3.878 \(\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=381 \[ -\frac{\tan (c+d x) \left (-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B+6 a^4 b B-2 a^5 C-24 a b^4 (5 A+4 C)-32 b^5 B\right )}{60 b^2 d}+\frac{\left (6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{120 b^2 d}-\frac{\tan (c+d x) \left (6 a^2 b B-2 a^3 C-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{120 b^2 d}-\frac{\tan (c+d x) \sec (c+d x) \left (-12 a^2 b^2 (5 A+3 C)+12 a^3 b B-4 a^4 C-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{240 b d}+\frac{(3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{15 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d} \]

[Out]

((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) - ((6*a^4*b*B -
104*a^2*b^3*B - 32*b^5*B - 2*a^5*C - 24*a*b^4*(5*A + 4*C) - a^3*b^2*(30*A + 17*C))*Tan[c + d*x])/(60*b^2*d) -
((12*a^3*b*B - 142*a*b^3*B - 4*a^4*C - 12*a^2*b^2*(5*A + 3*C) - 15*b^4*(6*A + 5*C))*Sec[c + d*x]*Tan[c + d*x])
/(240*b*d) - ((6*a^2*b*B - 32*b^3*B - 2*a^3*C - 3*a*b^2*(10*A + 7*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12
0*b^2*d) + ((30*A*b^2 - 6*a*b*B + 2*a^2*C + 25*b^2*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b^2*d) + ((3*b
*B - a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(15*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^4*Tan[c + d*x
])/(6*b*d)

________________________________________________________________________________________

Rubi [A]  time = 0.869532, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {4092, 4082, 4002, 3997, 3787, 3770, 3767, 8} \[ -\frac{\tan (c+d x) \left (-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B+6 a^4 b B-2 a^5 C-24 a b^4 (5 A+4 C)-32 b^5 B\right )}{60 b^2 d}+\frac{\left (6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{120 b^2 d}-\frac{\tan (c+d x) \left (6 a^2 b B-2 a^3 C-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{120 b^2 d}-\frac{\tan (c+d x) \sec (c+d x) \left (-12 a^2 b^2 (5 A+3 C)+12 a^3 b B-4 a^4 C-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{240 b d}+\frac{(3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{15 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) - ((6*a^4*b*B -
104*a^2*b^3*B - 32*b^5*B - 2*a^5*C - 24*a*b^4*(5*A + 4*C) - a^3*b^2*(30*A + 17*C))*Tan[c + d*x])/(60*b^2*d) -
((12*a^3*b*B - 142*a*b^3*B - 4*a^4*C - 12*a^2*b^2*(5*A + 3*C) - 15*b^4*(6*A + 5*C))*Sec[c + d*x]*Tan[c + d*x])
/(240*b*d) - ((6*a^2*b*B - 32*b^3*B - 2*a^3*C - 3*a*b^2*(10*A + 7*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12
0*b^2*d) + ((30*A*b^2 - 6*a*b*B + 2*a^2*C + 25*b^2*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b^2*d) + ((3*b
*B - a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(15*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^4*Tan[c + d*x
])/(6*b*d)

