3.6.13 \(\int \frac {\cos ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) [513]

Optimal. Leaf size=296 \[ \frac {\left (a^2+12 b^2\right ) x}{2 a^5}-\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (7 a^2-4 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]

[Out]

1/2*(a^2+12*b^2)*x/a^5-b^3*(20*a^4-29*a^2*b^2+12*b^4)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/
(a-b)^(5/2)/(a+b)^(5/2)/d-3/2*b*(2*a^4-7*a^2*b^2+4*b^4)*sin(d*x+c)/a^4/(a^2-b^2)^2/d+1/2*(a^4-10*a^2*b^2+6*b^4
)*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*b^2*cos(d*x+c)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+1/2*b
^2*(7*a^2-4*b^2)*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3932, 4185, 4189, 4004, 3916, 2738, 214} \begin {gather*} \frac {b^2 \left (7 a^2-4 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {b^2 \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {x \left (a^2+12 b^2\right )}{2 a^5}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^4 d \left (a^2-b^2\right )^2}-\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]

[Out]

((a^2 + 12*b^2)*x)/(2*a^5) - (b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a
+ b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(2*a^4*(a^2 - b^2
)^2*d) + ((a^4 - 10*a^2*b^2 + 6*b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Cos[c + d*x]*Si
n[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(7*a^2 - 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^
2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[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 3916

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

Rule 3932

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

Rule 4004

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

Rule 4185

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(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)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I
nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*
(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] &
& ILtQ[n, 0])

Rule 4189

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

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (-2 a^2+4 b^2+2 a b \sec (c+d x)-3 b^2 \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (7 a^2-4 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (a^4-10 a^2 b^2+6 b^4\right )-a b \left (4 a^2-b^2\right ) \sec (c+d x)+2 b^2 \left (7 a^2-4 b^2\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (7 a^2-4 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (6 b \left (2 a^4-7 a^2 b^2+4 b^4\right )-2 a \left (a^4+4 a^2 b^2-2 b^4\right ) \sec (c+d x)-2 b \left (a^4-10 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (7 a^2-4 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {2 \left (a^2-b^2\right )^2 \left (a^2+12 b^2\right )+2 a b \left (a^4-10 a^2 b^2+6 b^4\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (a^2+12 b^2\right ) x}{2 a^5}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (7 a^2-4 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (a^2+12 b^2\right ) x}{2 a^5}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (7 a^2-4 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (b^2 \left (20 a^4-29 a^2 b^2+12 b^4\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (a^2+12 b^2\right ) x}{2 a^5}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (7 a^2-4 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (b^2 \left (20 a^4-29 a^2 b^2+12 b^4\right )\right ) \text {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 )}{a^5 \left (a^2-b^2\right )^2 d}\\ &=\frac {\left (a^2+12 b^2\right ) x}{2 a^5}-\frac {b^3 \left (20 a^4-29 a^2 b^2+12 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {3 b \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4-10 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b^2 \left (7 a^2-4 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.13, size = 199, normalized size = 0.67 \begin {gather*} \frac {2 \left (a^2+12 b^2\right ) (c+d x)+\frac {4 b^3 \left (20 a^4-29 a^2 b^2+12 b^4\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 )^{5/2}}-12 a b \sin (c+d x)+\frac {2 a b^5 \sin (c+d x)}{(-a+b) (a+b) (b+a \cos (c+d x))^2}+\frac {2 a b^4 \left (10 a^2-7 b^2\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}+a^2 \sin (2 (c+d x))}{4 a^5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]

[Out]

(2*(a^2 + 12*b^2)*(c + d*x) + (4*b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a
^2 - b^2]])/(a^2 - b^2)^(5/2) - 12*a*b*Sin[c + d*x] + (2*a*b^5*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*Cos[c +
d*x])^2) + (2*a*b^4*(10*a^2 - 7*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])) + a^2*Sin[2*(c +
d*x)])/(4*a^5*d)

