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

Optimal. Leaf size=170 \[ \frac{a \left (a^2 (-C)+12 A b^2+8 b^2 C\right ) \tan (c+d x)}{6 b d}+\frac{\left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 b d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^2}{12 b d} \]

[Out]

((4*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(12*A*b^2 - a^2*C + 8*b^2*C)*Tan[c + d*
x])/(6*b*d) - ((2*a^2*C - 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (a*C*(a + b*Sec[c + d*x])^2*T
an[c + d*x])/(12*b*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)

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Rubi [A]  time = 0.308335, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4083, 4002, 3997, 3787, 3770, 3767, 8} \[ \frac{a \left (a^2 (-C)+12 A b^2+8 b^2 C\right ) \tan (c+d x)}{6 b d}+\frac{\left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 b d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^2}{12 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(12*A*b^2 - a^2*C + 8*b^2*C)*Tan[c + d*
x])/(6*b*d) - ((2*a^2*C - 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (a*C*(a + b*Sec[c + d*x])^2*T
an[c + d*x])/(12*b*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)

Rule 4083

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + 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) - a*C*Csc[e + f*x], x], x], x] /; FreeQ
[{a, b, e, f, A, 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))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^2 (b (4 A+3 C)-a C \sec (c+d x)) \, dx}{4 b}\\ &=-\frac{a C (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x)) \left (a b (12 A+7 C)-\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \sec (c+d x)\right ) \, dx}{12 b}\\ &=-\frac{\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{a C (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}+\frac{\int \sec (c+d x) \left (3 b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+4 a \left (12 A b^2-\left (a^2-8 b^2\right ) C\right ) \sec (c+d x)\right ) \, dx}{24 b}\\ &=-\frac{\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{a C (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}+\frac{\left (a \left (12 A b^2-a^2 C+8 b^2 C\right )\right ) \int \sec ^2(c+d x) \, dx}{6 b}+\frac{1}{8} \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac{\left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{a C (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}-\frac{\left (a \left (12 A b^2-a^2 C+8 b^2 C\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 b d}\\ &=\frac{\left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left (12 A b^2-a^2 C+8 b^2 C\right ) \tan (c+d x)}{6 b d}-\frac{\left (2 a^2 C-3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{a C (a+b \sec (c+d x))^2 \tan (c+d x)}{12 b d}+\frac{C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 b d}\\ \end{align*}

Mathematica [B]  time = 6.31943, size = 1123, normalized size = 6.61 \[ \frac{\left (-8 A a^2-4 C a^2-4 A b^2-3 b^2 C\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \cos ^4(c+d x)}{4 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C)}+\frac{\left (8 A a^2+4 C a^2+4 A b^2+3 b^2 C\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \cos ^4(c+d x)}{4 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C)}+\frac{2 a b C (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^4(c+d x)}{3 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{4 (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \left (3 a A b \sin \left (\frac{1}{2} (c+d x)\right )+2 a b C \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^4(c+d x)}{3 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{4 (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \left (3 a A b \sin \left (\frac{1}{2} (c+d x)\right )+2 a b C \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^4(c+d x)}{3 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{\left (12 C a^2+8 b C a+12 A b^2+9 b^2 C\right ) (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \cos ^4(c+d x)}{24 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{\left (-12 C a^2-8 b C a-12 A b^2-9 b^2 C\right ) (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \cos ^4(c+d x)}{24 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{2 a b C (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^4(c+d x)}{3 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{b^2 C (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \cos ^4(c+d x)}{8 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^4}-\frac{b^2 C (a+b \sec (c+d x))^2 \left (C \sec ^2(c+d x)+A\right ) \cos ^4(c+d x)}{8 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*a^2*A - 4*A*b^2 - 4*a^2*C - 3*b^2*C)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c
+ d*x])^2*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((8*a^2*A + 4
*A*b^2 + 4*a^2*C + 3*b^2*C)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^2*(A
+ C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + (b^2*C*Cos[c + d*x]^4*(a +
b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c +
d*x)/2] - Sin[(c + d*x)/2])^4) + ((12*A*b^2 + 12*a^2*C + 8*a*b*C + 9*b^2*C)*Cos[c + d*x]^4*(a + b*Sec[c + d*x
])^2*(A + C*Sec[c + d*x]^2))/(24*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - S
in[(c + d*x)/2])^2) + (2*a*b*C*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/
(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (b^2*C*C
os[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c
+ 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (2*a*b*C*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec
[c + d*x]^2)*Sin[(c + d*x)/2])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^3) + ((-12*A*b^2 - 12*a^2*C - 8*a*b*C - 9*b^2*C)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A +
C*Sec[c + d*x]^2))/(24*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*
x)/2])^2) + (4*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(3*a*A*b*Sin[(c + d*x)/2] + 2*a*b*
C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d
*x)/2])) + (4*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(3*a*A*b*Sin[(c + d*x)/2] + 2*a*b*C
*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*
x)/2]))

