3.9.80 \(\int (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [880]

Optimal. Leaf size=207 \[ a^3 A x+\frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d} \]

[Out]

a^3*A*x+1/8*(8*a^3*B+12*a*b^2*B+12*a^2*b*(2*A+C)+b^3*(4*A+3*C))*arctanh(sin(d*x+c))/d+1/6*(16*a^2*b*B+4*b^3*B+
3*a^3*C+6*a*b^2*(3*A+2*C))*tan(d*x+c)/d+1/24*b*(12*A*b^2+20*B*a*b+6*C*a^2+9*C*b^2)*sec(d*x+c)*tan(d*x+c)/d+1/1
2*(4*B*b+3*C*a)*(a+b*sec(d*x+c))^2*tan(d*x+c)/d+1/4*C*(a+b*sec(d*x+c))^3*tan(d*x+c)/d

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4141, 4133, 3855, 3852, 8} \begin {gather*} a^3 A x+\frac {b \tan (c+d x) \sec (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac {\tan (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac {\left (8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(3 a C+4 b B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*A*x + ((8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((16*a
^2*b*B + 4*b^3*B + 3*a^3*C + 6*a*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) + (b*(12*A*b^2 + 20*a*b*B + 6*a^2*C + 9*
b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(
a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*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 4133

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*(Cot[e + f*x]/(2*f)), x] + Dist[1/2, Int[Simp[2*A*a + (2*B*a + b
*(2*A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x]

Rule 4141

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

Rubi steps

\begin {align*} \int (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \sec (c+d x))^2 \left (4 a A+(4 A b+4 a B+3 b C) \sec (c+d x)+(4 b B+3 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \sec (c+d x)) \left (12 a^2 A+\left (24 a A b+12 a^2 B+8 b^2 B+15 a b C\right ) \sec (c+d x)+\left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^3 A+3 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \sec (c+d x)+4 \left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=a^3 A x+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac {1}{6} \left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 A x+\frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=a^3 A x+\frac {\left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}+\frac {C (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(525\) vs. \(2(207)=414\).
time = 5.32, size = 525, normalized size = 2.54 \begin {gather*} \frac {\cos (c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (36 a^3 A (c+d x)+48 a^3 A (c+d x) \cos (2 (c+d x))+12 a^3 A (c+d x) \cos (4 (c+d x))-12 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 A b^3 \sin (c+d x)+36 a b^2 B \sin (c+d x)+36 a^2 b C \sin (c+d x)+33 b^3 C \sin (c+d x)+72 a A b^2 \sin (2 (c+d x))+72 a^2 b B \sin (2 (c+d x))+32 b^3 B \sin (2 (c+d x))+24 a^3 C \sin (2 (c+d x))+96 a b^2 C \sin (2 (c+d x))+12 A b^3 \sin (3 (c+d x))+36 a b^2 B \sin (3 (c+d x))+36 a^2 b C \sin (3 (c+d x))+9 b^3 C \sin (3 (c+d x))+36 a A b^2 \sin (4 (c+d x))+36 a^2 b B \sin (4 (c+d x))+8 b^3 B \sin (4 (c+d x))+12 a^3 C \sin (4 (c+d x))+24 a b^2 C \sin (4 (c+d x))\right )}{48 d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(36*a^3*A*(c + d*x) + 48*a^3*A*(c
+ d*x)*Cos[2*(c + d*x)] + 12*a^3*A*(c + d*x)*Cos[4*(c + d*x)] - 12*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C)
+ b^3*(4*A + 3*C))*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*(8*a^3*B + 12*a*b^2*B + 12*a^
2*b*(2*A + C) + b^3*(4*A + 3*C))*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*A*b^3*Sin[c + d*
x] + 36*a*b^2*B*Sin[c + d*x] + 36*a^2*b*C*Sin[c + d*x] + 33*b^3*C*Sin[c + d*x] + 72*a*A*b^2*Sin[2*(c + d*x)] +
72*a^2*b*B*Sin[2*(c + d*x)] + 32*b^3*B*Sin[2*(c + d*x)] + 24*a^3*C*Sin[2*(c + d*x)] + 96*a*b^2*C*Sin[2*(c + d
*x)] + 12*A*b^3*Sin[3*(c + d*x)] + 36*a*b^2*B*Sin[3*(c + d*x)] + 36*a^2*b*C*Sin[3*(c + d*x)] + 9*b^3*C*Sin[3*(
c + d*x)] + 36*a*A*b^2*Sin[4*(c + d*x)] + 36*a^2*b*B*Sin[4*(c + d*x)] + 8*b^3*B*Sin[4*(c + d*x)] + 12*a^3*C*Si
n[4*(c + d*x)] + 24*a*b^2*C*Sin[4*(c + d*x)]))/(48*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Co
s[2*(c + d*x)]))

