\(\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [438]

Optimal result

Integrand size = 41, antiderivative size = 252 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d} \]

[Out]

1/16*a^4*(56*A+49*B+44*C)*arctanh(sin(d*x+c))/d+4/35*a^4*(56*A+49*B+44*C)*tan(d*x+c)/d+27/560*a^4*(56*A+49*B+4
4*C)*sec(d*x+c)*tan(d*x+c)/d+1/280*a^4*(56*A+49*B+44*C)*sec(d*x+c)^3*tan(d*x+c)/d+1/210*(42*A-7*B+8*C)*(a+a*se
c(d*x+c))^4*tan(d*x+c)/d+1/7*C*sec(d*x+c)^2*(a+a*sec(d*x+c))^4*tan(d*x+c)/d+1/42*(7*B+4*C)*(a+a*sec(d*x+c))^5*
tan(d*x+c)/a/d+2/105*a^4*(56*A+49*B+44*C)*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {a^4 (56 A+49 B+44 C) \tan (c+d x) \sec ^3(c+d x)}{280 d}+\frac {27 a^4 (56 A+49 B+44 C) \tan (c+d x) \sec (c+d x)}{560 d}+\frac {(42 A-7 B+8 C) \tan (c+d x) (a \sec (c+d x)+a)^4}{210 d}+\frac {(7 B+4 C) \tan (c+d x) (a \sec (c+d x)+a)^5}{42 a d}+\frac {C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^4}{7 d} \]

[In]

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

[Out]

(a^4*(56*A + 49*B + 44*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(56*A + 49*B + 44*C)*Tan[c + d*x])/(35*d) + (
27*a^4*(56*A + 49*B + 44*C)*Sec[c + d*x]*Tan[c + d*x])/(560*d) + (a^4*(56*A + 49*B + 44*C)*Sec[c + d*x]^3*Tan[
c + d*x])/(280*d) + ((42*A - 7*B + 8*C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(210*d) + (C*Sec[c + d*x]^2*(a +
a*Sec[c + d*x])^4*Tan[c + d*x])/(7*d) + ((7*B + 4*C)*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(42*a*d) + (2*a^4*(5
6*A + 49*B + 44*C)*Tan[c + d*x]^3)/(105*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))^4 \tan (c+d x)}{7 d}+\frac {\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (a (7 A+2 C)+a (7 B+4 C) \sec (c+d x)) \, dx}{7 a} \\ & = \frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 \left (5 a^2 (7 B+4 C)+a^2 (42 A-7 B+8 C) \sec (c+d x)\right ) \, dx}{42 a^2} \\ & = \frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} (56 A+49 B+44 C) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx \\ & = \frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} (56 A+49 B+44 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 {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{70} \left (a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{70} \left (a^4 (56 A+49 B+44 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{35} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{70 d}+\frac {3 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{70 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {1}{280} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{70} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (2 a^4 (56 A+49 B+44 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{35 d}-\frac {\left (2 a^4 (56 A+49 B+44 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d} \\ & = \frac {2 a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{35 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d}+\frac {1}{560} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \sec (c+d x) \, dx \\ & = \frac {a^4 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \sec (c+d x) \tan (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{210 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \sec (c+d x))^5 \tan (c+d x)}{42 a d}+\frac {2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.61 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \left (105 (56 A+49 B+44 C) \text {arctanh}(\sin (c+d x))+\left (16 (581 A+504 B+454 C)+105 (56 A+49 B+44 C) \sec (c+d x)+16 (238 A+252 B+227 C) \sec ^2(c+d x)+70 (24 A+41 B+44 C) \sec ^3(c+d x)+48 (7 A+28 B+48 C) \sec ^4(c+d x)+280 (B+4 C) \sec ^5(c+d x)+240 C \sec ^6(c+d x)\right ) \tan (c+d x)\right )}{1680 d} \]

[In]

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

[Out]

(a^4*(105*(56*A + 49*B + 44*C)*ArcTanh[Sin[c + d*x]] + (16*(581*A + 504*B + 454*C) + 105*(56*A + 49*B + 44*C)*
Sec[c + d*x] + 16*(238*A + 252*B + 227*C)*Sec[c + d*x]^2 + 70*(24*A + 41*B + 44*C)*Sec[c + d*x]^3 + 48*(7*A +
28*B + 48*C)*Sec[c + d*x]^4 + 280*(B + 4*C)*Sec[c + d*x]^5 + 240*C*Sec[c + d*x]^6)*Tan[c + d*x]))/(1680*d)

