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

Optimal. Leaf size=384 \[ \frac{\tan (c+d x) \left (a^3 b^2 (190 A+121 C)+224 a^2 b^3 B+24 a^4 b B-4 a^5 C+32 a b^4 (5 A+4 C)+32 b^5 B\right )}{60 b d}+\frac{\left (12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+32 a^3 b B+24 a b^3 B+b^4 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \left (24 a^2 b B-4 a^3 C+a b^2 (70 A+53 C)+32 b^3 B\right ) (a+b \sec (c+d x))^2}{120 b d}+\frac{\tan (c+d x) \sec (c+d x) \left (2 a^2 b^2 (130 A+89 C)+48 a^3 b B-8 a^4 C+232 a b^3 B+15 b^4 (6 A+5 C)\right )}{240 d}+\frac{\tan (c+d x) \left (4 a (6 b B-a C)+5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3}{120 b d}+\frac{(6 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d} \]

[Out]

((32*a^3*b*B + 24*a*b^3*B + 8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])
/(16*d) + ((24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 32*a*b^4*(5*A + 4*C) + a^3*b^2*(190*A + 121*C))*
Tan[c + d*x])/(60*b*d) + ((48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 15*b^4*(6*A + 5*C) + 2*a^2*b^2*(130*A + 89*C))
*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((24*a^2*b*B + 32*b^3*B - 4*a^3*C + a*b^2*(70*A + 53*C))*(a + b*Sec[c +
d*x])^2*Tan[c + d*x])/(120*b*d) + ((5*b^2*(6*A + 5*C) + 4*a*(6*b*B - a*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x]
)/(120*b*d) + ((6*b*B - a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + (C*(a + b*Sec[c + d*x])^5*Tan[c +
d*x])/(6*b*d)

________________________________________________________________________________________

Rubi [A]  time = 0.875709, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.18, Rules used = {4082, 4002, 3997, 3787, 3770, 3767, 8} \[ \frac{\tan (c+d x) \left (a^3 b^2 (190 A+121 C)+224 a^2 b^3 B+24 a^4 b B-4 a^5 C+32 a b^4 (5 A+4 C)+32 b^5 B\right )}{60 b d}+\frac{\left (12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+32 a^3 b B+24 a b^3 B+b^4 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \left (24 a^2 b B-4 a^3 C+a b^2 (70 A+53 C)+32 b^3 B\right ) (a+b \sec (c+d x))^2}{120 b d}+\frac{\tan (c+d x) \sec (c+d x) \left (2 a^2 b^2 (130 A+89 C)+48 a^3 b B-8 a^4 C+232 a b^3 B+15 b^4 (6 A+5 C)\right )}{240 d}+\frac{\tan (c+d x) \left (4 a (6 b B-a C)+5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3}{120 b d}+\frac{(6 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((32*a^3*b*B + 24*a*b^3*B + 8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])
/(16*d) + ((24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 32*a*b^4*(5*A + 4*C) + a^3*b^2*(190*A + 121*C))*
Tan[c + d*x])/(60*b*d) + ((48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 15*b^4*(6*A + 5*C) + 2*a^2*b^2*(130*A + 89*C))
*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((24*a^2*b*B + 32*b^3*B - 4*a^3*C + a*b^2*(70*A + 53*C))*(a + b*Sec[c +
d*x])^2*Tan[c + d*x])/(120*b*d) + ((5*b^2*(6*A + 5*C) + 4*a*(6*b*B - a*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x]
)/(120*b*d) + ((6*b*B - a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + (C*(a + b*Sec[c + d*x])^5*Tan[c +
d*x])/(6*b*d)

