*a^3*b^6 + A*a^2*b^7)*cos(d
*x + c)^3 + 2*(3*B*a^8*b - A*a^7*b^2 - 9*B*a^6*b^3 + 3*A*a^5*b^4 + 9*B*a^4*b^5 - 3*A*a^3*b^6 - 3*B*a^2*b^7 + A
*a*b^8)*cos(d*x + c)^2 + (3*B*a^7*b^2 - A*a^6*b^3 - 9*B*a^5*b^4 + 3*A*a^4*b^5 + 9*B*a^3*b^6 - 3*A*a^2*b^7 - 3*
B*a*b^8 + A*b^9)*cos(d*x + c))*log(-sin(d*x + c) + 1) + (2*B*a^6*b^3 - 6*B*a^4*b^5 + 6*B*a^2*b^7 - 2*B*b^9 + (
6*B*a^8*b - 2*A*a^7*b^2 - 17*B*a^6*b^3 + 7*A*a^5*b^4 + 13*B*a^4*b^5 - 5*A*a^3*b^6 - 2*B*a^2*b^7)*cos(d*x + c)^
2 + (9*B*a^7*b^2 - 3*A*a^6*b^3 - 25*B*a^5*b^4 + 9*A*a^4*b^5 + 20*B*a^3*b^6 - 6*A*a^2*b^7 - 4*B*a*b^8)*cos(d*x
+ c))*sin(d*x + c))/((a^8*b^4 - 3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10)*d*cos(d*x + c)^3 + 2*(a^7*b^5 - 3*a^5*b^7 +
3*a^3*b^9 - a*b^11)*d*cos(d*x + c)^2 + (a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*d*cos(d*x + c))]

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giac [B]  time = 0.46, size = 581, normalized size = 2.01 \[ \frac {\frac {{\left (6 \, B a^{6} - 2 \, A a^{5} b - 15 \, B a^{4} b^{2} + 5 \, A a^{3} b^{3} + 12 \, B a^{2} b^{4} - 6 \, A a b^{5}\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^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {4 \, B a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\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 )}^{2}} - \frac {{\left (3 \, B a - A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} + \frac {{\left (3 \, B a - A b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac {2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} b^{3}}}{d} \]

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))^3,x, algorithm="giac")

[Out]

((6*B*a^6 - 2*A*a^5*b - 15*B*a^4*b^2 + 5*A*a^3*b^3 + 12*B*a^2*b^4 - 6*A*a*b^5)*(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^4*b^4 -
2*a^2*b^6 + b^8)*sqrt(-a^2 + b^2)) - (4*B*a^6*tan(1/2*d*x + 1/2*c)^3 - 2*A*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 5*B*
a^5*b*tan(1/2*d*x + 1/2*c)^3 + 3*A*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 - 7*B*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 + 5*A*a
^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 8*B*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 6*A*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 - 4*B*
a^6*tan(1/2*d*x + 1/2*c) + 2*A*a^5*b*tan(1/2*d*x + 1/2*c) - 5*B*a^5*b*tan(1/2*d*x + 1/2*c) + 3*A*a^4*b^2*tan(1
/2*d*x + 1/2*c) + 7*B*a^4*b^2*tan(1/2*d*x + 1/2*c) - 5*A*a^3*b^3*tan(1/2*d*x + 1/2*c) + 8*B*a^3*b^3*tan(1/2*d*
x + 1/2*c) - 6*A*a^2*b^4*tan(1/2*d*x + 1/2*c))/((a^4*b^3 - 2*a^2*b^5 + b^7)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(
1/2*d*x + 1/2*c)^2 - a - b)^2) - (3*B*a - A*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^4 + (3*B*a - A*b)*log(abs(
tan(1/2*d*x + 1/2*c) - 1))/b^4 - 2*B*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 - 1)*b^3))/d

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maple [B]  time = 0.69, size = 1406, normalized size = 4.87 \[ \text {result too large to display} \]

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))^3,x)

[Out]