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.11

method	result	size
norman	\(\frac {\frac {128 a^{4} \left (56 A +49 B +44 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {283 a^{4} \left (56 A +49 B +44 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}+\frac {5 a^{4} \left (56 A +49 B +44 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}-\frac {a^{4} \left (56 A +49 B +44 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}-\frac {a^{4} \left (200 A +207 B +212 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{4} \left (616 A +523 B +420 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}-\frac {a^{4} \left (22808 A +19157 B +18012 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}-\frac {a^{4} \left (56 A +49 B +44 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {a^{4} \left (56 A +49 B +44 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\)	\(280\)
parallelrisch	\(-\frac {7 a^{4} \left (\left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right ) \left (A +\frac {7 B}{8}+\frac {11 C}{14}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right ) \left (A +\frac {7 B}{8}+\frac {11 C}{14}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {67 C}{21}-\frac {102 A}{49}-\frac {31 B}{12}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {848 B}{245}-\frac {124 C}{35}-\frac {802 A}{245}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {220 C}{147}-\frac {72 A}{49}-\frac {229 B}{147}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {48 B}{35}-\frac {1102 A}{735}-\frac {908 C}{735}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {B}{4}-\frac {2 A}{7}-\frac {11 C}{49}\right ) \sin \left (6 d x +6 c \right )+\left (-\frac {908 C}{5145}-\frac {166 A}{735}-\frac {48 B}{245}\right ) \sin \left (7 d x +7 c \right )-2 \sin \left (d x +c \right ) \left (A +\frac {8 B}{7}+\frac {10 C}{7}\right )\right )}{2 d \left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right )}\)	\(311\)
parts	\(\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (B \,a^{4}+4 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (a^{4} A +4 B \,a^{4}+6 a^{4} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 a^{4} A +4 B \,a^{4}+a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} A \tan \left (d x +c \right )}{d}-\frac {a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\)	\(339\)
risch	\(-\frac {i a^{4} \left (-7264 C -9296 A -8064 B -121520 A \,{\mathrm e}^{8 i \left (d x +c \right )}-42840 A \,{\mathrm e}^{5 i \left (d x +c \right )}-30240 A \,{\mathrm e}^{3 i \left (d x +c \right )}-5880 A \,{\mathrm e}^{i \left (d x +c \right )}-155904 B \,{\mathrm e}^{4 i \left (d x +c \right )}-203840 A \,{\mathrm e}^{6 i \left (d x +c \right )}-185920 C \,{\mathrm e}^{6 i \left (d x +c \right )}-164976 A \,{\mathrm e}^{4 i \left (d x +c \right )}-149184 C \,{\mathrm e}^{4 i \left (d x +c \right )}-63392 A \,{\mathrm e}^{2 i \left (d x +c \right )}-50848 C \,{\mathrm e}^{2 i \left (d x +c \right )}-65660 C \,{\mathrm e}^{5 i \left (d x +c \right )}-30800 C \,{\mathrm e}^{3 i \left (d x +c \right )}-56448 B \,{\mathrm e}^{2 i \left (d x +c \right )}-4620 C \,{\mathrm e}^{i \left (d x +c \right )}-1680 A \,{\mathrm e}^{12 i \left (d x +c \right )}+4620 C \,{\mathrm e}^{13 i \left (d x +c \right )}+5880 A \,{\mathrm e}^{13 i \left (d x +c \right )}-3360 C \,{\mathrm e}^{10 i \left (d x +c \right )}+30240 A \,{\mathrm e}^{11 i \left (d x +c \right )}+30800 C \,{\mathrm e}^{11 i \left (d x +c \right )}-30240 A \,{\mathrm e}^{10 i \left (d x +c \right )}+42840 A \,{\mathrm e}^{9 i \left (d x +c \right )}+65660 C \,{\mathrm e}^{9 i \left (d x +c \right )}-68320 C \,{\mathrm e}^{8 i \left (d x +c \right )}-5145 B \,{\mathrm e}^{i \left (d x +c \right )}-53165 B \,{\mathrm e}^{5 i \left (d x +c \right )}-188160 B \,{\mathrm e}^{6 i \left (d x +c \right )}+32060 B \,{\mathrm e}^{11 i \left (d x +c \right )}+5145 B \,{\mathrm e}^{13 i \left (d x +c \right )}-32060 B \,{\mathrm e}^{3 i \left (d x +c \right )}-13440 B \,{\mathrm e}^{10 i \left (d x +c \right )}+53165 B \,{\mathrm e}^{9 i \left (d x +c \right )}-94080 B \,{\mathrm e}^{8 i \left (d x +c \right )}\right )}{840 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{16 d}+\frac {11 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{4 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{16 d}-\frac {11 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{4 d}\)	\(574\)
derivativedivides	\(\frac {-a^{4} A \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 )+B \,a^{4} \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 )-a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+4 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 )-4 B \,a^{4} \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} 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 )-6 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 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 )-6 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 )+a^{4} A \tan \left (d x +c \right )+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 )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(577\)
default	\(\frac {-a^{4} A \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 )+B \,a^{4} \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 )-a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+4 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 )-4 B \,a^{4} \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} 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 )-6 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 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 )-6 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 )+a^{4} A \tan \left (d x +c \right )+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 )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(577\)