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 (c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^4 (b (6 A+5 C)+(6 b B-a C) \sec (c+d x)) \, dx}{6 b}\\ &=\frac{(6 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 b (10 a A+8 b B+7 a C)+\left (5 b^2 (6 A+5 C)+4 a (6 b B-a C)\right ) \sec (c+d x)\right ) \, dx}{30 b}\\ &=\frac{\left (5 b^2 (6 A+5 C)+4 a (6 b B-a C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (3 b \left (56 a b B+8 a^2 (5 A+3 C)+5 b^2 (6 A+5 C)\right )+3 \left (24 a^2 b B+32 b^3 B-4 a^3 C+a b^2 (70 A+53 C)\right ) \sec (c+d x)\right ) \, dx}{120 b}\\ &=\frac{\left (24 a^2 b B+32 b^3 B-4 a^3 C+a b^2 (70 A+53 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (5 b^2 (6 A+5 C)+4 a (6 b B-a C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x)) \left (3 b \left (216 a^2 b B+64 b^3 B+8 a^3 (15 A+8 C)+a b^2 (230 A+181 C)\right )+3 \left (48 a^3 b B+232 a b^3 B-8 a^4 C+15 b^4 (6 A+5 C)+2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x)\right ) \, dx}{360 b}\\ &=\frac{\left (48 a^3 b B+232 a b^3 B-8 a^4 C+15 b^4 (6 A+5 C)+2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{\left (24 a^2 b B+32 b^3 B-4 a^3 C+a b^2 (70 A+53 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (5 b^2 (6 A+5 C)+4 a (6 b B-a C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) \left (45 b \left (32 a^3 b B+24 a b^3 B+8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right )+12 \left (24 a^4 b B+224 a^2 b^3 B+32 b^5 B-4 a^5 C+32 a b^4 (5 A+4 C)+a^3 b^2 (190 A+121 C)\right ) \sec (c+d x)\right ) \, dx}{720 b}\\ &=\frac{\left (48 a^3 b B+232 a b^3 B-8 a^4 C+15 b^4 (6 A+5 C)+2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{\left (24 a^2 b B+32 b^3 B-4 a^3 C+a b^2 (70 A+53 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (5 b^2 (6 A+5 C)+4 a (6 b B-a C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{1}{16} \left (32 a^3 b B+24 a b^3 B+8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \int \sec (c+d x) \, dx+\frac{\left (24 a^4 b B+224 a^2 b^3 B+32 b^5 B-4 a^5 C+32 a b^4 (5 A+4 C)+a^3 b^2 (190 A+121 C)\right ) \int \sec ^2(c+d x) \, dx}{60 b}\\ &=\frac{\left (32 a^3 b B+24 a b^3 B+8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (48 a^3 b B+232 a b^3 B-8 a^4 C+15 b^4 (6 A+5 C)+2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{\left (24 a^2 b B+32 b^3 B-4 a^3 C+a b^2 (70 A+53 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (5 b^2 (6 A+5 C)+4 a (6 b B-a C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}-\frac{\left (24 a^4 b B+224 a^2 b^3 B+32 b^5 B-4 a^5 C+32 a b^4 (5 A+4 C)+a^3 b^2 (190 A+121 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{60 b d}\\ &=\frac{\left (32 a^3 b B+24 a b^3 B+8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (24 a^4 b B+224 a^2 b^3 B+32 b^5 B-4 a^5 C+32 a b^4 (5 A+4 C)+a^3 b^2 (190 A+121 C)\right ) \tan (c+d x)}{60 b d}+\frac{\left (48 a^3 b B+232 a b^3 B-8 a^4 C+15 b^4 (6 A+5 C)+2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{\left (24 a^2 b B+32 b^3 B-4 a^3 C+a b^2 (70 A+53 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (5 b^2 (6 A+5 C)+4 a (6 b B-a C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}\\ \end{align*}

