\(\int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [440]

Optimal result

Integrand size = 33, antiderivative size = 195 \[ \int (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^4 A x+\frac {a^4 (48 A+35 B+28 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 (40 A+35 B+28 C) \tan (c+d x)}{8 d}+\frac {a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d} \]

[Out]

a^4*A*x+1/8*a^4*(48*A+35*B+28*C)*arctanh(sin(d*x+c))/d+1/8*a^4*(40*A+35*B+28*C)*tan(d*x+c)/d+1/20*a*(5*B+4*C)*
(a+a*sec(d*x+c))^3*tan(d*x+c)/d+1/5*C*(a+a*sec(d*x+c))^4*tan(d*x+c)/d+1/60*(20*A+35*B+28*C)*(a^2+a^2*sec(d*x+c
))^2*tan(d*x+c)/d+1/24*(32*A+35*B+28*C)*(a^4+a^4*sec(d*x+c))*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4139, 4002, 3999, 3852, 8, 3855} \[ \int (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (48 A+35 B+28 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 (40 A+35 B+28 C) \tan (c+d x)}{8 d}+\frac {(32 A+35 B+28 C) \tan (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+a^4 A x+\frac {(20 A+35 B+28 C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{60 d}+\frac {a (5 B+4 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac {C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d} \]

[In]

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

[Out]

a^4*A*x + (a^4*(48*A + 35*B + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(40*A + 35*B + 28*C)*Tan[c + d*x])/(8*
d) + (a*(5*B + 4*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d
) + ((20*A + 35*B + 28*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((32*A + 35*B + 28*C)*(a^4 + a^4*S
ec[c + d*x])*Tan[c + d*x])/(24*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 3855

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

Rule 3999

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[a*c*x, x]
+ (Dist[b*d, Int[Csc[e + f*x]^2, x], x] + Dist[b*c + a*d, Int[Csc[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 4002

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[(-b)
*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Dist[1/m, Int[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*c
*m + (b*c*m + a*d*(2*m - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] &&
GtQ[m, 1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rule 4139

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

Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {\int (a+a \sec (c+d x))^4 (5 a A+a (5 B+4 C) \sec (c+d x)) \, dx}{5 a} \\ & = \frac {a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {\int (a+a \sec (c+d x))^3 \left (20 a^2 A+a^2 (20 A+35 B+28 C) \sec (c+d x)\right ) \, dx}{20 a} \\ & = \frac {a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac {\int (a+a \sec (c+d x))^2 \left (60 a^3 A+5 a^3 (32 A+35 B+28 C) \sec (c+d x)\right ) \, dx}{60 a} \\ & = \frac {a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac {\int (a+a \sec (c+d x)) \left (120 a^4 A+15 a^4 (40 A+35 B+28 C) \sec (c+d x)\right ) \, dx}{120 a} \\ & = a^4 A x+\frac {a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}+\frac {1}{8} \left (a^4 (40 A+35 B+28 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (a^4 (48 A+35 B+28 C)\right ) \int \sec (c+d x) \, dx \\ & = a^4 A x+\frac {a^4 (48 A+35 B+28 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d}-\frac {\left (a^4 (40 A+35 B+28 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{8 d} \\ & = a^4 A x+\frac {a^4 (48 A+35 B+28 C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 (40 A+35 B+28 C) \tan (c+d x)}{8 d}+\frac {a (5 B+4 C) (a+a \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {C (a+a \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \sec (c+d x)\right ) \tan (c+d x)}{24 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.58 \[ \int (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \left (120 A d x+15 (48 A+35 B+28 C) \text {arctanh}(\sin (c+d x))+15 \left (56 A+64 (B+C)+(16 A+27 B+28 C) \sec (c+d x)+2 (B+4 C) \sec ^3(c+d x)\right ) \tan (c+d x)+40 (A+4 B+8 C) \tan ^3(c+d x)+24 C \tan ^5(c+d x)\right )}{120 d} \]

[In]

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

[Out]