________________________________________________________________________________________

Maple [A]
time = 0.25, size = 310, normalized size = 1.05

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2}-3 b a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 b a +\frac {1}{2} a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (a^{2}+12 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}+\frac {2 b^{3} \left (\frac {-\frac {\left (10 a^{2}+b a -6 b^{2}\right ) b a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 b a +b^{2}\right )}+\frac {\left (10 a^{2}-b a -6 b^{2}\right ) b a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 b a +b^{2}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (20 a^{4}-29 b^{2} a^{2}+12 b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\)	\(310\)
default	\(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2}-3 b a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3 b a +\frac {1}{2} a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (a^{2}+12 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}+\frac {2 b^{3} \left (\frac {-\frac {\left (10 a^{2}+b a -6 b^{2}\right ) b a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 b a +b^{2}\right )}+\frac {\left (10 a^{2}-b a -6 b^{2}\right ) b a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 b a +b^{2}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (20 a^{4}-29 b^{2} a^{2}+12 b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\)	\(310\)
risch	\(\frac {x}{2 a^{3}}+\frac {6 x \,b^{2}}{a^{5}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {3 i b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{4}}-\frac {3 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{4}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {i b^{4} \left (-11 b \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+8 a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-10 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-13 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+14 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-29 a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+20 b^{3} a \,{\mathrm e}^{i \left (d x +c \right )}-10 a^{4}+7 b^{2} a^{2}\right )}{a^{5} \left (-a^{2}+b^{2}\right )^{2} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {10 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}-\frac {29 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}+\frac {6 b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{5}}-\frac {10 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d a}+\frac {29 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{3}}-\frac {6 b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} d \,a^{5}}\)	\(776\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2/a^5*(((-1/2*a^2-3*b*a)*tan(1/2*d*x+1/2*c)^3+(-3*b*a+1/2*a^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^
2)^2+1/2*(a^2+12*b^2)*arctan(tan(1/2*d*x+1/2*c)))+2*b^3/a^5*((-1/2*(10*a^2+a*b-6*b^2)*b*a/(a-b)/(a^2+2*a*b+b^2
)*tan(1/2*d*x+1/2*c)^3+1/2*(10*a^2-a*b-6*b^2)*b*a/(a+b)/(a^2-2*a*b+b^2)*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)^2-1/2*(20*a^4-29*a^2*b^2+12*b^4)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arc
tanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))))

________________________________________________________________________________________

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(cos(d*x+c)^2/(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 de