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Maple [A]  time = 0.044, size = 229, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+2\,{\frac{Aab\tan \left ( dx+c \right ) }{d}}+{\frac{4\,abC\tan \left ( dx+c \right ) }{3\,d}}+{\frac{2\,abC\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{A{b}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{b}^{2}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{b}^{2}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{b}^{2}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*a^2*A*ln(sec(d*x+c)+tan(d*x+c))+1/2/d*a^2*C*sec(d*x+c)*tan(d*x+c)+1/2/d*a^2*C*ln(sec(d*x+c)+tan(d*x+c))+2/
d*A*a*b*tan(d*x+c)+4/3/d*a*b*C*tan(d*x+c)+2/3/d*a*b*C*tan(d*x+c)*sec(d*x+c)^2+1/2/d*A*b^2*sec(d*x+c)*tan(d*x+c
)+1/2/d*A*b^2*ln(sec(d*x+c)+tan(d*x+c))+1/4/d*b^2*C*tan(d*x+c)*sec(d*x+c)^3+3/8/d*b^2*C*sec(d*x+c)*tan(d*x+c)+
3/8/d*b^2*C*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 0.988969, size = 304, normalized size = 1.79 \begin{align*} \frac{32 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b - 3 \, C b^{2}{\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 )} - 12 \, C a^{2}{\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 )} - 12 \, A b^{2}{\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 )} + 48 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, A a b \tan \left (d x + c\right )}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(32*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a*b - 3*C*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)) - 12*C*a^2*(2*sin(d*x + c)/(s
in(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 12*A*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)) + 48*A*a^2*log(sec(d*x + c) + tan(d*x + c)) + 96*A*a*b*
tan(d*x + c))/d

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Fricas [A]  time = 0.537459, size = 424, normalized size = 2.49 \begin{align*} \frac{3 \,{\left (4 \,{\left (2 \, A + C\right )} a^{2} +{\left (4 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (4 \,{\left (2 \, A + C\right )} a^{2} +{\left (4 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (3 \, A + 2 \, C\right )} a b \cos \left (d x + c\right )^{3} + 16 \, C a b \cos \left (d x + c\right ) + 6 \, C b^{2} + 3 \,{\left (4 \, C a^{2} +{\left (4 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(3*(4*(2*A + C)*a^2 + (4*A + 3*C)*b^2)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(4*(2*A + C)*a^2 + (4*A +
3*C)*b^2)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(16*(3*A + 2*C)*a*b*cos(d*x + c)^3 + 16*C*a*b*cos(d*x + c
) + 6*C*b^2 + 3*(4*C*a^2 + (4*A + 3*C)*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^4)

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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 )^{2} \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))**2*(A+C*sec(d*x+c)**2),x)

[Out]

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

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Giac [B]  time = 1.21912, size = 575, normalized size = 3.38 \begin{align*} \frac{3 \,{\left (8 \, A a^{2} + 4 \, C a^{2} + 4 \, A b^{2} + 3 \, C b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (8 \, A a^{2} + 4 \, C a^{2} + 4 \, A b^{2} + 3 \, C b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 48 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 48 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 144 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 80 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 144 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 80 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 48 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 48 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(8*A*a^2 + 4*C*a^2 + 4*A*b^2 + 3*C*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(8*A*a^2 + 4*C*a^2 + 4*
A*b^2 + 3*C*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(12*C*a^2*tan(1/2*d*x + 1/2*c)^7 - 48*A*a*b*tan(1/2*d*
x + 1/2*c)^7 - 48*C*a*b*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^2*tan(1/2*d*x + 1/2*c)^7 + 15*C*b^2*tan(1/2*d*x + 1/2*
c)^7 - 12*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 144*A*a*b*tan(1/2*d*x + 1/2*c)^5 + 80*C*a*b*tan(1/2*d*x + 1/2*c)^5 -
12*A*b^2*tan(1/2*d*x + 1/2*c)^5 + 9*C*b^2*tan(1/2*d*x + 1/2*c)^5 - 12*C*a^2*tan(1/2*d*x + 1/2*c)^3 - 144*A*a*b
*tan(1/2*d*x + 1/2*c)^3 - 80*C*a*b*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 9*C*b^2*tan(1/2*
d*x + 1/2*c)^3 + 12*C*a^2*tan(1/2*d*x + 1/2*c) + 48*A*a*b*tan(1/2*d*x + 1/2*c) + 48*C*a*b*tan(1/2*d*x + 1/2*c)
+ 12*A*b^2*tan(1/2*d*x + 1/2*c) + 15*C*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d