\(\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [663]

Optimal result

Integrand size = 33, antiderivative size = 381 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {\left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \tan (c+d x)}{105 b^2 d}+\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d} \]

[Out]

1/4*a*b*(b^2*(6*A+5*C)+a^2*(8*A+6*C))*arctanh(sin(d*x+c))/d+1/105*(2*a^6*C+8*b^6*(7*A+6*C)+a^4*b^2*(42*A+23*C)
+8*a^2*b^4*(49*A+39*C))*tan(d*x+c)/b^2/d+1/420*a*(4*a^4*C+12*a^2*b^2*(7*A+4*C)+b^4*(406*A+333*C))*sec(d*x+c)*t
an(d*x+c)/b/d+1/210*(2*a^4*C+8*b^4*(7*A+6*C)+3*a^2*b^2*(14*A+9*C))*(a+b*sec(d*x+c))^2*tan(d*x+c)/b^2/d+1/210*a
*(42*A*b^2+2*C*a^2+31*C*b^2)*(a+b*sec(d*x+c))^3*tan(d*x+c)/b^2/d+1/105*(C*a^2+3*b^2*(7*A+6*C))*(a+b*sec(d*x+c)
)^4*tan(d*x+c)/b^2/d-1/21*a*C*(a+b*sec(d*x+c))^5*tan(d*x+c)/b^2/d+1/7*C*sec(d*x+c)*(a+b*sec(d*x+c))^5*tan(d*x+
c)/b/d

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4178, 4167, 4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^4}{105 b^2 d}+\frac {a \left (2 a^2 C+42 A b^2+31 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{210 b^2 d}+\frac {\left (2 a^4 C+3 a^2 b^2 (14 A+9 C)+8 b^4 (7 A+6 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{210 b^2 d}+\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \tan (c+d x) \sec (c+d x)}{420 b d}+\frac {\left (2 a^6 C+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)+8 b^6 (7 A+6 C)\right ) \tan (c+d x)}{105 b^2 d}-\frac {a C \tan (c+d x) (a+b \sec (c+d x))^5}{21 b^2 d}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d} \]

[In]

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

[Out]

(a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*ArcTanh[Sin[c + d*x]])/(4*d) + ((2*a^6*C + 8*b^6*(7*A + 6*C) + a^4*b^
2*(42*A + 23*C) + 8*a^2*b^4*(49*A + 39*C))*Tan[c + d*x])/(105*b^2*d) + (a*(4*a^4*C + 12*a^2*b^2*(7*A + 4*C) +
b^4*(406*A + 333*C))*Sec[c + d*x]*Tan[c + d*x])/(420*b*d) + ((2*a^4*C + 8*b^4*(7*A + 6*C) + 3*a^2*b^2*(14*A +
9*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(210*b^2*d) + (a*(42*A*b^2 + 2*a^2*C + 31*b^2*C)*(a + b*Sec[c + d*x
])^3*Tan[c + d*x])/(210*b^2*d) + ((a^2*C + 3*b^2*(7*A + 6*C))*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(105*b^2*d)
- (a*C*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(21*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^5*Tan[c + d*x])
/(7*b*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 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4082

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

Rule 4087

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[1/(m + 1), Int[Csc[e + f
*x]*(a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1))*Csc[e + f*x], x], x], x] /;
FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]

Rule 4167

Int[csc[(e_.) + (f_.)*(x_)]*((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 + 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) + (b*
B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 4178