________________________________________________________________________________________

Fricas [A]
time = 3.47, size = 1158, normalized size = 3.91 \begin {gather*} \left [\frac {2 \, {\left (a^{10} + 9 \, a^{8} b^{2} - 33 \, a^{6} b^{4} + 35 \, a^{4} b^{6} - 12 \, a^{2} b^{8}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (a^{8} b^{2} + 9 \, a^{6} b^{4} - 33 \, a^{4} b^{6} + 35 \, a^{2} b^{8} - 12 \, b^{10}\right )} d x + {\left (20 \, a^{4} b^{5} - 29 \, a^{2} b^{7} + 12 \, b^{9} + {\left (20 \, a^{6} b^{3} - 29 \, a^{4} b^{5} + 12 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (20 \, a^{5} b^{4} - 29 \, a^{3} b^{6} + 12 \, a b^{8}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - 2 \, {\left (6 \, a^{7} b^{3} - 27 \, a^{5} b^{5} + 33 \, a^{3} b^{7} - 12 \, a b^{9} - {\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (11 \, a^{8} b^{2} - 43 \, a^{6} b^{4} + 50 \, a^{4} b^{6} - 18 \, a^{2} b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{13} - 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} - a^{7} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{12} b - 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} - a^{6} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{11} b^{2} - 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} - a^{5} b^{8}\right )} d\right )}}, \frac {{\left (a^{10} + 9 \, a^{8} b^{2} - 33 \, a^{6} b^{4} + 35 \, a^{4} b^{6} - 12 \, a^{2} b^{8}\right )} d x \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9}\right )} d x \cos \left (d x + c\right ) + {\left (a^{8} b^{2} + 9 \, a^{6} b^{4} - 33 \, a^{4} b^{6} + 35 \, a^{2} b^{8} - 12 \, b^{10}\right )} d x - {\left (20 \, a^{4} b^{5} - 29 \, a^{2} b^{7} + 12 \, b^{9} + {\left (20 \, a^{6} b^{3} - 29 \, a^{4} b^{5} + 12 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (20 \, a^{5} b^{4} - 29 \, a^{3} b^{6} + 12 \, a b^{8}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (6 \, a^{7} b^{3} - 27 \, a^{5} b^{5} + 33 \, a^{3} b^{7} - 12 \, a b^{9} - {\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (11 \, a^{8} b^{2} - 43 \, a^{6} b^{4} + 50 \, a^{4} b^{6} - 18 \, a^{2} b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{13} - 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} - a^{7} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{12} b - 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} - a^{6} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{11} b^{2} - 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} - a^{5} b^{8}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/4*(2*(a^10 + 9*a^8*b^2 - 33*a^6*b^4 + 35*a^4*b^6 - 12*a^2*b^8)*d*x*cos(d*x + c)^2 + 4*(a^9*b + 9*a^7*b^3 -
33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*d*x*cos(d*x + c) + 2*(a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b
^10)*d*x + (20*a^4*b^5 - 29*a^2*b^7 + 12*b^9 + (20*a^6*b^3 - 29*a^4*b^5 + 12*a^2*b^7)*cos(d*x + c)^2 + 2*(20*a
^5*b^4 - 29*a^3*b^6 + 12*a*b^8)*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*(6*a^7*b^3 - 27*a^5*b^5 + 33*a^3*b^7 - 12*a*b^9 - (a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*
cos(d*x + c)^3 + 4*(a^9*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*cos(d*x + c)^2 + (11*a^8*b^2 - 43*a^6*b^4 + 50*a^
4*b^6 - 18*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^13 - 3*a^11*b^2 + 3*a^9*b^4 - a^7*b^6)*d*cos(d*x + c)^2 +
2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*d*cos(d*x + c) + (a^11*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*b^8)*d)
, 1/2*((a^10 + 9*a^8*b^2 - 33*a^6*b^4 + 35*a^4*b^6 - 12*a^2*b^8)*d*x*cos(d*x + c)^2 + 2*(a^9*b + 9*a^7*b^3 - 3
3*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*d*x*cos(d*x + c) + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b^10
)*d*x - (20*a^4*b^5 - 29*a^2*b^7 + 12*b^9 + (20*a^6*b^3 - 29*a^4*b^5 + 12*a^2*b^7)*cos(d*x + c)^2 + 2*(20*a^5*
b^4 - 29*a^3*b^6 + 12*a*b^8)*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))) - (6*a^7*b^3 - 27*a^5*b^5 + 33*a^3*b^7 - 12*a*b^9 - (a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^
4*b^6)*cos(d*x + c)^3 + 4*(a^9*b - 3*a^7*b^3 + 3*a^5*b^5 - a^3*b^7)*cos(d*x + c)^2 + (11*a^8*b^2 - 43*a^6*b^4
+ 50*a^4*b^6 - 18*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^13 - 3*a^11*b^2 + 3*a^9*b^4 - a^7*b^6)*d*cos(d*x +
c)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*d*cos(d*x + c) + (a^11*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*
b^8)*d)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2/(a+b*sec(d*x+c))**3,x)

[Out]