2/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A
-1/d*a^3/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-
6/d*a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-4/d
*a^5/b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+1/
d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+8
/d*a^3/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-2/
d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-1/d*a^3/b/(
a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A+6/d*a^2/(a*tan(1/2*d*x
+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A+4/d*a^5/b^3/(a*tan(1/2*d*x+1/2*c)^2
-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+1/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*
d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-8/d*a^3/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^
2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-2/d*a^5/b^3/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(
1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+5/d*a^3/b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*
d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-6/d*a*b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/
2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+6/d*a^6/b^4/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*
c)*(a-b)/((a-b)*(a+b))^(1/2))*B-15/d*a^4/b^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c
)*(a-b)/((a-b)*(a+b))^(1/2))*B+12/d*a^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-
b)/((a-b)*(a+b))^(1/2))*B-1/d/b^3/(tan(1/2*d*x+1/2*c)-1)*B-1/d/b^3*ln(tan(1/2*d*x+1/2*c)-1)*A+3/d/b^4*ln(tan(1
/2*d*x+1/2*c)-1)*a*B-1/d/b^3/(tan(1/2*d*x+1/2*c)+1)*B+1/d/b^3*ln(tan(1/2*d*x+1/2*c)+1)*A-3/d/b^4*ln(tan(1/2*d*
x+1/2*c)+1)*a*B

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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))^3,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 = 14.54, size = 9286, normalized size = 32.13 \[ \text {result too large to display} \]

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))^3),x)

[Out]