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))
^(m_), x_Symbol] :> Simp[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dis
t[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - 2
*a*C*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^4 \left (a C+b (7 A+6 C) \sec (c+d x)-2 a C \sec ^2(c+d x)\right ) \, dx}{7 b} \\ & = -\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^4 \left (-4 a b C+2 \left (a^2 C+3 b^2 (7 A+6 C)\right ) \sec (c+d x)\right ) \, dx}{42 b^2} \\ & = \frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (12 b \left (14 A b^2-a^2 C+12 b^2 C\right )+4 a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) \sec (c+d x)\right ) \, dx}{210 b^2} \\ & = \frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (12 a b \left (98 A b^2-2 a^2 C+79 b^2 C\right )+12 \left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) \sec (c+d x)\right ) \, dx}{840 b^2} \\ & = \frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x)) \left (-12 b \left (2 a^4 C-16 b^4 (7 A+6 C)-3 a^2 b^2 (126 A+97 C)\right )+12 a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x)\right ) \, dx}{2520 b^2} \\ & = \frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) \left (1260 a b^3 \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right )+48 \left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \sec (c+d x)\right ) \, dx}{5040 b^2} \\ & = \frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {1}{4} \left (a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right )\right ) \int \sec (c+d x) \, dx+\frac {\left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \int \sec ^2(c+d x) \, dx}{105 b^2} \\ & = \frac {a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}-\frac {\left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 b^2 d} \\ & = \frac {a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {\left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \tan (c+d x)}{105 b^2 d}+\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.62 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.61 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (420 \left (a^4+6 a^2 b^2+b^4\right ) (A+C)+105 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \sec (c+d x)+70 a b \left (6 A b^2+6 a^2 C+5 b^2 C\right ) \sec ^3(c+d x)+280 a b^3 C \sec ^5(c+d x)+140 \left (a^4 C+6 a^2 b^2 (A+2 C)+b^4 (2 A+3 C)\right ) \tan ^2(c+d x)+84 b^2 \left (A b^2+3 \left (2 a^2+b^2\right ) C\right ) \tan ^4(c+d x)+60 b^4 C \tan ^6(c+d x)\right )}{420 d} \]

[In]

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

[Out]

(105*a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(420*(a^4 + 6*a^2*b^2 + b^4)
*(A + C) + 105*a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Sec[c + d*x] + 70*a*b*(6*A*b^2 + 6*a^2*C + 5*b^2*C)*Sec
[c + d*x]^3 + 280*a*b^3*C*Sec[c + d*x]^5 + 140*(a^4*C + 6*a^2*b^2*(A + 2*C) + b^4*(2*A + 3*C))*Tan[c + d*x]^2
+ 84*b^2*(A*b^2 + 3*(2*a^2 + b^2)*C)*Tan[c + d*x]^4 + 60*b^4*C*Tan[c + d*x]^6))/(420*d)

Maple [A] (verified)