Integral(cos(c + d*x)**2/(a + b*sec(c + d*x))**3, x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1723 vs. \(2 (277) = 554\).
time = 0.74, size = 1723, normalized size = 5.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(((a^6 - a^5*b + 10*a^4*b^2 + 10*a^3*b^3 - 23*a^2*b^4 - 6*a*b^5 + 12*b^6)*sqrt(-a^2 + b^2)*abs(a^9 - 2*a^
7*b^2 + a^5*b^4)*abs(-a + b) - (a^15 - a^14*b + 8*a^13*b^2 - 28*a^12*b^3 - 42*a^11*b^4 + 111*a^10*b^5 + 68*a^9
*b^6 - 158*a^8*b^7 - 47*a^7*b^8 + 100*a^6*b^9 + 12*a^5*b^10 - 24*a^4*b^11)*sqrt(-a^2 + b^2)*abs(-a + b))*(pi*f
loor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^8*b - 2*a^6*b^3 + a^4*b^5 + sqrt((a^9 + a^
8*b - 2*a^7*b^2 - 2*a^6*b^3 + a^5*b^4 + a^4*b^5)*(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5) + (
a^8*b - 2*a^6*b^3 + a^4*b^5)^2))/(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5))))/((a^9 - 2*a^7*b^
2 + a^5*b^4)^2*(a^2 - 2*a*b + b^2) + (a^10*b - 2*a^9*b^2 - a^8*b^3 + 4*a^7*b^4 - a^6*b^5 - 2*a^5*b^6 + a^4*b^7
)*abs(a^9 - 2*a^7*b^2 + a^5*b^4)) + (a^15 - a^14*b + 8*a^13*b^2 - 28*a^12*b^3 - 42*a^11*b^4 + 111*a^10*b^5 + 6
8*a^9*b^6 - 158*a^8*b^7 - 47*a^7*b^8 + 100*a^6*b^9 + 12*a^5*b^10 - 24*a^4*b^11 + a^6*abs(a^9 - 2*a^7*b^2 + a^5
*b^4) - a^5*b*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 10*a^4*b^2*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 10*a^3*b^3*abs(a^9
- 2*a^7*b^2 + a^5*b^4) - 23*a^2*b^4*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 6*a*b^5*abs(a^9 - 2*a^7*b^2 + a^5*b^4) +
12*b^6*abs(a^9 - 2*a^7*b^2 + a^5*b^4))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(
a^8*b - 2*a^6*b^3 + a^4*b^5 - sqrt((a^9 + a^8*b - 2*a^7*b^2 - 2*a^6*b^3 + a^5*b^4 + a^4*b^5)*(a^9 - a^8*b - 2*
a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5) + (a^8*b - 2*a^6*b^3 + a^4*b^5)^2))/(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*
b^3 + a^5*b^4 - a^4*b^5))))/(a^8*b*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 2*a^6*b^3*abs(a^9 - 2*a^7*b^2 + a^5*b^4) +
a^4*b^5*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - (a^9 - 2*a^7*b^2 + a^5*b^4)^2) + 2*(a^7*tan(1/2*d*x + 1/2*c)^7 + 4*a
^6*b*tan(1/2*d*x + 1/2*c)^7 - 13*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 2*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*a^3*b^
4*tan(1/2*d*x + 1/2*c)^7 - 17*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 18*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 12*b^7*tan(1/
2*d*x + 1/2*c)^7 - 3*a^7*tan(1/2*d*x + 1/2*c)^5 - 4*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 5*a^5*b^2*tan(1/2*d*x + 1/2
*c)^5 + 26*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 29*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 67*a^2*b^5*tan(1/2*d*x + 1/2*c
)^5 - 18*a*b^6*tan(1/2*d*x + 1/2*c)^5 + 36*b^7*tan(1/2*d*x + 1/2*c)^5 + 3*a^7*tan(1/2*d*x + 1/2*c)^3 - 4*a^6*b
*tan(1/2*d*x + 1/2*c)^3 + 5*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 26*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*a^3*b^4*ta
n(1/2*d*x + 1/2*c)^3 - 67*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 18*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 36*b^7*tan(1/2*d*
x + 1/2*c)^3 - a^7*tan(1/2*d*x + 1/2*c) + 4*a^6*b*tan(1/2*d*x + 1/2*c) + 13*a^5*b^2*tan(1/2*d*x + 1/2*c) - 2*a
^4*b^3*tan(1/2*d*x + 1/2*c) - 33*a^3*b^4*tan(1/2*d*x + 1/2*c) - 17*a^2*b^5*tan(1/2*d*x + 1/2*c) + 18*a*b^6*tan
(1/2*d*x + 1/2*c) + 12*b^7*tan(1/2*d*x + 1/2*c))/((a^8 - 2*a^6*b^2 + a^4*b^4)*(a*tan(1/2*d*x + 1/2*c)^4 - b*ta
n(1/2*d*x + 1/2*c)^4 - 2*b*tan(1/2*d*x + 1/2*c)^2 - a - b)^2))/d

________________________________________________________________________________________

Mupad [B]
time = 9.29, size = 2500, normalized size = 8.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^2/(a + b/cos(c + d*x))^3,x)

[Out]