(a^4*(120*A*d*x + 15*(48*A + 35*B + 28*C)*ArcTanh[Sin[c + d*x]] + 15*(56*A + 64*(B + C) + (16*A + 27*B + 28*C)
*Sec[c + d*x] + 2*(B + 4*C)*Sec[c + d*x]^3)*Tan[c + d*x] + 40*(A + 4*B + 8*C)*Tan[c + d*x]^3 + 24*C*Tan[c + d*
x]^5))/(120*d)

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.32

method	result	size
parts	\(a^{4} A x +\frac {\left (4 a^{4} A +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (B \,a^{4}+4 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (a^{4} A +4 B \,a^{4}+6 a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+a^{4} C \right ) \tan \left (d x +c \right )}{d}-\frac {a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\)	\(258\)
parallelrisch	\(-\frac {6 a^{4} \left (\left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right ) \left (A +\frac {35 B}{48}+\frac {7 C}{12}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right ) \left (A +\frac {35 B}{48}+\frac {7 C}{12}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {5 d x A \cos \left (3 d x +3 c \right )}{6}-\frac {d x A \cos \left (5 d x +5 c \right )}{6}+\frac {\left (-4 A -\frac {31 B}{4}-11 C \right ) \sin \left (2 d x +2 c \right )}{3}+\frac {\left (-\frac {77 C}{2}-32 A -38 B \right ) \sin \left (3 d x +3 c \right )}{9}+\left (-\frac {9 B}{8}-\frac {7 C}{6}-\frac {2 A}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {\left (-\frac {83 C}{10}-10 A -10 B \right ) \sin \left (5 d x +5 c \right )}{9}-\frac {5 d x A \cos \left (d x +c \right )}{3}-\frac {22 \left (A +\frac {14 B}{11}+\frac {35 C}{22}\right ) \sin \left (d x +c \right )}{9}\right )}{d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\)	\(280\)
norman	\(\frac {a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-a^{4} A x +5 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-10 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+10 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-5 a^{4} A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {a^{4} \left (40 A +35 B +28 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {a^{4} \left (72 A +93 B +100 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{4} \left (272 A +245 B +196 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}-\frac {4 a^{4} \left (295 A +280 B +224 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {a^{4} \left (368 A +395 B +316 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {a^{4} \left (48 A +35 B +28 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {a^{4} \left (48 A +35 B +28 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\)	\(318\)
derivativedivides	\(\frac {-a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+6 a^{4} A \tan \left (d x +c \right )+6 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \tan \left (d x +c \right )+4 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \tan \left (d x +c \right )}{d}\)	\(406\)
default	\(\frac {-a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+6 a^{4} A \tan \left (d x +c \right )+6 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{4} \tan \left (d x +c \right )+4 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} A \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \tan \left (d x +c \right )}{d}\)	\(406\)
risch	\(a^{4} A x -\frac {i a^{4} \left (-664 C -800 A -800 B -720 A \,{\mathrm e}^{8 i \left (d x +c \right )}+480 A \,{\mathrm e}^{7 i \left (d x +c \right )}-480 A \,{\mathrm e}^{3 i \left (d x +c \right )}-240 A \,{\mathrm e}^{i \left (d x +c \right )}-5120 B \,{\mathrm e}^{4 i \left (d x +c \right )}-3120 A \,{\mathrm e}^{6 i \left (d x +c \right )}-1920 C \,{\mathrm e}^{6 i \left (d x +c \right )}-4880 A \,{\mathrm e}^{4 i \left (d x +c \right )}-4720 C \,{\mathrm e}^{4 i \left (d x +c \right )}-3280 A \,{\mathrm e}^{2 i \left (d x +c \right )}-3200 C \,{\mathrm e}^{2 i \left (d x +c \right )}+1320 C \,{\mathrm e}^{7 i \left (d x +c \right )}-1320 C \,{\mathrm e}^{3 i \left (d x +c \right )}-3520 B \,{\mathrm e}^{2 i \left (d x +c \right )}-420 C \,{\mathrm e}^{i \left (d x +c \right )}+240 A \,{\mathrm e}^{9 i \left (d x +c \right )}+420 C \,{\mathrm e}^{9 i \left (d x +c \right )}-120 C \,{\mathrm e}^{8 i \left (d x +c \right )}-405 B \,{\mathrm e}^{i \left (d x +c \right )}-2880 B \,{\mathrm e}^{6 i \left (d x +c \right )}+930 B \,{\mathrm e}^{7 i \left (d x +c \right )}-930 B \,{\mathrm e}^{3 i \left (d x +c \right )}+405 B \,{\mathrm e}^{9 i \left (d x +c \right )}-480 B \,{\mathrm e}^{8 i \left (d x +c \right )}\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{8 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{8 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}\)	\(460\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.01 \[ \int (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {240 \, A a^{4} d x \cos \left (d x + c\right )^{5} + 15 \, {\left (48 \, A + 35 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (48 \, A + 35 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (100 \, A + 100 \, B + 83 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 15 \, {\left (16 \, A + 27 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 20 \, B + 34 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 30 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, C a^{4}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]