Rule 4092

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + 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 - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, 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))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (a C+b (6 A+5 C) \sec (c+d x)+2 (3 b B-a C) \sec ^2(c+d x)\right ) \, dx}{6 b}\\ &=\frac{(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 b (8 b B-a C)+\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) \sec (c+d x)\right ) \, dx}{30 b^2}\\ &=\frac{\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac{(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (3 b \left (30 A b^2+26 a b B-2 a^2 C+25 b^2 C\right )-3 \left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) \sec (c+d x)\right ) \, dx}{120 b^2}\\ &=-\frac{\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac{\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac{(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x)) \left (3 b \left (66 a^2 b B+64 b^3 B-2 a^3 C+3 a b^2 (50 A+39 C)\right )-3 \left (12 a^3 b B-142 a b^3 B-4 a^4 C-12 a^2 b^2 (5 A+3 C)-15 b^4 (6 A+5 C)\right ) \sec (c+d x)\right ) \, dx}{360 b^2}\\ &=-\frac{\left (12 a^3 b B-142 a b^3 B-4 a^4 C-12 a^2 b^2 (5 A+3 C)-15 b^4 (6 A+5 C)\right ) \sec (c+d x) \tan (c+d x)}{240 b d}-\frac{\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac{\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac{(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) \left (45 b^2 \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right )-12 \left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \sec (c+d x)\right ) \, dx}{720 b^2}\\ &=-\frac{\left (12 a^3 b B-142 a b^3 B-4 a^4 C-12 a^2 b^2 (5 A+3 C)-15 b^4 (6 A+5 C)\right ) \sec (c+d x) \tan (c+d x)}{240 b d}-\frac{\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac{\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac{(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}+\frac{1}{16} \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \int \sec ^2(c+d x) \, dx}{60 b^2}\\ &=\frac{\left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\left (12 a^3 b B-142 a b^3 B-4 a^4 C-12 a^2 b^2 (5 A+3 C)-15 b^4 (6 A+5 C)\right ) \sec (c+d x) \tan (c+d x)}{240 b d}-\frac{\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac{\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac{(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}+\frac{\left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{60 b^2 d}\\ &=\frac{\left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \tan (c+d x)}{60 b^2 d}-\frac{\left (12 a^3 b B-142 a b^3 B-4 a^4 C-12 a^2 b^2 (5 A+3 C)-15 b^4 (6 A+5 C)\right ) \sec (c+d x) \tan (c+d x)}{240 b d}-\frac{\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac{\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac{(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}\\ \end{align*}

Mathematica [A]  time = 3.17543, size = 384, normalized size = 1.01 \[ -\frac{\sec ^5(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (-b \left (5 \sin (2 (c+d x)) \left (18 a^2 C+18 a b B+6 A b^2+5 b^2 C\right )+48 b (3 a C+b B) \sin (c+d x)+40 b^2 C \tan (c+d x)\right )-16 \sin (c+d x) \cos ^4(c+d x) \left (5 a^3 (3 A+2 C)+30 a^2 b B+6 a b^2 (5 A+4 C)+8 b^3 B\right )-15 \sin (c+d x) \cos ^3(c+d x) \left (6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right )-16 \sin (c+d x) \cos ^2(c+d x) \left (15 a^2 b B+5 a^3 C+3 a b^2 (5 A+4 C)+4 b^3 B\right )+15 \cos ^5(c+d x) \left (6 a^2 b (4 A+3 C)+8 a^3 B+18 a b^2 B+b^3 (6 A+5 C)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{120 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

-((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(15*(8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3
*(6*A + 5*C))*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/
2]]) - 16*(15*a^2*b*B + 4*b^3*B + 5*a^3*C + 3*a*b^2*(5*A + 4*C))*Cos[c + d*x]^2*Sin[c + d*x] - 15*(8*a^3*B + 1
8*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*Cos[c + d*x]^3*Sin[c + d*x] - 16*(30*a^2*b*B + 8*b^3*B + 5*
a^3*(3*A + 2*C) + 6*a*b^2*(5*A + 4*C))*Cos[c + d*x]^4*Sin[c + d*x] - b*(48*b*(b*B + 3*a*C)*Sin[c + d*x] + 5*(6
*A*b^2 + 18*a*b*B + 18*a^2*C + 5*b^2*C)*Sin[2*(c + d*x)] + 40*b^2*C*Tan[c + d*x])))/(120*d*(A + 2*C + 2*B*Cos[
c + d*x] + A*Cos[2*(c + d*x)]))

________________________________________________________________________________________

Maple [A]  time = 0.062, size = 644, normalized size = 1.7 \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))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