Time = 1.68 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.83

method	result	size
parts	\(-\frac {\left (A \,b^{4}+6 C \,a^{2} b^{2}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a A \,b^{3}+4 a^{3} b C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 A \,a^{2} b^{2}+a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}-\frac {C \,b^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} A \tan \left (d x +c \right )}{d}+\frac {4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(317\)
derivativedivides	\(\frac {a^{4} A \tan \left (d x +c \right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{3} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 C \,a^{2} b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-A \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-C \,b^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\)	\(384\)
default	\(\frac {a^{4} A \tan \left (d x +c \right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{3} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 C \,a^{2} b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-A \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-C \,b^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\)	\(384\)
parallelrisch	\(\frac {-5880 a b \left (a^{2} \left (A +\frac {3 C}{4}\right )+\frac {3 \left (A +\frac {5 C}{6}\right ) b^{2}}{4}\right ) \left (\frac {\cos \left (7 d x +7 c \right )}{7}+\cos \left (5 d x +5 c \right )+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+5880 a b \left (a^{2} \left (A +\frac {3 C}{4}\right )+\frac {3 \left (A +\frac {5 C}{6}\right ) b^{2}}{4}\right ) \left (\frac {\cos \left (7 d x +7 c \right )}{7}+\cos \left (5 d x +5 c \right )+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (3780 A +4200 C \right ) a^{4}+25200 \left (A +\frac {28 C}{25}\right ) b^{2} a^{2}+4704 b^{4} \left (A +\frac {6 C}{7}\right )\right ) \sin \left (3 d x +3 c \right )+\left (\left (2100 A +1960 C \right ) a^{4}+11760 b^{2} \left (A +\frac {4 C}{5}\right ) a^{2}+1568 b^{4} \left (A +\frac {6 C}{7}\right )\right ) \sin \left (5 d x +5 c \right )+\left (\left (420 A +280 C \right ) a^{4}+1680 b^{2} \left (A +\frac {4 C}{5}\right ) a^{2}+224 b^{4} \left (A +\frac {6 C}{7}\right )\right ) \sin \left (7 d x +7 c \right )+8400 a b \left (\left (A +\frac {31 C}{20}\right ) a^{2}+\frac {31 \left (A +\frac {283 C}{186}\right ) b^{2}}{20}\right ) \sin \left (2 d x +2 c \right )+6720 a b \left (\left (A +\frac {5 C}{4}\right ) a^{2}+\frac {5 \left (A +\frac {5 C}{6}\right ) b^{2}}{4}\right ) \sin \left (4 d x +4 c \right )+1680 a b \left (a^{2} \left (A +\frac {3 C}{4}\right )+\frac {3 \left (A +\frac {5 C}{6}\right ) b^{2}}{4}\right ) \sin \left (6 d x +6 c \right )+2100 \left (\left (A +\frac {6 C}{5}\right ) a^{4}+\frac {36 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}}{5}+\frac {8 b^{4} \left (A +2 C \right )}{5}\right ) \sin \left (d x +c \right )}{2940 d \left (\frac {\cos \left (7 d x +7 c \right )}{7}+\cos \left (5 d x +5 c \right )+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right )}\)	\(476\)
norman	\(\frac {\frac {8 \left (175 a^{4} A +630 A \,a^{2} b^{2}+91 A \,b^{4}+105 a^{4} C +546 C \,a^{2} b^{2}+53 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {\left (4 a^{4} A -8 A \,a^{3} b +24 A \,a^{2} b^{2}-10 a A \,b^{3}+4 A \,b^{4}+4 a^{4} C -10 a^{3} b C +24 C \,a^{2} b^{2}-11 C a \,b^{3}+4 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{2 d}-\frac {\left (4 a^{4} A +8 A \,a^{3} b +24 A \,a^{2} b^{2}+10 a A \,b^{3}+4 A \,b^{4}+4 a^{4} C +10 a^{3} b C +24 C \,a^{2} b^{2}+11 C a \,b^{3}+4 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {2 \left (18 a^{4} A -24 A \,a^{3} b +84 A \,a^{2} b^{2}-18 a A \,b^{3}+10 A \,b^{4}+14 a^{4} C -18 a^{3} b C +60 C \,a^{2} b^{2}-7 C a \,b^{3}+6 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}+\frac {2 \left (18 a^{4} A +24 A \,a^{3} b +84 A \,a^{2} b^{2}+18 a A \,b^{3}+10 A \,b^{4}+14 a^{4} C +18 a^{3} b C +60 C \,a^{2} b^{2}+7 C a \,b^{3}+6 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {\left (900 a^{4} A -600 A \,a^{3} b +3480 A \,a^{2} b^{2}-270 a A \,b^{3}+452 A \,b^{4}+580 a^{4} C -270 a^{3} b C +2712 C \,a^{2} b^{2}-425 C a \,b^{3}+516 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{30 d}-\frac {\left (900 a^{4} A +600 A \,a^{3} b +3480 A \,a^{2} b^{2}+270 a A \,b^{3}+452 A \,b^{4}+580 a^{4} C +270 a^{3} b C +2712 C \,a^{2} b^{2}+425 C a \,b^{3}+516 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{30 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}-\frac {a b \left (8 a^{2} A +6 A \,b^{2}+6 C \,a^{2}+5 C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 d}+\frac {a b \left (8 a^{2} A +6 A \,b^{2}+6 C \,a^{2}+5 C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d}\)	\(686\)
risch	\(\text {Expression too large to display}\)	\(1064\)