[In]

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

[Out]

1/240*(240*A*a^4*d*x*cos(d*x + c)^5 + 15*(48*A + 35*B + 28*C)*a^4*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(4
8*A + 35*B + 28*C)*a^4*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(8*(100*A + 100*B + 83*C)*a^4*cos(d*x + c)^4
+ 15*(16*A + 27*B + 28*C)*a^4*cos(d*x + c)^3 + 8*(5*A + 20*B + 34*C)*a^4*cos(d*x + c)^2 + 30*(B + 4*C)*a^4*cos
(d*x + c) + 24*C*a^4)*sin(d*x + c))/(d*cos(d*x + c)^5)

Sympy [F]

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

[In]

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

[Out]

a**4*(Integral(A, x) + Integral(4*A*sec(c + d*x), x) + Integral(6*A*sec(c + d*x)**2, x) + Integral(4*A*sec(c +
d*x)**3, x) + Integral(A*sec(c + d*x)**4, x) + Integral(B*sec(c + d*x), x) + Integral(4*B*sec(c + d*x)**2, x)
+ Integral(6*B*sec(c + d*x)**3, x) + Integral(4*B*sec(c + d*x)**4, x) + Integral(B*sec(c + d*x)**5, x) + Inte
gral(C*sec(c + d*x)**2, x) + Integral(4*C*sec(c + d*x)**3, x) + Integral(6*C*sec(c + d*x)**4, x) + Integral(4*
C*sec(c + d*x)**5, x) + Integral(C*sec(c + d*x)**6, x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (183) = 366\).

Time = 0.23 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.47 \[ \int (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 240 \, {\left (d x + c\right )} A a^{4} + 320 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 16 \, {\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} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 15 \, B 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 )} - 60 \, 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 )} - 240 \, 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 )} - 360 \, B 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 )} - 240 \, 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 )} + 960 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 240 \, B a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 1440 \, A a^{4} \tan \left (d x + c\right ) + 960 \, B a^{4} \tan \left (d x + c\right ) + 240 \, C a^{4} \tan \left (d x + c\right )}{240 \, d} \]

[In]

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

[Out]

1/240*(80*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 240*(d*x + c)*A*a^4 + 320*(tan(d*x + c)^3 + 3*tan(d*x + c)
)*B*a^4 + 16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 480*(tan(d*x + c)^3 + 3*tan(d*x
+ c))*C*a^4 - 15*B*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)) - 60*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)) - 240*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)) - 360*B*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)) - 240*C*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(s
in(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 960*A*a^4*log(sec(d*x + c) + tan(d*x + c)) + 240*B*a^4*log(sec(d*x
+ c) + tan(d*x + c)) + 1440*A*a^4*tan(d*x + c) + 960*B*a^4*tan(d*x + c) + 240*C*a^4*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.81 \[ \int (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {120 \, {\left (d x + c\right )} A a^{4} + 15 \, {\left (48 \, A a^{4} + 35 \, B a^{4} + 28 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (48 \, A a^{4} + 35 \, B a^{4} + 28 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (600 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 420 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2720 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2450 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1960 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4720 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3584 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3680 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3950 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3160 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1080 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1395 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1500 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]

[In]

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

[Out]