((tan(c/2 + (d*x)/2)^5*(6*B*a^5 - 2*B*b^5 + 6*A*a^2*b^3 + A*a^3*b^2 + 4*B*a^2*b^3 - 12*B*a^3*b^2 - 2*A*a^4*b +
2*B*a*b^4 - 3*B*a^4*b))/((a*b^3 - b^4)*(a + b)^2) + (tan(c/2 + (d*x)/2)*(6*B*a^5 + 2*B*b^5 + 6*A*a^2*b^3 - A*
a^3*b^2 - 4*B*a^2*b^3 - 12*B*a^3*b^2 - 2*A*a^4*b + 2*B*a*b^4 + 3*B*a^4*b))/((a + b)*(b^5 - 2*a*b^4 + a^2*b^3))
- (2*tan(c/2 + (d*x)/2)^3*(6*B*a^6 - 2*B*b^6 + 5*A*a^3*b^3 + 6*B*a^2*b^4 - 13*B*a^4*b^2 - 2*A*a^5*b))/(b*(a*b
^2 - b^3)*(a + b)^2*(a - b)))/(d*(2*a*b - tan(c/2 + (d*x)/2)^2*(2*a*b + 3*a^2 - b^2) - tan(c/2 + (d*x)/2)^6*(a
^2 - 2*a*b + b^2) + a^2 + b^2 - tan(c/2 + (d*x)/2)^4*(2*a*b - 3*a^2 + b^2))) + (atan(((((8*tan(c/2 + (d*x)/2)*
(4*A^2*b^12 + 72*B^2*a^12 - 8*A^2*a*b^11 - 72*B^2*a^11*b + 24*A^2*a^2*b^10 + 32*A^2*a^3*b^9 - 52*A^2*a^4*b^8 -
48*A^2*a^5*b^7 + 57*A^2*a^6*b^6 + 32*A^2*a^7*b^5 - 32*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 8*A^2*a^10*b^2 + 36*B^2*a
^2*b^10 - 72*B^2*a^3*b^9 + 36*B^2*a^4*b^8 + 288*B^2*a^5*b^7 - 288*B^2*a^6*b^6 - 432*B^2*a^7*b^5 + 441*B^2*a^8*
b^4 + 288*B^2*a^9*b^3 - 288*B^2*a^10*b^2 - 24*A*B*a*b^11 - 48*A*B*a^11*b + 48*A*B*a^2*b^10 - 72*A*B*a^3*b^9 -
192*A*B*a^4*b^8 + 252*A*B*a^5*b^7 + 288*A*B*a^6*b^6 - 318*A*B*a^7*b^5 - 192*A*B*a^8*b^4 + 192*A*B*a^9*b^3 + 48
*A*B*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (((8*(
4*A*b^18 - 8*A*a^2*b^16 + 34*A*a^3*b^15 + 6*A*a^4*b^14 - 36*A*a^5*b^13 - 4*A*a^6*b^12 + 18*A*a^7*b^11 + 2*A*a^
8*b^10 - 4*A*a^9*b^9 + 24*B*a^2*b^16 + 36*B*a^3*b^15 - 78*B*a^4*b^14 - 42*B*a^5*b^13 + 96*B*a^6*b^12 + 24*B*a^
7*b^11 - 54*B*a^8*b^10 - 6*B*a^9*b^9 + 12*B*a^10*b^8 - 12*A*a*b^17 - 12*B*a*b^17))/(a*b^15 + b^16 - 3*a^2*b^14
- 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (8*tan(c/2 + (d*x)/2)*(A*b - 3*B*a)*(8*a*b^17
- 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 -
8*a^10*b^8))/(b^4*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*(A*
b - 3*B*a))/b^4)*(A*b - 3*B*a)*1i)/b^4 + (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 72*B^2*a^12 - 8*A^2*a*b^11 - 72
*B^2*a^11*b + 24*A^2*a^2*b^10 + 32*A^2*a^3*b^9 - 52*A^2*a^4*b^8 - 48*A^2*a^5*b^7 + 57*A^2*a^6*b^6 + 32*A^2*a^7
*b^5 - 32*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 8*A^2*a^10*b^2 + 36*B^2*a^2*b^10 - 72*B^2*a^3*b^9 + 36*B^2*a^4*b^8 + 2
88*B^2*a^5*b^7 - 288*B^2*a^6*b^6 - 432*B^2*a^7*b^5 + 441*B^2*a^8*b^4 + 288*B^2*a^9*b^3 - 288*B^2*a^10*b^2 - 24
*A*B*a*b^11 - 48*A*B*a^11*b + 48*A*B*a^2*b^10 - 72*A*B*a^3*b^9 - 192*A*B*a^4*b^8 + 252*A*B*a^5*b^7 + 288*A*B*a
^6*b^6 - 318*A*B*a^7*b^5 - 192*A*B*a^8*b^4 + 192*A*B*a^9*b^3 + 48*A*B*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 -
3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (((8*(4*A*b^18 - 8*A*a^2*b^16 + 34*A*a^3*b^15 + 6*A
*a^4*b^14 - 36*A*a^5*b^13 - 4*A*a^6*b^12 + 18*A*a^7*b^11 + 2*A*a^8*b^10 - 4*A*a^9*b^9 + 24*B*a^2*b^16 + 36*B*a