[In]

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

[Out]

(128/35*a^4*(56*A+49*B+44*C)/d*tan(1/2*d*x+1/2*c)^7-283/120*a^4*(56*A+49*B+44*C)/d*tan(1/2*d*x+1/2*c)^9+5/6*a^
4*(56*A+49*B+44*C)/d*tan(1/2*d*x+1/2*c)^11-1/8*a^4*(56*A+49*B+44*C)/d*tan(1/2*d*x+1/2*c)^13-1/8*a^4*(200*A+207
*B+212*C)/d*tan(1/2*d*x+1/2*c)+1/6*a^4*(616*A+523*B+420*C)/d*tan(1/2*d*x+1/2*c)^3-1/120*a^4*(22808*A+19157*B+1
8012*C)/d*tan(1/2*d*x+1/2*c)^5)/(tan(1/2*d*x+1/2*c)^2-1)^7-1/16*a^4*(56*A+49*B+44*C)/d*ln(tan(1/2*d*x+1/2*c)-1
)+1/16*a^4*(56*A+49*B+44*C)/d*ln(tan(1/2*d*x+1/2*c)+1)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.90 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (581 \, A + 504 \, B + 454 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 16 \, {\left (238 \, A + 252 \, B + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, A + 41 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 48 \, {\left (7 \, A + 28 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 240 \, C a^{4}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]

[In]

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

[Out]

1/3360*(105*(56*A + 49*B + 44*C)*a^4*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 105*(56*A + 49*B + 44*C)*a^4*cos(d
*x + c)^7*log(-sin(d*x + c) + 1) + 2*(16*(581*A + 504*B + 454*C)*a^4*cos(d*x + c)^6 + 105*(56*A + 49*B + 44*C)
*a^4*cos(d*x + c)^5 + 16*(238*A + 252*B + 227*C)*a^4*cos(d*x + c)^4 + 70*(24*A + 41*B + 44*C)*a^4*cos(d*x + c)
^3 + 48*(7*A + 28*B + 48*C)*a^4*cos(d*x + c)^2 + 280*(B + 4*C)*a^4*cos(d*x + c) + 240*C*a^4)*sin(d*x + c))/(d*
cos(d*x + c)^7)

Sympy [F]

\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{6}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{8}{\left (c + d x \right )}\, dx\right ) \]

[In]

integrate(sec(d*x+c)**2*(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*sec(c + d*x)**2, x) + Integral(4*A*sec(c + d*x)**3, x) + Integral(6*A*sec(c + d*x)**4, x) + I
ntegral(4*A*sec(c + d*x)**5, x) + Integral(A*sec(c + d*x)**6, x) + Integral(B*sec(c + d*x)**3, x) + Integral(4
*B*sec(c + d*x)**4, x) + Integral(6*B*sec(c + d*x)**5, x) + Integral(4*B*sec(c + d*x)**6, x) + Integral(B*sec(
c + d*x)**7, x) + Integral(C*sec(c + d*x)**4, x) + Integral(4*C*sec(c + d*x)**5, x) + Integral(6*C*sec(c + d*x
)**6, x) + Integral(4*C*sec(c + d*x)**7, x) + Integral(C*sec(c + d*x)**8, x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (236) = 472\).