3/8/d*A*b^3*ln(sec(d*x+c)+tan(d*x+c))+5/16/d*C*b^3*ln(sec(d*x+c)+tan(d*x+c))+9/8/d*a^2*b*C*ln(sec(d*x+c)+tan(d
*x+c))+8/5/d*C*a*b^2*tan(d*x+c)+5/16/d*C*b^3*sec(d*x+c)*tan(d*x+c)+1/d*A*a^3*tan(d*x+c)+1/2/d*B*a^3*ln(sec(d*x
+c)+tan(d*x+c))+8/15/d*B*b^3*tan(d*x+c)+1/3/d*a^3*C*tan(d*x+c)*sec(d*x+c)^2+1/5/d*B*b^3*tan(d*x+c)*sec(d*x+c)^
4+4/15/d*B*b^3*tan(d*x+c)*sec(d*x+c)^2+1/2/d*B*a^3*sec(d*x+c)*tan(d*x+c)+3/2/d*A*a^2*b*ln(sec(d*x+c)+tan(d*x+c
))+9/8/d*B*a*b^2*ln(sec(d*x+c)+tan(d*x+c))+1/4/d*A*b^3*tan(d*x+c)*sec(d*x+c)^3+3/2/d*A*a^2*b*sec(d*x+c)*tan(d*
x+c)+3/5/d*C*a*b^2*tan(d*x+c)*sec(d*x+c)^4+4/5/d*C*a*b^2*tan(d*x+c)*sec(d*x+c)^2+3/4/d*a^2*b*C*tan(d*x+c)*sec(
d*x+c)^3+2/3*a^3*C*tan(d*x+c)/d+3/8/d*A*b^3*sec(d*x+c)*tan(d*x+c)+9/8/d*a^2*b*C*sec(d*x+c)*tan(d*x+c)+3/4/d*B*
a*b^2*tan(d*x+c)*sec(d*x+c)^3+9/8/d*B*a*b^2*sec(d*x+c)*tan(d*x+c)+1/d*B*a^2*b*tan(d*x+c)*sec(d*x+c)^2+1/d*A*a*
b^2*tan(d*x+c)*sec(d*x+c)^2+1/6/d*C*b^3*tan(d*x+c)*sec(d*x+c)^5+2/d*B*a^2*b*tan(d*x+c)+2/d*A*a*b^2*tan(d*x+c)+
5/24/d*C*b^3*tan(d*x+c)*sec(d*x+c)^3

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Maxima [A]  time = 1.10175, size = 763, normalized size = 2. \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))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*(160*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3 + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^2*b + 480*(tan(
d*x + c)^3 + 3*tan(d*x + c))*A*a*b^2 + 96*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a*b^2 + 3
2*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*b^3 - 5*C*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)) - 90*C*a^2*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)) - 90*B*a*b^2*(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)) - 30*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)) - 120*B*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x
+ c) - 1)) - 360*A*a^2*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1))
+ 480*A*a^3*tan(d*x + c))/d

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Fricas [A]  time = 0.612898, size = 824, normalized size = 2.16 \begin{align*} \frac{15 \,{\left (8 \, B a^{3} + 6 \,{\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} +{\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (8 \, B a^{3} + 6 \,{\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} +{\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (5 \,{\left (3 \, A + 2 \, C\right )} a^{3} + 30 \, B a^{2} b + 6 \,{\left (5 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{5} + 15 \,{\left (8 \, B a^{3} + 6 \,{\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} +{\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 40 \, C b^{3} + 16 \,{\left (5 \, C a^{3} + 15 \, B a^{2} b + 3 \,{\left (5 \, A + 4 \, C\right )} a b^{2} + 4 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \,{\left (18 \, C a^{2} b + 18 \, B a b^{2} +{\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 48 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \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))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/480*(15*(8*B*a^3 + 6*(4*A + 3*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c)^6*log(sin(d*x + c) + 1)
- 15*(8*B*a^3 + 6*(4*A + 3*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*
(16*(5*(3*A + 2*C)*a^3 + 30*B*a^2*b + 6*(5*A + 4*C)*a*b^2 + 8*B*b^3)*cos(d*x + c)^5 + 15*(8*B*a^3 + 6*(4*A + 3
*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c)^4 + 40*C*b^3 + 16*(5*C*a^3 + 15*B*a^2*b + 3*(5*A + 4*C)
*a*b^2 + 4*B*b^3)*cos(d*x + c)^3 + 10*(18*C*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c)^2 + 48*(3*C*a*b
^2 + B*b^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (c + d x \right )}\right )^{3} \left (A + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \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))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Integral((a + b*sec(c + d*x))**3*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x)**2, x)

