\(\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+C \sec ^2(c+d x)) \, dx\) [102]

Optimal result

Integrand size = 33, antiderivative size = 197 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (30 A+23 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (30 A+23 C) \tan (c+d x)}{10 d}+\frac {3 a^3 (30 A+23 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {(30 A+7 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{120 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {a^3 (30 A+23 C) \tan ^3(c+d x)}{120 d} \]

[Out]

1/16*a^3*(30*A+23*C)*arctanh(sin(d*x+c))/d+1/10*a^3*(30*A+23*C)*tan(d*x+c)/d+3/80*a^3*(30*A+23*C)*sec(d*x+c)*t
an(d*x+c)/d+1/120*(30*A+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*C*(a+a*sec(d*x+c))^4*tan(d*x+c)/a/d+1/120*a^3*(30*A+23*C)*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4174, 4095, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (30 A+23 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (30 A+23 C) \tan ^3(c+d x)}{120 d}+\frac {a^3 (30 A+23 C) \tan (c+d x)}{10 d}+\frac {3 a^3 (30 A+23 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac {(30 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{120 d}+\frac {C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{10 a d} \]

[In]

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

[Out]

(a^3*(30*A + 23*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(30*A + 23*C)*Tan[c + d*x])/(10*d) + (3*a^3*(30*A + 23
*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + ((30*A + 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) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(10*a*d) + (a^3*
(30*A + 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 4174

Int[((A_.) + 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*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, 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 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 {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 C+a^2 (30 A+7 C) \sec (c+d x)\right ) \, dx}{30 a^2} \\ & = \frac {(30 A+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 {C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {1}{40} (30 A+23 C) \int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx \\ & = \frac {(30 A+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 {C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {1}{40} (30 A+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+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 {C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {1}{40} \left (a^3 (30 A+23 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{40} \left (a^3 (30 A+23 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{40} \left (3 a^3 (30 A+23 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{40} \left (3 a^3 (30 A+23 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^3 (30 A+23 C) \text {arctanh}(\sin (c+d x))}{40 d}+\frac {3 a^3 (30 A+23 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {(30 A+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 {C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {1}{80} \left (3 a^3 (30 A+23 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^3 (30 A+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+23 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{40 d} \\ & = \frac {a^3 (30 A+23 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (30 A+23 C) \tan (c+d x)}{10 d}+\frac {3 a^3 (30 A+23 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {(30 A+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 {C (a+a \sec (c+d x))^4 \tan (c+d x)}{10 a d}+\frac {a^3 (30 A+23 C) \tan ^3(c+d x)}{120 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.77 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.57 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \left (15 (30 A+23 C) \text {arctanh}(\sin (c+d x))+\left (720 A+544 C+15 (30 A+23 C) \sec (c+d x)+16 (15 A+17 C) \sec ^2(c+d x)+10 (6 A+23 C) \sec ^3(c+d x)+144 C \sec ^4(c+d x)+40 C \sec ^5(c+d x)\right ) \tan (c+d x)\right )}{240 d} \]

[In]

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

[Out]

(a^3*(15*(30*A + 23*C)*ArcTanh[Sin[c + d*x]] + (720*A + 544*C + 15*(30*A + 23*C)*Sec[c + d*x] + 16*(15*A + 17*
C)*Sec[c + d*x]^2 + 10*(6*A + 23*C)*Sec[c + d*x]^3 + 144*C*Sec[c + d*x]^4 + 40*C*Sec[c + d*x]^5)*Tan[c + d*x])
)/(240*d)

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.15

method	result	size
norman	\(\frac {\frac {7 a^{3} \left (14 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {33 a^{3} \left (30 A +23 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {17 a^{3} \left (30 A +23 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {a^{3} \left (30 A +23 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}-\frac {a^{3} \left (342 A +211 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 d}+\frac {a^{3} \left (1250 A +969 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {a^{3} \left (30 A +23 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {a^{3} \left (30 A +23 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\)	\(227\)
parallelrisch	\(\frac {19 a^{3} \left (-\frac {225 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {23 C}{30}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{76}+\frac {225 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {23 C}{30}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{76}+\left (\frac {46 A}{19}+\frac {60 C}{19}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {53 A}{38}+\frac {391 C}{228}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {32 A}{19}+\frac {136 C}{95}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {15 A}{38}+\frac {23 C}{76}\right ) \sin \left (5 d x +5 c \right )+\left (\frac {6 A}{19}+\frac {68 C}{285}\right ) \sin \left (6 d x +6 c \right )+\sin \left (d x +c \right ) \left (\frac {75 C}{38}+A \right )\right )}{2 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(246\)
parts	\(\frac {\left (a^{3} A +3 a^{3} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 a^{3} A +a^{3} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{3} A \tan \left (d x +c \right )}{d}+\frac {a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {3 a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(250\)
derivativedivides	\(\frac {a^{3} A \tan \left (d x +c \right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-3 a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{3} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(294\)
default	\(\frac {a^{3} A \tan \left (d x +c \right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-3 a^{3} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{3} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(294\)
risch	\(-\frac {i a^{3} \left (450 A \,{\mathrm e}^{11 i \left (d x +c \right )}+345 C \,{\mathrm e}^{11 i \left (d x +c \right )}-240 A \,{\mathrm e}^{10 i \left (d x +c \right )}+1590 A \,{\mathrm e}^{9 i \left (d x +c \right )}+1955 C \,{\mathrm e}^{9 i \left (d x +c \right )}-2640 A \,{\mathrm e}^{8 i \left (d x +c \right )}-480 C \,{\mathrm e}^{8 i \left (d x +c \right )}+1140 A \,{\mathrm e}^{7 i \left (d x +c \right )}+2250 C \,{\mathrm e}^{7 i \left (d x +c \right )}-7200 A \,{\mathrm e}^{6 i \left (d x +c \right )}-5440 C \,{\mathrm e}^{6 i \left (d x +c \right )}-1140 A \,{\mathrm e}^{5 i \left (d x +c \right )}-2250 C \,{\mathrm e}^{5 i \left (d x +c \right )}-8160 A \,{\mathrm e}^{4 i \left (d x +c \right )}-7680 C \,{\mathrm e}^{4 i \left (d x +c \right )}-1590 A \,{\mathrm e}^{3 i \left (d x +c \right )}-1955 C \,{\mathrm e}^{3 i \left (d x +c \right )}-4080 A \,{\mathrm e}^{2 i \left (d x +c \right )}-3264 C \,{\mathrm e}^{2 i \left (d x +c \right )}-450 A \,{\mathrm e}^{i \left (d x +c \right )}-345 C \,{\mathrm e}^{i \left (d x +c \right )}-720 A -544 C \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}\)	\(371\)