Mathematica [A]  time = 3.68672, size = 424, normalized size = 1.1 \[ -\frac{\sec ^6(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (-16 \sin (c+d x) \cos ^5(c+d x) \left (20 a^3 b (3 A+2 C)+60 a^2 b^2 B+15 a^4 B+8 a b^3 (5 A+4 C)+8 b^4 B\right )-15 \sin (c+d x) \cos ^4(c+d x) \left (12 a^2 b^2 (4 A+3 C)+32 a^3 b B+8 a^4 C+24 a b^3 B+b^4 (6 A+5 C)\right )-32 b \sin (c+d x) \cos ^3(c+d x) \left (15 a^2 b B+10 a^3 C+2 a b^2 (5 A+4 C)+2 b^3 B\right )-10 b^2 \sin (c+d x) \cos ^2(c+d x) \left (36 a^2 C+24 a b B+6 A b^2+5 b^2 C\right )+15 \cos ^6(c+d x) \left (12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+32 a^3 b B+24 a b^3 B+b^4 (6 A+5 C)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-8 b^3 \sin (c+d x) (6 (4 a C+b B) \cos (c+d x)+5 b C)\right )}{120 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^4*(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]^6*(15*(32*a^3*b*B + 24*a*b^3*B + 8*a^4*(2*A + C) + 12*a
^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*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]]) - 10*b^2*(6*A*b^2 + 24*a*b*B + 36*a^2*C + 5*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x] -
32*b*(15*a^2*b*B + 2*b^3*B + 10*a^3*C + 2*a*b^2*(5*A + 4*C))*Cos[c + d*x]^3*Sin[c + d*x] - 15*(32*a^3*b*B + 24
*a*b^3*B + 8*a^4*C + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*Cos[c + d*x]^4*Sin[c + d*x] - 16*(15*a^4*B + 60
*a^2*b^2*B + 8*b^4*B + 20*a^3*b*(3*A + 2*C) + 8*a*b^3*(5*A + 4*C))*Cos[c + d*x]^5*Sin[c + d*x] - 8*b^3*(5*b*C
+ 6*(b*B + 4*a*C)*Cos[c + d*x])*Sin[c + d*x]))/(120*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))

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Maple [B]  time = 0.069, size = 745, normalized size = 1.9 \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)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

3/8/d*A*b^4*ln(sec(d*x+c)+tan(d*x+c))+5/16/d*C*b^4*ln(sec(d*x+c)+tan(d*x+c))+8/15/d*B*b^4*tan(d*x+c)+1/d*B*a^4
*tan(d*x+c)+1/2/d*a^4*C*ln(sec(d*x+c)+tan(d*x+c))+1/d*A*a^4*ln(sec(d*x+c)+tan(d*x+c))+4/d*A*a^3*b*tan(d*x+c)+8
/3/d*a^3*b*C*tan(d*x+c)+3/d*A*a^2*b^2*ln(sec(d*x+c)+tan(d*x+c))+9/4/d*C*a^2*b^2*ln(sec(d*x+c)+tan(d*x+c))+8/3/
d*A*a*b^3*tan(d*x+c)+32/15/d*C*a*b^3*tan(d*x+c)+1/4/d*A*b^4*tan(d*x+c)*sec(d*x+c)^3+3/8/d*A*b^4*sec(d*x+c)*tan
(d*x+c)+1/6/d*C*b^4*tan(d*x+c)*sec(d*x+c)^5+5/24/d*C*b^4*tan(d*x+c)*sec(d*x+c)^3+5/16/d*C*b^4*sec(d*x+c)*tan(d
*x+c)+1/5/d*B*b^4*tan(d*x+c)*sec(d*x+c)^4+4/15/d*B*b^4*tan(d*x+c)*sec(d*x+c)^2+3/d*A*a^2*b^2*sec(d*x+c)*tan(d*
x+c)+3/2/d*C*a^2*b^2*tan(d*x+c)*sec(d*x+c)^3+9/4/d*C*a^2*b^2*sec(d*x+c)*tan(d*x+c)+4/3/d*A*a*b^3*tan(d*x+c)*se
c(d*x+c)^2+3/2/d*a*b^3*B*sec(d*x+c)*tan(d*x+c)+2/d*a^2*b^2*B*tan(d*x+c)*sec(d*x+c)^2+2/d*B*a^3*b*sec(d*x+c)*ta
n(d*x+c)+1/d*a*b^3*B*tan(d*x+c)*sec(d*x+c)^3+4/5/d*C*a*b^3*tan(d*x+c)*sec(d*x+c)^4+16/15/d*C*a*b^3*tan(d*x+c)*
sec(d*x+c)^2+4/3/d*a^3*b*C*tan(d*x+c)*sec(d*x+c)^2+2/d*B*a^3*b*ln(sec(d*x+c)+tan(d*x+c))+4/d*a^2*b^2*B*tan(d*x
+c)+3/2/d*a*b^3*B*ln(sec(d*x+c)+tan(d*x+c))+1/2/d*a^4*C*sec(d*x+c)*tan(d*x+c)