[In]

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

[Out]

-(A*b^4+6*C*a^2*b^2)/d*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+(4*A*a*b^3+4*C*a^3*b)/d*(-(-1/4*s
ec(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)))-(6*A*a^2*b^2+C*a^4)/d*(-2/3-1/3*sec(d*x+
c)^2)*tan(d*x+c)-C*b^4/d*(-16/35-1/7*sec(d*x+c)^6-6/35*sec(d*x+c)^4-8/35*sec(d*x+c)^2)*tan(d*x+c)+a^4*A/d*tan(
d*x+c)+4*A*a^3*b/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+4*C*a*b^3/d*(-(-1/6*sec(d*x+c)^5-
5/24*sec(d*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c)+5/16*ln(sec(d*x+c)+tan(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.85 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, {\left (35 \, {\left (3 \, A + 2 \, C\right )} a^{4} + 84 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 280 \, C a b^{3} \cos \left (d x + c\right ) + 105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} + 60 \, C b^{4} + 4 \, {\left (35 \, C a^{4} + 42 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 4 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, C a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 12 \, {\left (42 \, C a^{2} b^{2} + {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{7}} \]

[In]

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

[Out]

1/840*(105*(2*(4*A + 3*C)*a^3*b + (6*A + 5*C)*a*b^3)*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 105*(2*(4*A + 3*C)
*a^3*b + (6*A + 5*C)*a*b^3)*cos(d*x + c)^7*log(-sin(d*x + c) + 1) + 2*(4*(35*(3*A + 2*C)*a^4 + 84*(5*A + 4*C)*
a^2*b^2 + 8*(7*A + 6*C)*b^4)*cos(d*x + c)^6 + 280*C*a*b^3*cos(d*x + c) + 105*(2*(4*A + 3*C)*a^3*b + (6*A + 5*C
)*a*b^3)*cos(d*x + c)^5 + 60*C*b^4 + 4*(35*C*a^4 + 42*(5*A + 4*C)*a^2*b^2 + 4*(7*A + 6*C)*b^4)*cos(d*x + c)^4
+ 70*(6*C*a^3*b + (6*A + 5*C)*a*b^3)*cos(d*x + c)^3 + 12*(42*C*a^2*b^2 + (7*A + 6*C)*b^4)*cos(d*x + c)^2)*sin(
d*x + c))/(d*cos(d*x + c)^7)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.24 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {280 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} + 1680 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b^{2} + 336 \, {\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^{2} b^{2} + 56 \, {\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 b^{4} + 24 \, {\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 b^{4} - 35 \, C a b^{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 )} - 210 \, C a^{3} b {\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 )} - 210 \, A a b^{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 )} - 840 \, A a^{3} b {\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 \, A a^{4} \tan \left (d x + c\right )}{840 \, d} \]

[In]

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

[Out]

1/840*(280*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 + 1680*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^2*b^2 + 336*(3
*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^2*b^2 + 56*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 +
15*tan(d*x + c))*A*b^4 + 24*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*C*b^4
- 35*C*a*b^3*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4
+ 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 210*C*a^3*b*(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)) - 210*A*a*b^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)) - 840*A*a^3*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c)
+ 1) + log(sin(d*x + c) - 1)) + 840*A*a^4*tan(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1280 vs. \(2 (366) = 732\).