[In]

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

[Out]

(7/8*a^3*(14*A+15*C)/d*tan(1/2*d*x+1/2*c)-33/20*a^3*(30*A+23*C)/d*tan(1/2*d*x+1/2*c)^7+17/24*a^3*(30*A+23*C)/d
*tan(1/2*d*x+1/2*c)^9-1/8*a^3*(30*A+23*C)/d*tan(1/2*d*x+1/2*c)^11-1/8*a^3*(342*A+211*C)/d*tan(1/2*d*x+1/2*c)^3
+1/20*a^3*(1250*A+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+23*C)/d*ln(tan(1/2*
d*x+1/2*c)-1)+1/16*a^3*(30*A+23*C)/d*ln(tan(1/2*d*x+1/2*c)+1)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.92 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (30 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (30 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (45 \, A + 34 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \, {\left (30 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \, {\left (15 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \, {\left (6 \, A + 23 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 144 \, C a^{3} \cos \left (d x + c\right ) + 40 \, C a^{3}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]

[In]

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

[Out]

1/480*(15*(30*A + 23*C)*a^3*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 15*(30*A + 23*C)*a^3*cos(d*x + c)^6*log(-si
n(d*x + c) + 1) + 2*(16*(45*A + 34*C)*a^3*cos(d*x + c)^5 + 15*(30*A + 23*C)*a^3*cos(d*x + c)^4 + 16*(15*A + 17
*C)*a^3*cos(d*x + c)^3 + 10*(6*A + 23*C)*a^3*cos(d*x + c)^2 + 144*C*a^3*cos(d*x + c) + 40*C*a^3)*sin(d*x + c))
/(d*cos(d*x + c)^6)

Sympy [F]

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

[In]

integrate(sec(d*x+c)**2*(a+a*sec(d*x+c))**3*(A+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(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. 382 vs. \(2 (183) = 366\).

Time = 0.20 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.94 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{3} + 160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 5 \, C a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{3} \tan \left (d x + c\right )}{480 \, d} \]

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^3*(A+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 + 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 + 3
3*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*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)) + 480*A*a^3*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.42 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (30 \, A a^{3} + 23 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (30 \, A a^{3} + 23 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (450 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 345 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 2550 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1955 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5940 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4554 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7500 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5814 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5130 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3165 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1470 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]

[In]

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

[Out]

1/240*(15*(30*A*a^3 + 23*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(30*A*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 + 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 - 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 + 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 - 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 + 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) - 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 = 17.96 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.33 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (-\frac {15\,A\,a^3}{4}-\frac {23\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {85\,A\,a^3}{4}+\frac {391\,C\,a^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {99\,A\,a^3}{2}-\frac {759\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {125\,A\,a^3}{2}+\frac {969\,C\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {171\,A\,a^3}{4}-\frac {211\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {49\,A\,a^3}{4}+\frac {105\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (30\,A+23\,C\right )}{8\,d} \]

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*((49*A*a^3)/4 + (105*C*a^3)/8) - tan(c/2 + (d*x)/2)^11*((15*A*a^3)/4 + (23*C*a^3)/8) - tan
(c/2 + (d*x)/2)^3*((171*A*a^3)/4 + (211*C*a^3)/8) + tan(c/2 + (d*x)/2)^9*((85*A*a^3)/4 + (391*C*a^3)/24) - tan
(c/2 + (d*x)/2)^7*((99*A*a^3)/2 + (759*C*a^3)/20) + tan(c/2 + (d*x)/2)^5*((125*A*a^3)/2 + (969*C*a^3)/20))/(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)) + (a^3*atanh(tan(c/2 + (d*x)/2))*(30*A + 23*C))/(8*d)