\(\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [428]

Optimal result

Integrand size = 41, antiderivative size = 216 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (30 A+26 B+23 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (30 A+26 B+23 C) \tan (c+d x)}{10 d}+\frac {3 a^3 (30 A+26 B+23 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {(30 A-6 B+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {(2 B+C) (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {a^3 (30 A+26 B+23 C) \tan ^3(c+d x)}{120 d} \]

[Out]

1/16*a^3*(30*A+26*B+23*C)*arctanh(sin(d*x+c))/d+1/10*a^3*(30*A+26*B+23*C)*tan(d*x+c)/d+3/80*a^3*(30*A+26*B+23*
C)*sec(d*x+c)*tan(d*x+c)/d+1/120*(30*A-6*B+7*C)*(a+a*sec(d*x+c))^3*tan(d*x+c)/d+1/6*C*sec(d*x+c)^2*(a+a*sec(d*
x+c))^3*tan(d*x+c)/d+1/10*(2*B+C)*(a+a*sec(d*x+c))^4*tan(d*x+c)/a/d+1/120*a^3*(30*A+26*B+23*C)*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4173, 4095, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (30 A+26 B+23 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (30 A+26 B+23 C) \tan ^3(c+d x)}{120 d}+\frac {a^3 (30 A+26 B+23 C) \tan (c+d x)}{10 d}+\frac {3 a^3 (30 A+26 B+23 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac {(30 A-6 B+7 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{120 d}+\frac {(2 B+C) \tan (c+d x) (a \sec (c+d x)+a)^4}{10 a d}+\frac {C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^3}{6 d} \]

[In]

Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^3*(30*A + 26*B + 23*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(30*A + 26*B + 23*C)*Tan[c + d*x])/(10*d) + (3*
a^3*(30*A + 26*B + 23*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + ((30*A - 6*B + 7*C)*(a + a*Sec[c + d*x])^3*Tan[c
+ d*x])/(120*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(6*d) + ((2*B + C)*(a + a*Sec[c + d*x
])^4*Tan[c + d*x])/(10*a*d) + (a^3*(30*A + 26*B + 23*C)*Tan[c + d*x]^3)/(120*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

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

Rule 3876

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[m, 0] && RationalQ[n]

Rule 4086

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

Rule 4095

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

Rule 4173

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[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(
(d*Csc[e + f*x])^n/(f*(m + n + 1))), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^
n*Simp[A*b*(m + n + 1) + b*C*n + (a*C*m + b*B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A
, B, C, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)] &&  !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (2 a (3 A+C)+3 a (2 B+C) \sec (c+d x)) \, dx}{6 a} \\ & = \frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {(2 B+C) (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^3 \left (12 a^2 (2 B+C)+a^2 (30 A-6 B+7 C) \sec (c+d x)\right ) \, dx}{30 a^2} \\ & = \frac {(30 A-6 B+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {(2 B+C) (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {1}{40} (30 A+26 B+23 C) \int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx \\ & = \frac {(30 A-6 B+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {(2 B+C) (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {1}{40} (30 A+26 B+23 C) \int \left (a^3 \sec (c+d x)+3 a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+a^3 \sec ^4(c+d x)\right ) \, dx \\ & = \frac {(30 A-6 B+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {(2 B+C) (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {1}{40} \left (a^3 (30 A+26 B+23 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{40} \left (a^3 (30 A+26 B+23 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{40} \left (3 a^3 (30 A+26 B+23 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{40} \left (3 a^3 (30 A+26 B+23 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^3 (30 A+26 B+23 C) \text {arctanh}(\sin (c+d x))}{40 d}+\frac {3 a^3 (30 A+26 B+23 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {(30 A-6 B+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {(2 B+C) (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {1}{80} \left (3 a^3 (30 A+26 B+23 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^3 (30 A+26 B+23 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{40 d}-\frac {\left (3 a^3 (30 A+26 B+23 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{40 d} \\ & = \frac {a^3 (30 A+26 B+23 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (30 A+26 B+23 C) \tan (c+d x)}{10 d}+\frac {3 a^3 (30 A+26 B+23 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {(30 A-6 B+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {(2 B+C) (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {a^3 (30 A+26 B+23 C) \tan ^3(c+d x)}{120 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.02 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.62 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \left (15 (30 A+26 B+23 C) \text {arctanh}(\sin (c+d x))+\left (16 (45 A+38 B+34 C)+15 (30 A+26 B+23 C) \sec (c+d x)+16 (15 A+19 B+17 C) \sec ^2(c+d x)+10 (6 A+18 B+23 C) \sec ^3(c+d x)+48 (B+3 C) \sec ^4(c+d x)+40 C \sec ^5(c+d x)\right ) \tan (c+d x)\right )}{240 d} \]

