\(\int \sec (c+d x) (a+a \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [112]

Optimal result

Integrand size = 31, antiderivative size = 188 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {7 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d} \]

[Out]

7/16*a^4*(10*A+7*C)*arctanh(sin(d*x+c))/d+4/5*a^4*(10*A+7*C)*tan(d*x+c)/d+27/80*a^4*(10*A+7*C)*sec(d*x+c)*tan(
d*x+c)/d+1/40*a^4*(10*A+7*C)*sec(d*x+c)^3*tan(d*x+c)/d-1/30*C*(a+a*sec(d*x+c))^4*tan(d*x+c)/d+1/6*C*(a+a*sec(d
*x+c))^5*tan(d*x+c)/a/d+2/15*a^4*(10*A+7*C)*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4168, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {7 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {a^4 (10 A+7 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac {27 a^4 (10 A+7 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}-\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d} \]

[In]

Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]

[Out]

(7*a^4*(10*A + 7*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(10*A + 7*C)*Tan[c + d*x])/(5*d) + (27*a^4*(10*A +
7*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + (a^4*(10*A + 7*C)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) - (C*(a + a*Sec
[c + d*x])^4*Tan[c + d*x])/(30*d) + (C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + (2*a^4*(10*A + 7*C)*Tan[
c + d*x]^3)/(15*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 4168

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + 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) - a*C*Csc[e + f*x], x], x], x] /; Fre
eQ[{a, b, e, f, A, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 (a (6 A+5 C)-a C \sec (c+d x)) \, dx}{6 a} \\ & = -\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} (10 A+7 C) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx \\ & = -\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} (10 A+7 C) \int \left (a^4 \sec (c+d x)+4 a^4 \sec ^2(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^4(c+d x)+a^4 \sec ^5(c+d x)\right ) \, dx \\ & = -\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{10} \left (a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{10} \left (a^4 (10 A+7 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+7 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+7 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (10 A+7 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{10 d}+\frac {3 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{10 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {1}{40} \left (3 a^4 (10 A+7 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (2 a^4 (10 A+7 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{5 d}-\frac {\left (2 a^4 (10 A+7 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d} \\ & = \frac {2 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{5 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}+\frac {1}{80} \left (3 a^4 (10 A+7 C)\right ) \int \sec (c+d x) \, dx \\ & = \frac {7 a^4 (10 A+7 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \sec (c+d x) \tan (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \sec ^3(c+d x) \tan (c+d x)}{40 d}-\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{30 d}+\frac {C (a+a \sec (c+d x))^5 \tan (c+d x)}{6 a d}+\frac {2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.29 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {35 a^4 A \text {arctanh}(\sin (c+d x))}{8 d}+\frac {49 a^4 C \text {arctanh}(\sin (c+d x))}{16 d}+\frac {8 a^4 A \tan (c+d x)}{d}+\frac {8 a^4 C \tan (c+d x)}{d}+\frac {27 a^4 A \sec (c+d x) \tan (c+d x)}{8 d}+\frac {49 a^4 C \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^4 A \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {41 a^4 C \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 C \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 A \tan ^3(c+d x)}{3 d}+\frac {4 a^4 C \tan ^3(c+d x)}{d}+\frac {4 a^4 C \tan ^5(c+d x)}{5 d} \]

[In]

Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]

[Out]

(35*a^4*A*ArcTanh[Sin[c + d*x]])/(8*d) + (49*a^4*C*ArcTanh[Sin[c + d*x]])/(16*d) + (8*a^4*A*Tan[c + d*x])/d +
(8*a^4*C*Tan[c + d*x])/d + (27*a^4*A*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (49*a^4*C*Sec[c + d*x]*Tan[c + d*x])/(
16*d) + (a^4*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (41*a^4*C*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (a^4*C*Sec
[c + d*x]^5*Tan[c + d*x])/(6*d) + (4*a^4*A*Tan[c + d*x]^3)/(3*d) + (4*a^4*C*Tan[c + d*x]^3)/d + (4*a^4*C*Tan[c
+ d*x]^5)/(5*d)