Time = 0.41 (sec) , antiderivative size = 1280, normalized size of antiderivative = 3.36 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/420*(105*(8*A*a^3*b + 6*C*a^3*b + 6*A*a*b^3 + 5*C*a*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(8*A*a^3*b
+ 6*C*a^3*b + 6*A*a*b^3 + 5*C*a*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(420*A*a^4*tan(1/2*d*x + 1/2*c)^1
3 + 420*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 840*A*a^3*b*tan(1/2*d*x + 1/2*c)^13 - 1050*C*a^3*b*tan(1/2*d*x + 1/2*c
)^13 + 2520*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 + 2520*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 - 1050*A*a*b^3*tan(1/2*
d*x + 1/2*c)^13 - 1155*C*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 420*A*b^4*tan(1/2*d*x + 1/2*c)^13 + 420*C*b^4*tan(1/2
*d*x + 1/2*c)^13 - 2520*A*a^4*tan(1/2*d*x + 1/2*c)^11 - 1960*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 3360*A*a^3*b*tan(
1/2*d*x + 1/2*c)^11 + 2520*C*a^3*b*tan(1/2*d*x + 1/2*c)^11 - 11760*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 8400*C*
a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 2520*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 980*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 -
1400*A*b^4*tan(1/2*d*x + 1/2*c)^11 - 840*C*b^4*tan(1/2*d*x + 1/2*c)^11 + 6300*A*a^4*tan(1/2*d*x + 1/2*c)^9 +
4060*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 4200*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 1890*C*a^3*b*tan(1/2*d*x + 1/2*c)^9
+ 24360*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 18984*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 1890*A*a*b^3*tan(1/2*d*x +
1/2*c)^9 - 2975*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 3164*A*b^4*tan(1/2*d*x + 1/2*c)^9 + 3612*C*b^4*tan(1/2*d*x +
1/2*c)^9 - 8400*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 5040*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 30240*A*a^2*b^2*tan(1/2*d*
x + 1/2*c)^7 - 26208*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 4368*A*b^4*tan(1/2*d*x + 1/2*c)^7 - 2544*C*b^4*tan(1/2
*d*x + 1/2*c)^7 + 6300*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 4060*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 4200*A*a^3*b*tan(1/2
*d*x + 1/2*c)^5 + 1890*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 24360*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 18984*C*a^2*b
^2*tan(1/2*d*x + 1/2*c)^5 + 1890*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 2975*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 3164*A
*b^4*tan(1/2*d*x + 1/2*c)^5 + 3612*C*b^4*tan(1/2*d*x + 1/2*c)^5 - 2520*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 1960*C*a
^4*tan(1/2*d*x + 1/2*c)^3 - 3360*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 2520*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 11760*
A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 8400*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 2520*A*a*b^3*tan(1/2*d*x + 1/2*c)^3
- 980*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 1400*A*b^4*tan(1/2*d*x + 1/2*c)^3 - 840*C*b^4*tan(1/2*d*x + 1/2*c)^3 +
420*A*a^4*tan(1/2*d*x + 1/2*c) + 420*C*a^4*tan(1/2*d*x + 1/2*c) + 840*A*a^3*b*tan(1/2*d*x + 1/2*c) + 1050*C*a
^3*b*tan(1/2*d*x + 1/2*c) + 2520*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 2520*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 1050*A
*a*b^3*tan(1/2*d*x + 1/2*c) + 1155*C*a*b^3*tan(1/2*d*x + 1/2*c) + 420*A*b^4*tan(1/2*d*x + 1/2*c) + 420*C*b^4*t
an(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 = 20.08 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.98 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a\,b\,\mathrm {atanh}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{6\,A\,a\,b^3+8\,A\,a^3\,b+5\,C\,a\,b^3+6\,C\,a^3\,b}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{2\,d}-\frac {\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2-5\,A\,a\,b^3-4\,A\,a^3\,b-\frac {11\,C\,a\,b^3}{2}-5\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (12\,A\,a\,b^3-\frac {20\,A\,b^4}{3}-\frac {28\,C\,a^4}{3}-4\,C\,b^4-56\,A\,a^2\,b^2-40\,C\,a^2\,b^2-12\,A\,a^4+16\,A\,a^3\,b+\frac {14\,C\,a\,b^3}{3}+12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {452\,C\,a^2\,b^2}{5}-9\,A\,a\,b^3-20\,A\,a^3\,b-\frac {85\,C\,a\,b^3}{6}-9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-40\,A\,a^4-\frac {104\,A\,b^4}{5}-24\,C\,a^4-\frac {424\,C\,b^4}{35}-144\,A\,a^2\,b^2-\frac {624\,C\,a^2\,b^2}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {452\,C\,a^2\,b^2}{5}+9\,A\,a\,b^3+20\,A\,a^3\,b+\frac {85\,C\,a\,b^3}{6}+9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-12\,A\,a^4-\frac {20\,A\,b^4}{3}-\frac {28\,C\,a^4}{3}-4\,C\,b^4-56\,A\,a^2\,b^2-40\,C\,a^2\,b^2-12\,A\,a\,b^3-16\,A\,a^3\,b-\frac {14\,C\,a\,b^3}{3}-12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2+5\,A\,a\,b^3+4\,A\,a^3\,b+\frac {11\,C\,a\,b^3}{2}+5\,C\,a^3\,b\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 + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^4)/cos(c + d*x)^2,x)