[In]

Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(a^3*(15*(30*A + 26*B + 23*C)*ArcTanh[Sin[c + d*x]] + (16*(45*A + 38*B + 34*C) + 15*(30*A + 26*B + 23*C)*Sec[c
+ d*x] + 16*(15*A + 19*B + 17*C)*Sec[c + d*x]^2 + 10*(6*A + 18*B + 23*C)*Sec[c + d*x]^3 + 48*(B + 3*C)*Sec[c
+ d*x]^4 + 40*C*Sec[c + d*x]^5)*Tan[c + d*x]))/(240*d)

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.16

method	result	size
norman	\(\frac {-\frac {33 a^{3} \left (30 A +26 B +23 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {17 a^{3} \left (30 A +26 B +23 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {a^{3} \left (30 A +26 B +23 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {a^{3} \left (98 A +102 B +105 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{3} \left (1026 A +838 B +633 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {a^{3} \left (1250 A +998 B +969 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {a^{3} \left (30 A +26 B +23 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {a^{3} \left (30 A +26 B +23 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\)	\(251\)
parallelrisch	\(-\frac {15 a^{3} \left (\left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (A +\frac {13 B}{15}+\frac {23 C}{30}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (A +\frac {13 B}{15}+\frac {23 C}{30}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {184 A}{15}-\frac {208 B}{15}-16 C \right ) \sin \left (2 d x +2 c \right )+\left (-\frac {391 C}{45}-\frac {106 A}{15}-\frac {42 B}{5}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {608 B}{75}-\frac {544 C}{75}-\frac {128 A}{15}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {23 C}{15}-2 A -\frac {26 B}{15}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {304 B}{225}-\frac {8 A}{5}-\frac {272 C}{225}\right ) \sin \left (6 d x +6 c \right )-\frac {76 \sin \left (d x +c \right ) \left (A +\frac {25 B}{19}+\frac {75 C}{38}\right )}{15}\right )}{8 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(270\)
parts	\(\frac {\left (3 a^{3} A +B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (B \,a^{3}+3 a^{3} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (a^{3} A +3 B \,a^{3}+3 a^{3} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 a^{3} A +3 B \,a^{3}+a^{3} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{3} A \tan \left (d x +c \right )}{d}+\frac {a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(277\)
derivativedivides	\(\frac {a^{3} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(438\)
default	\(\frac {a^{3} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(438\)
risch	\(-\frac {i a^{3} \left (-544 C -720 A -608 B -2640 A \,{\mathrm e}^{8 i \left (d x +c \right )}+1140 A \,{\mathrm e}^{7 i \left (d x +c \right )}-1140 A \,{\mathrm e}^{5 i \left (d x +c \right )}-1590 A \,{\mathrm e}^{3 i \left (d x +c \right )}-450 A \,{\mathrm e}^{i \left (d x +c \right )}-7680 B \,{\mathrm e}^{4 i \left (d x +c \right )}-7200 A \,{\mathrm e}^{6 i \left (d x +c \right )}-5440 C \,{\mathrm e}^{6 i \left (d x +c \right )}-8160 A \,{\mathrm e}^{4 i \left (d x +c \right )}-7680 C \,{\mathrm e}^{4 i \left (d x +c \right )}-4080 A \,{\mathrm e}^{2 i \left (d x +c \right )}-3264 C \,{\mathrm e}^{2 i \left (d x +c \right )}+2250 C \,{\mathrm e}^{7 i \left (d x +c \right )}-2250 C \,{\mathrm e}^{5 i \left (d x +c \right )}-1955 C \,{\mathrm e}^{3 i \left (d x +c \right )}-3648 B \,{\mathrm e}^{2 i \left (d x +c \right )}-345 C \,{\mathrm e}^{i \left (d x +c \right )}+450 A \,{\mathrm e}^{11 i \left (d x +c \right )}+345 C \,{\mathrm e}^{11 i \left (d x +c \right )}-240 A \,{\mathrm e}^{10 i \left (d x +c \right )}+1590 A \,{\mathrm e}^{9 i \left (d x +c \right )}+1955 C \,{\mathrm e}^{9 i \left (d x +c \right )}-480 C \,{\mathrm e}^{8 i \left (d x +c \right )}-390 B \,{\mathrm e}^{i \left (d x +c \right )}-1500 B \,{\mathrm e}^{5 i \left (d x +c \right )}-1440 B \,{\mathrm e}^{8 i \left (d x +c \right )}+390 B \,{\mathrm e}^{11 i \left (d x +c \right )}+1890 B \,{\mathrm e}^{9 i \left (d x +c \right )}-6080 B \,{\mathrm e}^{6 i \left (d x +c \right )}-1890 B \,{\mathrm e}^{3 i \left (d x +c \right )}+1500 B \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{8 d}+\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{8 d}-\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}\)	\(538\)