1/120*(120*(d*x + c)*A*a^4 + 15*(48*A*a^4 + 35*B*a^4 + 28*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(48*A
*a^4 + 35*B*a^4 + 28*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(600*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 525*B*a
^4*tan(1/2*d*x + 1/2*c)^9 + 420*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 2720*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 2450*B*a^4*
tan(1/2*d*x + 1/2*c)^7 - 1960*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 4720*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 4480*B*a^4*ta
n(1/2*d*x + 1/2*c)^5 + 3584*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 3680*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 3950*B*a^4*tan(
1/2*d*x + 1/2*c)^3 - 3160*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 1080*A*a^4*tan(1/2*d*x + 1/2*c) + 1395*B*a^4*tan(1/2*
d*x + 1/2*c) + 1500*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^5)/d

Mupad [B] (verification not implemented)

Time = 18.00 (sec) , antiderivative size = 996, normalized size of antiderivative = 5.11 \[ \int (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(30*A*a^4*sin(2*c + 2*d*x) + 80*A*a^4*sin(3*c + 3*d*x) + 15*A*a^4*sin(4*c + 4*d*x) + 25*A*a^4*sin(5*c + 5*d*x)
+ (465*B*a^4*sin(2*c + 2*d*x))/8 + 95*B*a^4*sin(3*c + 3*d*x) + (405*B*a^4*sin(4*c + 4*d*x))/16 + 25*B*a^4*sin
(5*c + 5*d*x) + (165*C*a^4*sin(2*c + 2*d*x))/2 + (385*C*a^4*sin(3*c + 3*d*x))/4 + (105*C*a^4*sin(4*c + 4*d*x))
/4 + (83*C*a^4*sin(5*c + 5*d*x))/4 + 55*A*a^4*sin(c + d*x) + 70*B*a^4*sin(c + d*x) + (175*C*a^4*sin(c + d*x))/
2 + (75*A*a^4*atan((2368*A^2*sin(c/2 + (d*x)/2) + 1225*B^2*sin(c/2 + (d*x)/2) + 784*C^2*sin(c/2 + (d*x)/2) + 3
360*A*B*sin(c/2 + (d*x)/2) + 2688*A*C*sin(c/2 + (d*x)/2) + 1960*B*C*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(2
368*A^2 + 1225*B^2 + 784*C^2 + 3360*A*B + 2688*A*C + 1960*B*C)))*cos(3*c + 3*d*x))/2 + (15*A*a^4*atan((2368*A^
2*sin(c/2 + (d*x)/2) + 1225*B^2*sin(c/2 + (d*x)/2) + 784*C^2*sin(c/2 + (d*x)/2) + 3360*A*B*sin(c/2 + (d*x)/2)
+ 2688*A*C*sin(c/2 + (d*x)/2) + 1960*B*C*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(2368*A^2 + 1225*B^2 + 784*C^
2 + 3360*A*B + 2688*A*C + 1960*B*C)))*cos(5*c + 5*d*x))/2 + 450*A*a^4*cos(c + d*x)*atanh(sin(c/2 + (d*x)/2)/co
s(c/2 + (d*x)/2)) + (2625*B*a^4*cos(c + d*x)*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/8 + (525*C*a^4*cos(
c + d*x)*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + 225*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/
2))*cos(3*c + 3*d*x) + 45*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(5*c + 5*d*x) + (2625*B*a^4*at
anh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/16 + (525*B*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2
+ (d*x)/2))*cos(5*c + 5*d*x))/16 + (525*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x))/4
+ (105*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(5*c + 5*d*x))/4 + 75*A*a^4*atan((2368*A^2*sin(c
/2 + (d*x)/2) + 1225*B^2*sin(c/2 + (d*x)/2) + 784*C^2*sin(c/2 + (d*x)/2) + 3360*A*B*sin(c/2 + (d*x)/2) + 2688*
A*C*sin(c/2 + (d*x)/2) + 1960*B*C*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(2368*A^2 + 1225*B^2 + 784*C^2 + 336
0*A*B + 2688*A*C + 1960*B*C)))*cos(c + d*x))/(60*d*((5*cos(c + d*x))/8 + (5*cos(3*c + 3*d*x))/16 + cos(5*c + 5
*d*x)/16))