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

Optimal. Leaf size=381 \[ \frac{\left (a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)+2 a^6 C+8 b^6 (7 A+6 C)\right ) \tan (c+d x)}{105 b^2 d}+\frac{a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{\left (a^2 C+3 b^2 (7 A+6 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^4}{105 b^2 d}+\frac{a \left (2 a^2 C+42 A b^2+31 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{210 b^2 d}+\frac{\left (3 a^2 b^2 (14 A+9 C)+2 a^4 C+8 b^4 (7 A+6 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{210 b^2 d}+\frac{a \left (12 a^2 b^2 (7 A+4 C)+4 a^4 C+b^4 (406 A+333 C)\right ) \tan (c+d x) \sec (c+d x)}{420 b d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^5}{21 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.982717, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4093, 4082, 4002, 3997, 3787, 3770, 3767, 8} \[ \frac{\left (a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)+2 a^6 C+8 b^6 (7 A+6 C)\right ) \tan (c+d x)}{105 b^2 d}+\frac{a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{\left (a^2 C+3 b^2 (7 A+6 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^4}{105 b^2 d}+\frac{a \left (2 a^2 C+42 A b^2+31 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{210 b^2 d}+\frac{\left (3 a^2 b^2 (14 A+9 C)+2 a^4 C+8 b^4 (7 A+6 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{210 b^2 d}+\frac{a \left (12 a^2 b^2 (7 A+4 C)+4 a^4 C+b^4 (406 A+333 C)\right ) \tan (c+d x) \sec (c+d x)}{420 b d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^5}{21 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d} \]

Antiderivative was successfully verified.

[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 4093

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] + Dist[
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]

Rule 4082

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 4002

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] /; Fr
eeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]

Rule 3997

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*C
sc[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 3787

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 3770

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

Rule 3767

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\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 ) \tanh ^{-1}(\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 ) \operatorname{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 ) \tanh ^{-1}(\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]  time = 2.72815, size = 371, normalized size = 0.97 \[ -\frac{\sec ^6(c+d x) \left (A \cos ^2(c+d x)+C\right ) \left (-2 b^2 \left (3 \left (6 C \left (7 a^2+b^2\right )+7 A b^2\right ) \sin (2 (c+d x))+140 a b C \sin (c+d x)+30 b^2 C \tan (c+d x)\right )-4 \left (84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x) \cos ^5(c+d x)-105 a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos ^4(c+d x)-4 \left (42 a^2 b^2 (5 A+4 C)+35 a^4 C+4 b^4 (7 A+6 C)\right ) \sin (c+d x) \cos ^3(c+d x)-70 a b \left (6 a^2 C+6 A b^2+5 b^2 C\right ) \sin (c+d x) \cos ^2(c+d x)+105 a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{210 d (A \cos (2 (c+d x))+A+2 C)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((C + A*Cos[c + d*x]^2)*Sec[c + d*x]^6*(105*a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]^6*(Log[Cos[(
c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 70*a*b*(6*A*b^2 + 6*a^2*C + 5*b^
2*C)*Cos[c + d*x]^2*Sin[c + d*x] - 4*(35*a^4*C + 42*a^2*b^2*(5*A + 4*C) + 4*b^4*(7*A + 6*C))*Cos[c + d*x]^3*Si
n[c + d*x] - 105*a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]^4*Sin[c + d*x] - 4*(35*a^4*(3*A + 2*C) +
84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Cos[c + d*x]^5*Sin[c + d*x] - 2*b^2*(140*a*b*C*Sin[c + d*x] + 3*(
7*A*b^2 + 6*(7*a^2 + b^2)*C)*Sin[2*(c + d*x)] + 30*b^2*C*Tan[c + d*x])))/(210*d*(A + 2*C + A*Cos[2*(c + d*x)])
)

________________________________________________________________________________________

Maple [A]  time = 0.061, size = 591, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/15/d*A*b^4*tan(d*x+c)+16/35/d*C*b^4*tan(d*x+c)+2/3/d*a^4*C*tan(d*x+c)+1/d*A*a^4*tan(d*x+c)+1/3/d*a^4*C*tan(d
*x+c)*sec(d*x+c)^2+2/3/d*C*a*b^3*tan(d*x+c)*sec(d*x+c)^5+1/5/d*A*b^4*tan(d*x+c)*sec(d*x+c)^4+5/4/d*C*a*b^3*ln(
sec(d*x+c)+tan(d*x+c))+4/d*A*a^2*b^2*tan(d*x+c)+4/15/d*A*b^4*tan(d*x+c)*sec(d*x+c)^2+1/7/d*C*b^4*tan(d*x+c)*se
c(d*x+c)^6+6/35/d*C*b^4*tan(d*x+c)*sec(d*x+c)^4+8/35/d*C*b^4*tan(d*x+c)*sec(d*x+c)^2+16/5/d*C*a^2*b^2*tan(d*x+
c)+2/d*A*a^3*b*ln(sec(d*x+c)+tan(d*x+c))+3/2/d*a^3*b*C*ln(sec(d*x+c)+tan(d*x+c))+3/2/d*A*a*b^3*ln(sec(d*x+c)+t
an(d*x+c))+6/5/d*C*a^2*b^2*tan(d*x+c)*sec(d*x+c)^4+8/5/d*C*a^2*b^2*tan(d*x+c)*sec(d*x+c)^2+2/d*A*a^3*b*sec(d*x
+c)*tan(d*x+c)+1/d*a^3*b*C*tan(d*x+c)*sec(d*x+c)^3+3/2/d*a^3*b*C*sec(d*x+c)*tan(d*x+c)+5/6/d*C*a*b^3*tan(d*x+c
)*sec(d*x+c)^3+5/4/d*C*a*b^3*sec(d*x+c)*tan(d*x+c)+2/d*A*a^2*b^2*tan(d*x+c)*sec(d*x+c)^2+1/d*A*a*b^3*tan(d*x+c
)*sec(d*x+c)^3+3/2/d*A*a*b^3*sec(d*x+c)*tan(d*x+c)

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Maxima [A]  time = 1.00559, size = 637, normalized size = 1.67 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 0.609102, size = 783, normalized size = 2.06 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.28149, size = 1728, normalized size = 4.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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