[In]

int(sec(d*x+c)^2*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

(-33/20*a^3*(30*A+26*B+23*C)/d*tan(1/2*d*x+1/2*c)^7+17/24*a^3*(30*A+26*B+23*C)/d*tan(1/2*d*x+1/2*c)^9-1/8*a^3*
(30*A+26*B+23*C)/d*tan(1/2*d*x+1/2*c)^11+1/8*a^3*(98*A+102*B+105*C)/d*tan(1/2*d*x+1/2*c)-1/24*a^3*(1026*A+838*
B+633*C)/d*tan(1/2*d*x+1/2*c)^3+1/20*a^3*(1250*A+998*B+969*C)/d*tan(1/2*d*x+1/2*c)^5)/(tan(1/2*d*x+1/2*c)^2-1)
^6-1/16*a^3*(30*A+26*B+23*C)/d*ln(tan(1/2*d*x+1/2*c)-1)+1/16*a^3*(30*A+26*B+23*C)/d*ln(tan(1/2*d*x+1/2*c)+1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (30 \, A + 26 \, B + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (30 \, A + 26 \, B + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (45 \, A + 38 \, B + 34 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \, {\left (30 \, A + 26 \, B + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \, {\left (15 \, A + 19 \, B + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \, {\left (6 \, A + 18 \, B + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 48 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 40 \, C a^{3}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/480*(15*(30*A + 26*B + 23*C)*a^3*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 15*(30*A + 26*B + 23*C)*a^3*cos(d*x
+ c)^6*log(-sin(d*x + c) + 1) + 2*(16*(45*A + 38*B + 34*C)*a^3*cos(d*x + c)^5 + 15*(30*A + 26*B + 23*C)*a^3*co
s(d*x + c)^4 + 16*(15*A + 19*B + 17*C)*a^3*cos(d*x + c)^3 + 10*(6*A + 18*B + 23*C)*a^3*cos(d*x + c)^2 + 48*(B
+ 3*C)*a^3*cos(d*x + c) + 40*C*a^3)*sin(d*x + c))/(d*cos(d*x + c)^6)

Sympy [F]

\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{4}{\left (c + d x \right )}\, dx + \int A \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{5}{\left (c + d x \right )}\, dx + \int B \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{7}{\left (c + d x \right )}\, dx\right ) \]

[In]

integrate(sec(d*x+c)**2*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

a**3*(Integral(A*sec(c + d*x)**2, x) + Integral(3*A*sec(c + d*x)**3, x) + Integral(3*A*sec(c + d*x)**4, x) + I
ntegral(A*sec(c + d*x)**5, x) + Integral(B*sec(c + d*x)**3, x) + Integral(3*B*sec(c + d*x)**4, x) + Integral(3
*B*sec(c + d*x)**5, x) + Integral(B*sec(c + d*x)**6, x) + Integral(C*sec(c + d*x)**4, x) + Integral(3*C*sec(c
+ d*x)**5, x) + Integral(3*C*sec(c + d*x)**6, x) + Integral(C*sec(c + d*x)**7, x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (202) = 404\).