(atan(((((8*tan(c/2 + (d*x)/2)*(a^14 - 2*a^13*b - 288*a*b^13 + 288*b^14 - 1104*a^2*b^12 + 1104*a^3*b^11 + 1538
*a^4*b^10 - 1538*a^5*b^9 - 827*a^6*b^8 + 872*a^7*b^7 + 18*a^8*b^6 - 108*a^9*b^5 + 74*a^10*b^4 - 40*a^11*b^3 +
21*a^12*b^2))/(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2) + ((a^2*
1i + b^2*12i)*((4*(4*a^21 - 48*a^10*b^11 + 24*a^11*b^10 + 212*a^12*b^9 - 100*a^13*b^8 - 360*a^14*b^7 + 164*a^1
5*b^6 + 276*a^16*b^5 - 120*a^17*b^4 - 80*a^18*b^3 + 28*a^19*b^2))/(a^18*b + a^19 - a^12*b^7 - a^13*b^6 + 3*a^1
4*b^5 + 3*a^15*b^4 - 3*a^16*b^3 - 3*a^17*b^2) - (4*tan(c/2 + (d*x)/2)*(a^2*1i + b^2*12i)*(8*a^19*b - 8*a^10*b^
10 + 8*a^11*b^9 + 32*a^12*b^8 - 32*a^13*b^7 - 48*a^14*b^6 + 48*a^15*b^5 + 32*a^16*b^4 - 32*a^17*b^3 - 8*a^18*b
^2))/(a^5*(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2))))/(2*a^5))*
(a^2*1i + b^2*12i)*1i)/(2*a^5) + (((8*tan(c/2 + (d*x)/2)*(a^14 - 2*a^13*b - 288*a*b^13 + 288*b^14 - 1104*a^2*b
^12 + 1104*a^3*b^11 + 1538*a^4*b^10 - 1538*a^5*b^9 - 827*a^6*b^8 + 872*a^7*b^7 + 18*a^8*b^6 - 108*a^9*b^5 + 74
*a^10*b^4 - 40*a^11*b^3 + 21*a^12*b^2))/(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^12*
b^3 - 3*a^13*b^2) - ((a^2*1i + b^2*12i)*((4*(4*a^21 - 48*a^10*b^11 + 24*a^11*b^10 + 212*a^12*b^9 - 100*a^13*b^
8 - 360*a^14*b^7 + 164*a^15*b^6 + 276*a^16*b^5 - 120*a^17*b^4 - 80*a^18*b^3 + 28*a^19*b^2))/(a^18*b + a^19 - a
^12*b^7 - a^13*b^6 + 3*a^14*b^5 + 3*a^15*b^4 - 3*a^16*b^3 - 3*a^17*b^2) + (4*tan(c/2 + (d*x)/2)*(a^2*1i + b^2*
12i)*(8*a^19*b - 8*a^10*b^10 + 8*a^11*b^9 + 32*a^12*b^8 - 32*a^13*b^7 - 48*a^14*b^6 + 48*a^15*b^5 + 32*a^16*b^
4 - 32*a^17*b^3 - 8*a^18*b^2))/(a^5*(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^12*b^3
- 3*a^13*b^2))))/(2*a^5))*(a^2*1i + b^2*12i)*1i)/(2*a^5))/((8*(1728*b^15 - 864*a*b^14 - 7344*a^2*b^13 + 3456*a
^3*b^12 + 11700*a^4*b^11 - 4770*a^5*b^10 - 7829*a^6*b^9 + 2326*a^7*b^8 + 1314*a^8*b^7 - 11*a^9*b^6 + 411*a^10*
b^5 - 20*a^11*b^4 + 20*a^12*b^3))/(a^18*b + a^19 - a^12*b^7 - a^13*b^6 + 3*a^14*b^5 + 3*a^15*b^4 - 3*a^16*b^3
- 3*a^17*b^2) - (((8*tan(c/2 + (d*x)/2)*(a^14 - 2*a^13*b - 288*a*b^13 + 288*b^14 - 1104*a^2*b^12 + 1104*a^3*b^
11 + 1538*a^4*b^10 - 1538*a^5*b^9 - 827*a^6*b^8 + 872*a^7*b^7 + 18*a^8*b^6 - 108*a^9*b^5 + 74*a^10*b^4 - 40*a^
11*b^3 + 21*a^12*b^2))/(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2)
+ ((a^2*1i + b^2*12i)*((4*(4*a^21 - 48*a^10*b^11 + 24*a^11*b^10 + 212*a^12*b^9 - 100*a^13*b^8 - 360*a^14*b^7
+ 164*a^15*b^6 + 276*a^16*b^5 - 120*a^17*b^4 - 80*a^18*b^3 + 28*a^19*b^2))/(a^18*b + a^19 - a^12*b^7 - a^13*b^