^3*b^15 - 78*B*a^4*b^14 - 42*B*a^5*b^13 + 96*B*a^6*b^12 + 24*B*a^7*b^11 - 54*B*a^8*b^10 - 6*B*a^9*b^9 + 12*B*a
^10*b^8 - 12*A*a*b^17 - 12*B*a*b^17))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6
*b^10 - a^7*b^9) - (8*tan(c/2 + (d*x)/2)*(A*b - 3*B*a)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48
*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/(b^4*(a*b^12 + b^13 - 3*a^2*b^1
1 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*(A*b - 3*B*a))/b^4)*(A*b - 3*B*a)*1i)/b^4)/((16*
(108*B^3*a^12 - 12*A^3*a*b^11 - 54*B^3*a^11*b - 24*A^3*a^2*b^10 + 34*A^3*a^3*b^9 + 26*A^3*a^4*b^8 - 36*A^3*a^5
*b^7 - 13*A^3*a^6*b^6 + 18*A^3*a^7*b^5 + 2*A^3*a^8*b^4 - 4*A^3*a^9*b^3 + 216*B^3*a^4*b^8 + 216*B^3*a^5*b^7 - 7
02*B^3*a^6*b^6 - 378*B^3*a^7*b^5 + 864*B^3*a^8*b^4 + 243*B^3*a^9*b^3 - 486*B^3*a^10*b^2 - 108*A*B^2*a^11*b - 2
52*A*B^2*a^3*b^9 - 324*A*B^2*a^4*b^8 + 774*A*B^2*a^5*b^7 + 486*A*B^2*a^6*b^6 - 900*A*B^2*a^7*b^5 - 279*A*B^2*a
^8*b^4 + 486*A*B^2*a^9*b^3 + 54*A*B^2*a^10*b^2 + 96*A^2*B*a^2*b^10 + 156*A^2*B*a^3*b^9 - 282*A^2*B*a^4*b^8 - 1
98*A^2*B*a^5*b^7 + 312*A^2*B*a^6*b^6 + 105*A^2*B*a^7*b^5 - 162*A^2*B*a^8*b^4 - 18*A^2*B*a^9*b^3 + 36*A^2*B*a^1
0*b^2))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (((8*tan(c/
2 + (d*x)/2)*(4*A^2*b^12 + 72*B^2*a^12 - 8*A^2*a*b^11 - 72*B^2*a^11*b + 24*A^2*a^2*b^10 + 32*A^2*a^3*b^9 - 52*
A^2*a^4*b^8 - 48*A^2*a^5*b^7 + 57*A^2*a^6*b^6 + 32*A^2*a^7*b^5 - 32*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 8*A^2*a^10*b
^2 + 36*B^2*a^2*b^10 - 72*B^2*a^3*b^9 + 36*B^2*a^4*b^8 + 288*B^2*a^5*b^7 - 288*B^2*a^6*b^6 - 432*B^2*a^7*b^5 +
441*B^2*a^8*b^4 + 288*B^2*a^9*b^3 - 288*B^2*a^10*b^2 - 24*A*B*a*b^11 - 48*A*B*a^11*b + 48*A*B*a^2*b^10 - 72*A
*B*a^3*b^9 - 192*A*B*a^4*b^8 + 252*A*B*a^5*b^7 + 288*A*B*a^6*b^6 - 318*A*B*a^7*b^5 - 192*A*B*a^8*b^4 + 192*A*B
*a^9*b^3 + 48*A*B*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*
b^6) + (((8*(4*A*b^18 - 8*A*a^2*b^16 + 34*A*a^3*b^15 + 6*A*a^4*b^14 - 36*A*a^5*b^13 - 4*A*a^6*b^12 + 18*A*a^7*
b^11 + 2*A*a^8*b^10 - 4*A*a^9*b^9 + 24*B*a^2*b^16 + 36*B*a^3*b^15 - 78*B*a^4*b^14 - 42*B*a^5*b^13 + 96*B*a^6*b
^12 + 24*B*a^7*b^11 - 54*B*a^8*b^10 - 6*B*a^9*b^9 + 12*B*a^10*b^8 - 12*A*a*b^17 - 12*B*a*b^17))/(a*b^15 + b^16
- 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (8*tan(c/2 + (d*x)/2)*(A*b - 3*B*
a)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10
+ 8*a^9*b^9 - 8*a^10*b^8))/(b^4*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a
^7*b^6)))*(A*b - 3*B*a))/b^4)*(A*b - 3*B*a))/b^4 + (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 72*B^2*a^12 - 8*A^2*a
*b^11 - 72*B^2*a^11*b + 24*A^2*a^2*b^10 + 32*A^2*a^3*b^9 - 52*A^2*a^4*b^8 - 48*A^2*a^5*b^7 + 57*A^2*a^6*b^6 +
32*A^2*a^7*b^5 - 32*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 8*A^2*a^10*b^2 + 36*B^2*a^2*b^10 - 72*B^2*a^3*b^9 + 36*B^2*a