Time = 0.25 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.59 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 32 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{3} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{3} + 160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 5 \, C a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, B a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{3} \tan \left (d x + c\right )}{480 \, d} \]

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*(480*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^3 + 32*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c
))*B*a^3 + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^3 + 96*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x
+ c))*C*a^3 + 160*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3 - 5*C*a^3*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3
+ 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15
*log(sin(d*x + c) - 1)) - 30*A*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 +
1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 90*B*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(si
n(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 90*C*a^3*(2*(3*sin
(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*
x + c) - 1)) - 360*A*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1))
- 120*B*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 480*A*a^3
*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.81 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (30 \, A a^{3} + 26 \, B a^{3} + 23 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (30 \, A a^{3} + 26 \, B a^{3} + 23 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (450 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 390 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 345 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 2550 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2210 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1955 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5940 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5148 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4554 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7500 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5988 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5814 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5130 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4190 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3165 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1470 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1530 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/240*(15*(30*A*a^3 + 26*B*a^3 + 23*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(30*A*a^3 + 26*B*a^3 + 23*C
*a^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(450*A*a^3*tan(1/2*d*x + 1/2*c)^11 + 390*B*a^3*tan(1/2*d*x + 1/2*
c)^11 + 345*C*a^3*tan(1/2*d*x + 1/2*c)^11 - 2550*A*a^3*tan(1/2*d*x + 1/2*c)^9 - 2210*B*a^3*tan(1/2*d*x + 1/2*c
)^9 - 1955*C*a^3*tan(1/2*d*x + 1/2*c)^9 + 5940*A*a^3*tan(1/2*d*x + 1/2*c)^7 + 5148*B*a^3*tan(1/2*d*x + 1/2*c)^
7 + 4554*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 7500*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 5988*B*a^3*tan(1/2*d*x + 1/2*c)^5
- 5814*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 5130*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 4190*B*a^3*tan(1/2*d*x + 1/2*c)^3 +
3165*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 1470*A*a^3*tan(1/2*d*x + 1/2*c) - 1530*B*a^3*tan(1/2*d*x + 1/2*c) - 1575*C
*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d

Mupad [B] (verification not implemented)

Time = 20.43 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.56 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3\,\mathrm {atanh}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (30\,A+26\,B+23\,C\right )}{4\,\left (\frac {15\,A\,a^3}{2}+\frac {13\,B\,a^3}{2}+\frac {23\,C\,a^3}{4}\right )}\right )\,\left (30\,A+26\,B+23\,C\right )}{8\,d}-\frac {\left (\frac {15\,A\,a^3}{4}+\frac {13\,B\,a^3}{4}+\frac {23\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (-\frac {85\,A\,a^3}{4}-\frac {221\,B\,a^3}{12}-\frac {391\,C\,a^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {99\,A\,a^3}{2}+\frac {429\,B\,a^3}{10}+\frac {759\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-\frac {125\,A\,a^3}{2}-\frac {499\,B\,a^3}{10}-\frac {969\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {171\,A\,a^3}{4}+\frac {419\,B\,a^3}{12}+\frac {211\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-\frac {49\,A\,a^3}{4}-\frac {51\,B\,a^3}{4}-\frac {105\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/cos(c + d*x)^2,x)

[Out]

(a^3*atanh((a^3*tan(c/2 + (d*x)/2)*(30*A + 26*B + 23*C))/(4*((15*A*a^3)/2 + (13*B*a^3)/2 + (23*C*a^3)/4)))*(30
*A + 26*B + 23*C))/(8*d) - (tan(c/2 + (d*x)/2)^11*((15*A*a^3)/4 + (13*B*a^3)/4 + (23*C*a^3)/8) - tan(c/2 + (d*
x)/2)^9*((85*A*a^3)/4 + (221*B*a^3)/12 + (391*C*a^3)/24) + tan(c/2 + (d*x)/2)^3*((171*A*a^3)/4 + (419*B*a^3)/1
2 + (211*C*a^3)/8) + tan(c/2 + (d*x)/2)^7*((99*A*a^3)/2 + (429*B*a^3)/10 + (759*C*a^3)/20) - tan(c/2 + (d*x)/2
)^5*((125*A*a^3)/2 + (499*B*a^3)/10 + (969*C*a^3)/20) - tan(c/2 + (d*x)/2)*((49*A*a^3)/4 + (51*B*a^3)/4 + (105
*C*a^3)/8))/(d*(15*tan(c/2 + (d*x)/2)^4 - 6*tan(c/2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x
)/2)^8 - 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))