6 + 3*a^14*b^5 + 3*a^15*b^4 - 3*a^16*b^3 - 3*a^17*b^2) - (4*tan(c/2 + (d*x)/2)*(a^2*1i + b^2*12i)*(8*a^19*b -
8*a^10*b^10 + 8*a^11*b^9 + 32*a^12*b^8 - 32*a^13*b^7 - 48*a^14*b^6 + 48*a^15*b^5 + 32*a^16*b^4 - 32*a^17*b^3 -
8*a^18*b^2))/(a^5*(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2))))/
(2*a^5))*(a^2*1i + b^2*12i))/(2*a^5) + (((8*tan(c/2 + (d*x)/2)*(a^14 - 2*a^13*b - 288*a*b^13 + 288*b^14 - 1104
*a^2*b^12 + 1104*a^3*b^11 + 1538*a^4*b^10 - 1538*a^5*b^9 - 827*a^6*b^8 + 872*a^7*b^7 + 18*a^8*b^6 - 108*a^9*b^
5 + 74*a^10*b^4 - 40*a^11*b^3 + 21*a^12*b^2))/(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3
*a^12*b^3 - 3*a^13*b^2) - ((a^2*1i + b^2*12i)*((4*(4*a^21 - 48*a^10*b^11 + 24*a^11*b^10 + 212*a^12*b^9 - 100*a
^13*b^8 - 360*a^14*b^7 + 164*a^15*b^6 + 276*a^16*b^5 - 120*a^17*b^4 - 80*a^18*b^3 + 28*a^19*b^2))/(a^18*b + a^
19 - a^12*b^7 - a^13*b^6 + 3*a^14*b^5 + 3*a^15*b^4 - 3*a^16*b^3 - 3*a^17*b^2) + (4*tan(c/2 + (d*x)/2)*(a^2*1i
+ b^2*12i)*(8*a^19*b - 8*a^10*b^10 + 8*a^11*b^9 + 32*a^12*b^8 - 32*a^13*b^7 - 48*a^14*b^6 + 48*a^15*b^5 + 32*a
^16*b^4 - 32*a^17*b^3 - 8*a^18*b^2))/(a^5*(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^1
2*b^3 - 3*a^13*b^2))))/(2*a^5))*(a^2*1i + b^2*12i))/(2*a^5)))*(a^2*1i + b^2*12i)*1i)/(a^5*d) - ((tan(c/2 + (d*
x)/2)^3*(18*a*b^6 - 4*a^6*b + 3*a^7 + 36*b^7 - 67*a^2*b^5 - 29*a^3*b^4 + 26*a^4*b^3 + 5*a^5*b^2))/((a + b)^2*(
a^6 - 2*a^5*b + a^4*b^2)) - (tan(c/2 + (d*x)/2)^5*(18*a*b^6 + 4*a^6*b + 3*a^7 - 36*b^7 + 67*a^2*b^5 - 29*a^3*b
^4 - 26*a^4*b^3 + 5*a^5*b^2))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) - (tan(c/2 + (d*x)/2)^7*(6*a*b^5 + 5*a^5*b
+ a^6 - 12*b^6 + 23*a^2*b^4 - 10*a^3*b^3 - 8*a^4*b^2))/((a^4*b - a^5)*(a + b)^2) + (tan(c/2 + (d*x)/2)*(6*a*b
^5 + 5*a^5*b - a^6 + 12*b^6 - 23*a^2*b^4 - 10*a^3*b^3 + 8*a^4*b^2))/((a + b)*(a^6 - 2*a^5*b + a^4*b^2)))/(d*(2
*a*b - tan(c/2 + (d*x)/2)^4*(2*a^2 - 6*b^2) + tan(c/2 + (d*x)/2)^2*(4*a*b + 4*b^2) - tan(c/2 + (d*x)/2)^6*(4*a
*b - 4*b^2) + tan(c/2 + (d*x)/2)^8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) + (b^3*atan(((b^3*((8*tan(c/2 + (d*x)/2)*
(a^14 - 2*a^13*b - 288*a*b^13 + 288*b^14 - 1104*a^2*b^12 + 1104*a^3*b^11 + 1538*a^4*b^10 - 1538*a^5*b^9 - 827*
a^6*b^8 + 872*a^7*b^7 + 18*a^8*b^6 - 108*a^9*b^5 + 74*a^10*b^4 - 40*a^11*b^3 + 21*a^12*b^2))/(a^14*b + a^15 -
a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3...

________________________________________________________________________________________