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Maxima [A]  time = 1.05854, size = 882, normalized size = 2.3 \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)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*(640*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3*b + 960*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^2*b^2 + 640*(
tan(d*x + c)^3 + 3*tan(d*x + c))*A*a*b^3 + 128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a*b^
3 + 32*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*b^4 - 5*C*b^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)) - 180*C*a^2*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)) - 120*B*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)) -
30*A*b^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)) - 120*C*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)) - 480*B*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)) - 720*A*a^2*b^2*(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^4*log(sec(d*x + c) + tan(d*x + c)) + 480*B*a^4*tan(d*x + c) + 1920*A*a^3*b*tan(d*x + c))/d

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Fricas [A]  time = 0.659256, size = 941, normalized size = 2.45 \begin{align*} \frac{15 \,{\left (8 \,{\left (2 \, A + C\right )} a^{4} + 32 \, B a^{3} b + 12 \,{\left (4 \, A + 3 \, C\right )} a^{2} b^{2} + 24 \, B a b^{3} +{\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (8 \,{\left (2 \, A + C\right )} a^{4} + 32 \, B a^{3} b + 12 \,{\left (4 \, A + 3 \, C\right )} a^{2} b^{2} + 24 \, B a b^{3} +{\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (15 \, B a^{4} + 20 \,{\left (3 \, A + 2 \, C\right )} a^{3} b + 60 \, B a^{2} b^{2} + 8 \,{\left (5 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 40 \, C b^{4} + 15 \,{\left (8 \, C a^{4} + 32 \, B a^{3} b + 12 \,{\left (4 \, A + 3 \, C\right )} a^{2} b^{2} + 24 \, B a b^{3} +{\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 32 \,{\left (10 \, C a^{3} b + 15 \, B a^{2} b^{2} + 2 \,{\left (5 \, A + 4 \, C\right )} a b^{3} + 2 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 10 \,{\left (36 \, C a^{2} b^{2} + 24 \, B a b^{3} +{\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 48 \,{\left (4 \, C a b^{3} + B b^{4}\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)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/480*(15*(8*(2*A + C)*a^4 + 32*B*a^3*b + 12*(4*A + 3*C)*a^2*b^2 + 24*B*a*b^3 + (6*A + 5*C)*b^4)*cos(d*x + c)^
6*log(sin(d*x + c) + 1) - 15*(8*(2*A + C)*a^4 + 32*B*a^3*b + 12*(4*A + 3*C)*a^2*b^2 + 24*B*a*b^3 + (6*A + 5*C)
*b^4)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(16*(15*B*a^4 + 20*(3*A + 2*C)*a^3*b + 60*B*a^2*b^2 + 8*(5*A +
4*C)*a*b^3 + 8*B*b^4)*cos(d*x + c)^5 + 40*C*b^4 + 15*(8*C*a^4 + 32*B*a^3*b + 12*(4*A + 3*C)*a^2*b^2 + 24*B*a*
b^3 + (6*A + 5*C)*b^4)*cos(d*x + c)^4 + 32*(10*C*a^3*b + 15*B*a^2*b^2 + 2*(5*A + 4*C)*a*b^3 + 2*B*b^4)*cos(d*x
+ c)^3 + 10*(36*C*a^2*b^2 + 24*B*a*b^3 + (6*A + 5*C)*b^4)*cos(d*x + c)^2 + 48*(4*C*a*b^3 + B*b^4)*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 )^{4} \left (A + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.40354, size = 2238, normalized size = 5.83 \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)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/240*(15*(16*A*a^4 + 8*C*a^4 + 32*B*a^3*b + 48*A*a^2*b^2 + 36*C*a^2*b^2 + 24*B*a*b^3 + 6*A*b^4 + 5*C*b^4)*log
(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(16*A*a^4 + 8*C*a^4 + 32*B*a^3*b + 48*A*a^2*b^2 + 36*C*a^2*b^2 + 24*B*a*b
^3 + 6*A*b^4 + 5*C*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(240*B*a^4*tan(1/2*d*x + 1/2*c)^11 - 120*C*a^4*
tan(1/2*d*x + 1/2*c)^11 + 960*A*a^3*b*tan(1/2*d*x + 1/2*c)^11 - 480*B*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 960*C*a^