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.21

method	result	size
norman	\(\frac {\frac {281 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {231 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {119 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {7 a^{4} \left (10 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {3 a^{4} \left (62 A +69 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{4} \left (2138 A +1471 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {7 a^{4} \left (10 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {7 a^{4} \left (10 A +7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\)	\(227\)
parallelrisch	\(\frac {31 a^{4} \left (-\frac {525 \left (A +\frac {7 C}{10}\right ) \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{124}+\frac {525 \left (A +\frac {7 C}{10}\right ) \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{124}+\left (\frac {88 A}{31}+\frac {112 C}{31}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {89 A}{62}+\frac {769 C}{372}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {64 A}{31}+\frac {288 C}{155}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {27 A}{62}+\frac {49 C}{124}\right ) \sin \left (5 d x +5 c \right )+\left (\frac {40 A}{93}+\frac {48 C}{155}\right ) \sin \left (6 d x +6 c \right )+\sin \left (d x +c \right ) \left (\frac {125 C}{62}+A \right )\right )}{2 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(246\)
parts	\(\frac {\left (a^{4} A +6 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 a^{4} A +4 a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}+\frac {a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {4 a^{4} A \tan \left (d x +c \right )}{d}-\frac {4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(281\)
derivativedivides	\(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(350\)
default	\(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} A \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(350\)
risch	\(-\frac {i a^{4} \left (810 A \,{\mathrm e}^{11 i \left (d x +c \right )}+735 C \,{\mathrm e}^{11 i \left (d x +c \right )}-960 A \,{\mathrm e}^{10 i \left (d x +c \right )}+2670 A \,{\mathrm e}^{9 i \left (d x +c \right )}+3845 C \,{\mathrm e}^{9 i \left (d x +c \right )}-6720 A \,{\mathrm e}^{8 i \left (d x +c \right )}-1920 C \,{\mathrm e}^{8 i \left (d x +c \right )}+1860 A \,{\mathrm e}^{7 i \left (d x +c \right )}+3750 C \,{\mathrm e}^{7 i \left (d x +c \right )}-16000 A \,{\mathrm e}^{6 i \left (d x +c \right )}-11520 C \,{\mathrm e}^{6 i \left (d x +c \right )}-1860 A \,{\mathrm e}^{5 i \left (d x +c \right )}-3750 C \,{\mathrm e}^{5 i \left (d x +c \right )}-17280 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15360 C \,{\mathrm e}^{4 i \left (d x +c \right )}-2670 A \,{\mathrm e}^{3 i \left (d x +c \right )}-3845 C \,{\mathrm e}^{3 i \left (d x +c \right )}-8640 A \,{\mathrm e}^{2 i \left (d x +c \right )}-6912 C \,{\mathrm e}^{2 i \left (d x +c \right )}-810 A \,{\mathrm e}^{i \left (d x +c \right )}-735 C \,{\mathrm e}^{i \left (d x +c \right )}-1600 A -1152 C \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}\)	\(371\)

[In]

int(sec(d*x+c)*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

(281/20*a^4*(10*A+7*C)/d*tan(1/2*d*x+1/2*c)^5-231/20*a^4*(10*A+7*C)/d*tan(1/2*d*x+1/2*c)^7+119/24*a^4*(10*A+7*
C)/d*tan(1/2*d*x+1/2*c)^9-7/8*a^4*(10*A+7*C)/d*tan(1/2*d*x+1/2*c)^11+3/8*a^4*(62*A+69*C)/d*tan(1/2*d*x+1/2*c)-
1/24*a^4*(2138*A+1471*C)/d*tan(1/2*d*x+1/2*c)^3)/(tan(1/2*d*x+1/2*c)^2-1)^6-7/16*a^4*(10*A+7*C)/d*ln(tan(1/2*d
*x+1/2*c)-1)+7/16*a^4*(10*A+7*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 = 0.96 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (10 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (64 \, {\left (25 \, A + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (54 \, A + 49 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 64 \, {\left (5 \, A + 9 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (6 \, A + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 192 \, C a^{4} \cos \left (d x + c\right ) + 40 \, C a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]