Time = 0.24 (sec) , antiderivative size = 731, normalized size of antiderivative = 2.90 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {224 \, {\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 )} A a^{4} + 6720 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 896 \, {\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^{4} + 4480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 96 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} C a^{4} + 1344 \, {\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} + 1120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 35 \, B 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 )} - 140 \, 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 )} - 840 \, 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 )} - 1260 \, 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 )} - 840 \, 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 )} - 3360 \, 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 )} - 840 \, 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 )} + 3360 \, A a^{4} \tan \left (d x + c\right )}{3360 \, d} \]

[In]

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

[Out]

1/3360*(224*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^4 + 6720*(tan(d*x + c)^3 + 3*tan(d*x
+ c))*A*a^4 + 896*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^4 + 4480*(tan(d*x + c)^3 + 3*ta
n(d*x + c))*B*a^4 + 96*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*C*a^4 + 13
44*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 1120*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a
^4 - 35*B*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*log(sin(d*x + c) - 1)) - 140*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*log(sin(d*x + c) - 1)) - 840*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)) - 1260*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)) - 840*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)) - 3360*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)) - 840*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)) + 3360*A*a^4*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.76 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, 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 (5880 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 4620 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 39200 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 34300 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 30800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 110936 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 87164 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 172032 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150528 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 135168 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 159656 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 126084 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 86240 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 73220 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 58800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21000 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21735 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 22260 \, 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 )}^{7}}}{1680 \, d} \]

[In]

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

[Out]

1/1680*(105*(56*A*a^4 + 49*B*a^4 + 44*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(56*A*a^4 + 49*B*a^4 + 4
4*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(5880*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 5145*B*a^4*tan(1/2*d*x +
1/2*c)^13 + 4620*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 39200*A*a^4*tan(1/2*d*x + 1/2*c)^11 - 34300*B*a^4*tan(1/2*d*
x + 1/2*c)^11 - 30800*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 110936*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 97069*B*a^4*tan(1/
2*d*x + 1/2*c)^9 + 87164*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 172032*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 150528*B*a^4*tan
(1/2*d*x + 1/2*c)^7 - 135168*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 159656*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 134099*B*a^4
*tan(1/2*d*x + 1/2*c)^5 + 126084*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 86240*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 73220*B*a
^4*tan(1/2*d*x + 1/2*c)^3 - 58800*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 21000*A*a^4*tan(1/2*d*x + 1/2*c) + 21735*B*a^
4*tan(1/2*d*x + 1/2*c) + 22260*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d

Mupad [B] (verification not implemented)

Time = 20.05 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.51 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4\,\mathrm {atanh}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (56\,A+49\,B+44\,C\right )}{4\,\left (14\,A\,a^4+\frac {49\,B\,a^4}{4}+11\,C\,a^4\right )}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,d}-\frac {\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {140\,A\,a^4}{3}-\frac {245\,B\,a^4}{6}-\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {13867\,B\,a^4}{120}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {1024\,A\,a^4}{5}-\frac {896\,B\,a^4}{5}-\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {19157\,B\,a^4}{120}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {308\,A\,a^4}{3}-\frac {523\,B\,a^4}{6}-70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {207\,B\,a^4}{8}+\frac {53\,C\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

[In]

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

[Out]

(a^4*atanh((a^4*tan(c/2 + (d*x)/2)*(56*A + 49*B + 44*C))/(4*(14*A*a^4 + (49*B*a^4)/4 + 11*C*a^4)))*(56*A + 49*
B + 44*C))/(8*d) - (tan(c/2 + (d*x)/2)^13*(7*A*a^4 + (49*B*a^4)/8 + (11*C*a^4)/2) - tan(c/2 + (d*x)/2)^11*((14
0*A*a^4)/3 + (245*B*a^4)/6 + (110*C*a^4)/3) - tan(c/2 + (d*x)/2)^3*((308*A*a^4)/3 + (523*B*a^4)/6 + 70*C*a^4)
- tan(c/2 + (d*x)/2)^7*((1024*A*a^4)/5 + (896*B*a^4)/5 + (5632*C*a^4)/35) + tan(c/2 + (d*x)/2)^9*((1981*A*a^4)
/15 + (13867*B*a^4)/120 + (3113*C*a^4)/30) + tan(c/2 + (d*x)/2)^5*((2851*A*a^4)/15 + (19157*B*a^4)/120 + (1501
*C*a^4)/10) + tan(c/2 + (d*x)/2)*(25*A*a^4 + (207*B*a^4)/8 + (53*C*a^4)/2))/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*ta
n(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2
+ (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1))