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Giac [B]  time = 1.26334, size = 1850, normalized size = 4.86 \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))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/240*(15*(8*B*a^3 + 24*A*a^2*b + 18*C*a^2*b + 18*B*a*b^2 + 6*A*b^3 + 5*C*b^3)*log(abs(tan(1/2*d*x + 1/2*c) +
1)) - 15*(8*B*a^3 + 24*A*a^2*b + 18*C*a^2*b + 18*B*a*b^2 + 6*A*b^3 + 5*C*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1
)) - 2*(240*A*a^3*tan(1/2*d*x + 1/2*c)^11 - 120*B*a^3*tan(1/2*d*x + 1/2*c)^11 + 240*C*a^3*tan(1/2*d*x + 1/2*c)
^11 - 360*A*a^2*b*tan(1/2*d*x + 1/2*c)^11 + 720*B*a^2*b*tan(1/2*d*x + 1/2*c)^11 - 450*C*a^2*b*tan(1/2*d*x + 1/
2*c)^11 + 720*A*a*b^2*tan(1/2*d*x + 1/2*c)^11 - 450*B*a*b^2*tan(1/2*d*x + 1/2*c)^11 + 720*C*a*b^2*tan(1/2*d*x
+ 1/2*c)^11 - 150*A*b^3*tan(1/2*d*x + 1/2*c)^11 + 240*B*b^3*tan(1/2*d*x + 1/2*c)^11 - 165*C*b^3*tan(1/2*d*x +
1/2*c)^11 - 1200*A*a^3*tan(1/2*d*x + 1/2*c)^9 + 360*B*a^3*tan(1/2*d*x + 1/2*c)^9 - 880*C*a^3*tan(1/2*d*x + 1/2
*c)^9 + 1080*A*a^2*b*tan(1/2*d*x + 1/2*c)^9 - 2640*B*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 630*C*a^2*b*tan(1/2*d*x +
1/2*c)^9 - 2640*A*a*b^2*tan(1/2*d*x + 1/2*c)^9 + 630*B*a*b^2*tan(1/2*d*x + 1/2*c)^9 - 1680*C*a*b^2*tan(1/2*d*x
+ 1/2*c)^9 + 210*A*b^3*tan(1/2*d*x + 1/2*c)^9 - 560*B*b^3*tan(1/2*d*x + 1/2*c)^9 - 25*C*b^3*tan(1/2*d*x + 1/2
*c)^9 + 2400*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 240*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 1440*C*a^3*tan(1/2*d*x + 1/2*c)
^7 - 720*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 4320*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 180*C*a^2*b*tan(1/2*d*x + 1/2*
c)^7 + 4320*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 180*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 3744*C*a*b^2*tan(1/2*d*x + 1
/2*c)^7 - 60*A*b^3*tan(1/2*d*x + 1/2*c)^7 + 1248*B*b^3*tan(1/2*d*x + 1/2*c)^7 - 450*C*b^3*tan(1/2*d*x + 1/2*c)
^7 - 2400*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 240*B*a^3*tan(1/2*d*x + 1/2*c)^5 - 1440*C*a^3*tan(1/2*d*x + 1/2*c)^5
- 720*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 4320*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 180*C*a^2*b*tan(1/2*d*x + 1/2*c)^
5 - 4320*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 180*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 3744*C*a*b^2*tan(1/2*d*x + 1/2*
c)^5 - 60*A*b^3*tan(1/2*d*x + 1/2*c)^5 - 1248*B*b^3*tan(1/2*d*x + 1/2*c)^5 - 450*C*b^3*tan(1/2*d*x + 1/2*c)^5
+ 1200*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 360*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 880*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 10
80*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 2640*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 630*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 +
2640*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 630*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 1680*C*a*b^2*tan(1/2*d*x + 1/2*c)^
3 + 210*A*b^3*tan(1/2*d*x + 1/2*c)^3 + 560*B*b^3*tan(1/2*d*x + 1/2*c)^3 - 25*C*b^3*tan(1/2*d*x + 1/2*c)^3 - 24
0*A*a^3*tan(1/2*d*x + 1/2*c) - 120*B*a^3*tan(1/2*d*x + 1/2*c) - 240*C*a^3*tan(1/2*d*x + 1/2*c) - 360*A*a^2*b*t
an(1/2*d*x + 1/2*c) - 720*B*a^2*b*tan(1/2*d*x + 1/2*c) - 450*C*a^2*b*tan(1/2*d*x + 1/2*c) - 720*A*a*b^2*tan(1/
2*d*x + 1/2*c) - 450*B*a*b^2*tan(1/2*d*x + 1/2*c) - 720*C*a*b^2*tan(1/2*d*x + 1/2*c) - 150*A*b^3*tan(1/2*d*x +
1/2*c) - 240*B*b^3*tan(1/2*d*x + 1/2*c) - 165*C*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d