3*b*tan(1/2*d*x + 1/2*c)^11 - 720*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 1440*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 -
900*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 960*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 - 600*B*a*b^3*tan(1/2*d*x + 1/2*c
)^11 + 960*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 - 150*A*b^4*tan(1/2*d*x + 1/2*c)^11 + 240*B*b^4*tan(1/2*d*x + 1/2*c
)^11 - 165*C*b^4*tan(1/2*d*x + 1/2*c)^11 - 1200*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 360*C*a^4*tan(1/2*d*x + 1/2*c)^
9 - 4800*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 1440*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 3520*C*a^3*b*tan(1/2*d*x + 1/2
*c)^9 + 2160*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 5280*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 1260*C*a^2*b^2*tan(1/2
*d*x + 1/2*c)^9 - 3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 840*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 2240*C*a*b^3*tan(
1/2*d*x + 1/2*c)^9 + 210*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 560*B*b^4*tan(1/2*d*x + 1/2*c)^9 - 25*C*b^4*tan(1/2*d*
x + 1/2*c)^9 + 2400*B*a^4*tan(1/2*d*x + 1/2*c)^7 - 240*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 9600*A*a^3*b*tan(1/2*d*x
+ 1/2*c)^7 - 960*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 5760*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 1440*A*a^2*b^2*tan(1/
2*d*x + 1/2*c)^7 + 8640*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 360*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 5760*A*a*b^3
*tan(1/2*d*x + 1/2*c)^7 - 240*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 4992*C*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 60*A*b^4*
tan(1/2*d*x + 1/2*c)^7 + 1248*B*b^4*tan(1/2*d*x + 1/2*c)^7 - 450*C*b^4*tan(1/2*d*x + 1/2*c)^7 - 2400*B*a^4*tan
(1/2*d*x + 1/2*c)^5 - 240*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 9600*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 960*B*a^3*b*tan
(1/2*d*x + 1/2*c)^5 - 5760*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 1440*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 - 8640*B*a^2
*b^2*tan(1/2*d*x + 1/2*c)^5 - 360*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 - 5760*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 240
*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 4992*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 60*A*b^4*tan(1/2*d*x + 1/2*c)^5 - 1248
*B*b^4*tan(1/2*d*x + 1/2*c)^5 - 450*C*b^4*tan(1/2*d*x + 1/2*c)^5 + 1200*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 360*C*a
^4*tan(1/2*d*x + 1/2*c)^3 + 4800*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 1440*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 3520*C
*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 2160*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 5280*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3
+ 1260*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 840*B*a*b^3*tan(1/2*d*x + 1/2*
c)^3 + 2240*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 210*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 560*B*b^4*tan(1/2*d*x + 1/2*c)
^3 - 25*C*b^4*tan(1/2*d*x + 1/2*c)^3 - 240*B*a^4*tan(1/2*d*x + 1/2*c) - 120*C*a^4*tan(1/2*d*x + 1/2*c) - 960*A
*a^3*b*tan(1/2*d*x + 1/2*c) - 480*B*a^3*b*tan(1/2*d*x + 1/2*c) - 960*C*a^3*b*tan(1/2*d*x + 1/2*c) - 720*A*a^2*
b^2*tan(1/2*d*x + 1/2*c) - 1440*B*a^2*b^2*tan(1/2*d*x + 1/2*c) - 900*C*a^2*b^2*tan(1/2*d*x + 1/2*c) - 960*A*a*
b^3*tan(1/2*d*x + 1/2*c) - 600*B*a*b^3*tan(1/2*d*x + 1/2*c) - 960*C*a*b^3*tan(1/2*d*x + 1/2*c) - 150*A*b^4*tan
(1/2*d*x + 1/2*c) - 240*B*b^4*tan(1/2*d*x + 1/2*c) - 165*C*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 -
1)^6)/d