[Out]

(a*b*atanh((a*b*tan(c/2 + (d*x)/2)*(8*A*a^2 + 6*A*b^2 + 6*C*a^2 + 5*C*b^2))/(6*A*a*b^3 + 8*A*a^3*b + 5*C*a*b^3
+ 6*C*a^3*b))*(8*A*a^2 + 6*A*b^2 + 6*C*a^2 + 5*C*b^2))/(2*d) - (tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + 2*C*a
^4 + 2*C*b^4 + 12*A*a^2*b^2 + 12*C*a^2*b^2 + 5*A*a*b^3 + 4*A*a^3*b + (11*C*a*b^3)/2 + 5*C*a^3*b) - tan(c/2 + (
d*x)/2)^7*(40*A*a^4 + (104*A*b^4)/5 + 24*C*a^4 + (424*C*b^4)/35 + 144*A*a^2*b^2 + (624*C*a^2*b^2)/5) + tan(c/2
+ (d*x)/2)^13*(2*A*a^4 + 2*A*b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 + 12*C*a^2*b^2 - 5*A*a*b^3 - 4*A*a^3*b -
(11*C*a*b^3)/2 - 5*C*a^3*b) - tan(c/2 + (d*x)/2)^3*(12*A*a^4 + (20*A*b^4)/3 + (28*C*a^4)/3 + 4*C*b^4 + 56*A*a^
2*b^2 + 40*C*a^2*b^2 + 12*A*a*b^3 + 16*A*a^3*b + (14*C*a*b^3)/3 + 12*C*a^3*b) - tan(c/2 + (d*x)/2)^11*(12*A*a^
4 + (20*A*b^4)/3 + (28*C*a^4)/3 + 4*C*b^4 + 56*A*a^2*b^2 + 40*C*a^2*b^2 - 12*A*a*b^3 - 16*A*a^3*b - (14*C*a*b^
3)/3 - 12*C*a^3*b) + tan(c/2 + (d*x)/2)^5*(30*A*a^4 + (226*A*b^4)/15 + (58*C*a^4)/3 + (86*C*b^4)/5 + 116*A*a^2
*b^2 + (452*C*a^2*b^2)/5 + 9*A*a*b^3 + 20*A*a^3*b + (85*C*a*b^3)/6 + 9*C*a^3*b) + tan(c/2 + (d*x)/2)^9*(30*A*a
^4 + (226*A*b^4)/15 + (58*C*a^4)/3 + (86*C*b^4)/5 + 116*A*a^2*b^2 + (452*C*a^2*b^2)/5 - 9*A*a*b^3 - 20*A*a^3*b
- (85*C*a*b^3)/6 - 9*C*a^3*b))/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(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))