^4*b^8 + 288*B^2*a^5*b^7 - 288*B^2*a^6*b^6 - 432*B^2*a^7*b^5 + 441*B^2*a^8*b^4 + 288*B^2*a^9*b^3 - 288*B^2*a^1
0*b^2 - 24*A*B*a*b^11 - 48*A*B*a^11*b + 48*A*B*a^2*b^10 - 72*A*B*a^3*b^9 - 192*A*B*a^4*b^8 + 252*A*B*a^5*b^7 +
288*A*B*a^6*b^6 - 318*A*B*a^7*b^5 - 192*A*B*a^8*b^4 + 192*A*B*a^9*b^3 + 48*A*B*a^10*b^2))/(a*b^12 + b^13 - 3*
a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (((8*(4*A*b^18 - 8*A*a^2*b^16 + 34*A*a^3*
b^15 + 6*A*a^4*b^14 - 36*A*a^5*b^13 - 4*A*a^6*b^12 + 18*A*a^7*b^11 + 2*A*a^8*b^10 - 4*A*a^9*b^9 + 24*B*a^2*b^1
6 + 36*B*a^3*b^15 - 78*B*a^4*b^14 - 42*B*a^5*b^13 + 96*B*a^6*b^12 + 24*B*a^7*b^11 - 54*B*a^8*b^10 - 6*B*a^9*b^
9 + 12*B*a^10*b^8 - 12*A*a*b^17 - 12*B*a*b^17))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*
b^11 - a^6*b^10 - a^7*b^9) - (8*tan(c/2 + (d*x)/2)*(A*b - 3*B*a)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4
*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/(b^4*(a*b^12 + b^13 -
3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*(A*b - 3*B*a))/b^4)*(A*b - 3*B*a))/b^4
))*(A*b - 3*B*a)*2i)/(b^4*d) + (a*atan(((a*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 72*B^2*a^12 - 8*A^2*a*b^11 - 7
2*B^2*a^11*b + 24*A^2*a^2*b^10 + 32*A^2*a^3*b^9 - 52*A^2*a^4*b^8 - 48*A^2*a^5*b^7 + 57*A^2*a^6*b^6 + 32*A^2*a^
7*b^5 - 32*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 8*A^2*a^10*b^2 + 36*B^2*a^2*b^10 - 72*B^2*a^3*b^9 + 36*B^2*a^4*b^8 +
288*B^2*a^5*b^7 - 288*B^2*a^6*b^6 - 432*B^2*a^7*b^5 + 441*B^2*a^8*b^4 + 288*B^2*a^9*b^3 - 288*B^2*a^10*b^2 - 2
4*A*B*a*b^11 - 48*A*B*a^11*b + 48*A*B*a^2*b^10 - 72*A*B*a^3*b^9 - 192*A*B*a^4*b^8 + 252*A*B*a^5*b^7 + 288*A*B*
a^6*b^6 - 318*A*B*a^7*b^5 - 192*A*B*a^8*b^4 + 192*A*B*a^9*b^3 + 48*A*B*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11
- 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (a*((8*(4*A*b^18 - 8*A*a^2*b^16 + 34*A*a^3*b^15 +
6*A*a^4*b^14 - 36*A*a^5*b^13 - 4*A*a^6*b^12 + 18*A*a^7*b^11 + 2*A*a^8*b^10 - 4*A*a^9*b^9 + 24*B*a^2*b^16 + 36*
B*a^3*b^15 - 78*B*a^4*b^14 - 42*B*a^5*b^13 + 96*B*a^6*b^12 + 24*B*a^7*b^11 - 54*B*a^8*b^10 - 6*B*a^9*b^9 + 12*
B*a^10*b^8 - 12*A*a*b^17 - 12*B*a*b^17))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 -
a^6*b^10 - a^7*b^9) - (4*a*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 1
5*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^
6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 +
5*a^8*b^6 - a^10*b^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))
*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4))/(2*(b^
14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*
B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4)*1i)/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*
b^8 + 5*a^8*b^6 - a^10*b^4)) + (a*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 72*B^2*a^12 - 8*A^2*a*b^11 - 72*B^2*a^1