[In]

integrate(sec(d*x+c)*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/480*(105*(10*A + 7*C)*a^4*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 105*(10*A + 7*C)*a^4*cos(d*x + c)^6*log(-si
n(d*x + c) + 1) + 2*(64*(25*A + 18*C)*a^4*cos(d*x + c)^5 + 15*(54*A + 49*C)*a^4*cos(d*x + c)^4 + 64*(5*A + 9*C
)*a^4*cos(d*x + c)^3 + 10*(6*A + 41*C)*a^4*cos(d*x + c)^2 + 192*C*a^4*cos(d*x + c) + 40*C*a^4)*sin(d*x + c))/(
d*cos(d*x + c)^6)

Sympy [F]

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

[In]

integrate(sec(d*x+c)*(a+a*sec(d*x+c))**4*(A+C*sec(d*x+c)**2),x)

[Out]

a**4*(Integral(A*sec(c + d*x), x) + Integral(4*A*sec(c + d*x)**2, x) + Integral(6*A*sec(c + d*x)**3, x) + Inte
gral(4*A*sec(c + d*x)**4, x) + Integral(A*sec(c + d*x)**5, x) + Integral(C*sec(c + d*x)**3, x) + Integral(4*C*
sec(c + d*x)**4, x) + Integral(6*C*sec(c + d*x)**5, x) + Integral(4*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. 449 vs. \(2 (174) = 348\).

Time = 0.21 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.39 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 128 \, {\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^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 5 \, C a^{4} {\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^{4} {\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 )} - 180 \, C a^{4} {\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 )} - 720 \, A a^{4} {\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 \, C a^{4} {\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^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 1920 \, A a^{4} \tan \left (d x + c\right )}{480 \, d} \]

[In]

integrate(sec(d*x+c)*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*(640*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x +
c))*C*a^4 + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 - 5*C*a^4*(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*lo
g(sin(d*x + c) - 1)) - 30*A*a^4*(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)) - 180*C*a^4*(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)) - 720*A*a^4*(2*sin(d*x
+ c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 120*C*a^4*(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^4*log(sec(d*x + c) + tan(d*x + c)) +
1920*A*a^4*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.49 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (10 \, A a^{4} + 7 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1050 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 735 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 5950 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4165 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9702 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 16860 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11802 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 10690 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7355 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2790 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]

[In]

integrate(sec(d*x+c)*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/240*(105*(10*A*a^4 + 7*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(10*A*a^4 + 7*C*a^4)*log(abs(tan(1/2*
d*x + 1/2*c) - 1)) - 2*(1050*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 735*C*a^4*tan(1/2*d*x + 1/2*c)^11 - 5950*A*a^4*ta
n(1/2*d*x + 1/2*c)^9 - 4165*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 13860*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 9702*C*a^4*tan
(1/2*d*x + 1/2*c)^7 - 16860*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 11802*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 10690*A*a^4*ta
n(1/2*d*x + 1/2*c)^3 + 7355*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 2790*A*a^4*tan(1/2*d*x + 1/2*c) - 3105*C*a^4*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 = 18.20 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.39 \[ \int \sec (c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (-\frac {35\,A\,a^4}{4}-\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {595\,A\,a^4}{12}+\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {231\,A\,a^4}{2}-\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {281\,A\,a^4}{2}+\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {1069\,A\,a^4}{12}-\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (10\,A+7\,C\right )}{8\,d} \]

[In]

int(((A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^4)/cos(c + d*x),x)

[Out]

(tan(c/2 + (d*x)/2)*((93*A*a^4)/4 + (207*C*a^4)/8) - tan(c/2 + (d*x)/2)^11*((35*A*a^4)/4 + (49*C*a^4)/8) + tan
(c/2 + (d*x)/2)^9*((595*A*a^4)/12 + (833*C*a^4)/24) - tan(c/2 + (d*x)/2)^7*((231*A*a^4)/2 + (1617*C*a^4)/20) +
tan(c/2 + (d*x)/2)^5*((281*A*a^4)/2 + (1967*C*a^4)/20) - tan(c/2 + (d*x)/2)^3*((1069*A*a^4)/12 + (1471*C*a^4)
/24))/(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)) + (7*a^4*atanh(tan(c/2 + (d*x)/2))*(10*A + 7*C))/(8*d
)