1*b + 24*A^2*a^2*b^10 + 32*A^2*a^3*b^9 - 52*A^2*a^4*b^8 - 48*A^2*a^5*b^7 + 57*A^2*a^6*b^6 + 32*A^2*a^7*b^5 - 3
2*A^2*a^8*b^4 - 8*A^2*a^9*b^3 + 8*A^2*a^10*b^2 + 36*B^2*a^2*b^10 - 72*B^2*a^3*b^9 + 36*B^2*a^4*b^8 + 288*B^2*a
^5*b^7 - 288*B^2*a^6*b^6 - 432*B^2*a^7*b^5 + 441*B^2*a^8*b^4 + 288*B^2*a^9*b^3 - 288*B^2*a^10*b^2 - 24*A*B*a*b
^11 - 48*A*B*a^11*b + 48*A*B*a^2*b^10 - 72*A*B*a^3*b^9 - 192*A*B*a^4*b^8 + 252*A*B*a^5*b^7 + 288*A*B*a^6*b^6 -
318*A*B*a^7*b^5 - 192*A*B*a^8*b^4 + 192*A*B*a^9*b^3 + 48*A*B*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b
^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + (a*((8*(4*A*b^18 - 8*A*a^2*b^16 + 34*A*a^3*b^15 + 6*A*a^4*b
^14 - 36*A*a^5*b^13 - 4*A*a^6*b^12 + 18*A*a^7*b^11 + 2*A*a^8*b^10 - 4*A*a^9*b^9 + 24*B*a^2*b^16 + 36*B*a^3*b^1
5 - 78*B*a^4*b^14 - 42*B*a^5*b^13 + 96*B*a^6*b^12 + 24*B*a^7*b^11 - 54*B*a^8*b^10 - 6*B*a^9*b^9 + 12*B*a^10*b^
8 - 12*A*a*b^17 - 12*B*a*b^17))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10
- a^7*b^9) + (4*a*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b
^2 + 2*A*a^4*b - 12*B*a*b^4)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 -
32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^
6 - a^10*b^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*((a + b)
^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4))/(2*(b^14 - 5*a^
2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5
*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4)*1i)/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a
^8*b^6 - a^10*b^4)))/((16*(108*B^3*a^12 - 12*A^3*a*b^11 - 54*B^3*a^11*b - 24*A^3*a^2*b^10 + 34*A^3*a^3*b^9 + 2
6*A^3*a^4*b^8 - 36*A^3*a^5*b^7 - 13*A^3*a^6*b^6 + 18*A^3*a^7*b^5 + 2*A^3*a^8*b^4 - 4*A^3*a^9*b^3 + 216*B^3*a^4
*b^8 + 216*B^3*a^5*b^7 - 702*B^3*a^6*b^6 - 378*B^3*a^7*b^5 + 864*B^3*a^8*b^4 + 243*B^3*a^9*b^3 - 486*B^3*a^10*
b^2 - 108*A*B^2*a^11*b - 252*A*B^2*a^3*b^9 - 324*A*B^2*a^4*b^8 + 774*A*B^2*a^5*b^7 + 486*A*B^2*a^6*b^6 - 900*A
*B^2*a^7*b^5 - 279*A*B^2*a^8*b^4 + 486*A*B^2*a^9*b^3 + 54*A*B^2*a^10*b^2 + 96*A^2*B*a^2*b^10 + 156*A^2*B*a^3*b
^9 - 282*A^2*B*a^4*b^8 - 198*A^2*B*a^5*b^7 + 312*A^2*B*a^6*b^6 + 105*A^2*B*a^7*b^5 - 162*A^2*B*a^8*b^4 - 18*A^
2*B*a^9*b^3 + 36*A^2*B*a^10*b^2))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^1
0 - a^7*b^9) + (a*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 72*B^2*a^12 - 8*A^2*a*b^11 - 72*B^2*a^11*b + 24*A^2*a^2
*b^10 + 32*A^2*a^3*b^9 - 52*A^2*a^4*b^8 - 48*A^2*a^5*b^7 + 57*A^2*a^6*b^6 + 32*A^2*a^7*b^5 - 32*A^2*a^8*b^4 -
8*A^2*a^9*b^3 + 8*A^2*a^10*b^2 + 36*B^2*a^2*b^10 - 72*B^2*a^3*b^9 + 36*B^2*a^4*b^8 + 288*B^2*a^5*b^7 - 288*B^2
*a^6*b^6 - 432*B^2*a^7*b^5 + 441*B^2*a^8*b^4 + 288*B^2*a^9*b^3 - 288*B^2*a^10*b^2 - 24*A*B*a*b^11 - 48*A*B*a^1
1*b + 48*A*B*a^2*b^10 - 72*A*B*a^3*b^9 - 192*A*B*a^4*b^8 + 252*A*B*a^5*b^7 + 288*A*B*a^6*b^6 - 318*A*B*a^7*b^5
- 192*A*B*a^8*b^4 + 192*A*B*a^9*b^3 + 48*A*B*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9
+ 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - (a*((8*(4*A*b^18 - 8*A*a^2*b^16 + 34*A*a^3*b^15 + 6*A*a^4*b^14 - 36*A*a^5*b
^13 - 4*A*a^6*b^12 + 18*A*a^7*b^11 + 2*A*a^8*b^10 - 4*A*a^9*b^9 + 24*B*a^2*b^16 + 36*B*a^3*b^15 - 78*B*a^4*b^1
4 - 42*B*a^5*b^13 + 96*B*a^6*b^12 + 24*B*a^7*b^11 - 54*B*a^8*b^10 - 6*B*a^9*b^9 + 12*B*a^10*b^8 - 12*A*a*b^17
- 12*B*a*b^17))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (4*
a*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b -
12*B*a*b^4)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32
*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)*(a
*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*((a + b)^5*(a - b)^5)^(1
/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*
b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*
B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))
- (a*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 72*B^2*a^12 - 8*A^2*a*b^11 - 72*B^2*a^11*b + 24*A^2*a^2*b^10 + 32*A
^2*a^3*b^9 - 52*A^2*a^4*b^8 - 48*A^2*a^5*b^7 + 57*A^2*a^6*b^6 + 32*A^2*a^7*b^5 - 32*A^2*a^8*b^4 - 8*A^2*a^9*b^
3 + 8*A^2*a^10*b^2 + 36*B^2*a^2*b^10 - 72*B^2*a^3*b^9 + 36*B^2*a^4*b^8 + 288*B^2*a^5*b^7 - 288*B^2*a^6*b^6 - 4
32*B^2*a^7*b^5 + 441*B^2*a^8*b^4 + 288*B^2*a^9*b^3 - 288*B^2*a^10*b^2 - 24*A*B*a*b^11 - 48*A*B*a^11*b + 48*A*B
*a^2*b^10 - 72*A*B*a^3*b^9 - 192*A*B*a^4*b^8 + 252*A*B*a^5*b^7 + 288*A*B*a^6*b^6 - 318*A*B*a^7*b^5 - 192*A*B*a
^8*b^4 + 192*A*B*a^9*b^3 + 48*A*B*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8
- a^6*b^7 - a^7*b^6) + (a*((8*(4*A*b^18 - 8*A*a^2*b^16 + 34*A*a^3*b^15 + 6*A*a^4*b^14 - 36*A*a^5*b^13 - 4*A*a^
6*b^12 + 18*A*a^7*b^11 + 2*A*a^8*b^10 - 4*A*a^9*b^9 + 24*B*a^2*b^16 + 36*B*a^3*b^15 - 78*B*a^4*b^14 - 42*B*a^5
*b^13 + 96*B*a^6*b^12 + 24*B*a^7*b^11 - 54*B*a^8*b^10 - 6*B*a^9*b^9 + 12*B*a^10*b^8 - 12*A*a*b^17 - 12*B*a*b^1
7))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (4*a*tan(c/2 +
(d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4)
*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 +
8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)*(a*b^12 + b^13
- 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5
- 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^
6*b^8 + 5*a^8*b^6 - a^10*b^4)))*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 +
2*A*a^4*b - 12*B*a*b^4))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))))*((a + b)^
5*(a - b)^5)^(1/2)*(6*A*b^5 - 6*B*a^5 - 5*A*a^2*b^3 + 15*B*a^3*b^2 + 2*A*a^4*b - 12*B*a*b^4)*1i)/(d*(b^14 - 5*
a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]

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))**3,x)

[Out]

Integral((A + B*sec(c + d*x))*sec(c + d*x)**4/(a + b*sec(c + d*x))**3, x)

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