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 303, normalized size = 1.46 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 358, normalized size = 1.73 \begin {gather*} \frac {48 \, {\left (d x + c\right )} A a^{3} + 48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{2} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{3} - 3 \, C 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 )} - 36 \, C a^{2} 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 )} - 36 \, B 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 )} - 12 \, A b^{3} {\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 \, B a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 144 \, A a^{2} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 48 \, C a^{3} \tan \left (d x + c\right ) + 144 \, B a^{2} b \tan \left (d x + c\right ) + 144 \, A a b^{2} \tan \left (d x + c\right )}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/48*(48*(d*x + c)*A*a^3 + 48*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a*b^2 + 16*(tan(d*x + c)^3 + 3*tan(d*x + c))
*B*b^3 - 3*C*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)) - 36*C*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)) - 36*B*a*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(si
n(d*x + c) - 1)) - 12*A*b^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) -
1)) + 48*B*a^3*log(sec(d*x + c) + tan(d*x + c)) + 144*A*a^2*b*log(sec(d*x + c) + tan(d*x + c)) + 48*C*a^3*tan(
d*x + c) + 144*B*a^2*b*tan(d*x + c) + 144*A*a*b^2*tan(d*x + c))/d

________________________________________________________________________________________

Fricas [A]
time = 3.86, size = 257, normalized size = 1.24 \begin {gather*} \frac {48 \, A a^{3} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (8 \, B a^{3} + 12 \, {\left (2 \, A + C\right )} a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (8 \, B a^{3} + 12 \, {\left (2 \, A + C\right )} a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, C b^{3} + 8 \, {\left (3 \, C a^{3} + 9 \, B a^{2} b + 3 \, {\left (3 \, A + 2 \, C\right )} a b^{2} + 2 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (12 \, C a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/48*(48*A*a^3*d*x*cos(d*x + c)^4 + 3*(8*B*a^3 + 12*(2*A + C)*a^2*b + 12*B*a*b^2 + (4*A + 3*C)*b^3)*cos(d*x +
c)^4*log(sin(d*x + c) + 1) - 3*(8*B*a^3 + 12*(2*A + C)*a^2*b + 12*B*a*b^2 + (4*A + 3*C)*b^3)*cos(d*x + c)^4*lo
g(-sin(d*x + c) + 1) + 2*(6*C*b^3 + 8*(3*C*a^3 + 9*B*a^2*b + 3*(3*A + 2*C)*a*b^2 + 2*B*b^3)*cos(d*x + c)^3 + 3
*(12*C*a^2*b + 12*B*a*b^2 + (4*A + 3*C)*b^3)*cos(d*x + c)^2 + 8*(3*C*a*b^2 + B*b^3)*cos(d*x + c))*sin(d*x + c)
)/(d*cos(d*x + c)^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (197) = 394\).
time = 0.52, size = 759, normalized size = 3.67 \begin {gather*} \frac {24 \, {\left (d x + c\right )} A a^{3} + 3 \, {\left (8 \, B a^{3} + 24 \, A a^{2} b + 12 \, C a^{2} b + 12 \, B a b^{2} + 4 \, A b^{3} + 3 \, C b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, B a^{3} + 24 \, A a^{2} b + 12 \, C a^{2} b + 12 \, B a b^{2} + 4 \, A b^{3} + 3 \, C b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 216 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 216 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 120 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 216 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 216 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, C b^{3} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/24*(24*(d*x + c)*A*a^3 + 3*(8*B*a^3 + 24*A*a^2*b + 12*C*a^2*b + 12*B*a*b^2 + 4*A*b^3 + 3*C*b^3)*log(abs(tan(
1/2*d*x + 1/2*c) + 1)) - 3*(8*B*a^3 + 24*A*a^2*b + 12*C*a^2*b + 12*B*a*b^2 + 4*A*b^3 + 3*C*b^3)*log(abs(tan(1/
2*d*x + 1/2*c) - 1)) - 2*(24*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 72*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 36*C*a^2*b*tan
(1/2*d*x + 1/2*c)^7 + 72*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 72*C*a*b^2*tan(1
/2*d*x + 1/2*c)^7 - 12*A*b^3*tan(1/2*d*x + 1/2*c)^7 + 24*B*b^3*tan(1/2*d*x + 1/2*c)^7 - 15*C*b^3*tan(1/2*d*x +
1/2*c)^7 - 72*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 216*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^2*b*tan(1/2*d*x + 1/
2*c)^5 - 216*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 120*C*a*b^2*tan(1/2*d*x + 1/
2*c)^5 + 12*A*b^3*tan(1/2*d*x + 1/2*c)^5 - 40*B*b^3*tan(1/2*d*x + 1/2*c)^5 - 9*C*b^3*tan(1/2*d*x + 1/2*c)^5 +
72*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 216*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 216
*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*C*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 12*
A*b^3*tan(1/2*d*x + 1/2*c)^3 + 40*B*b^3*tan(1/2*d*x + 1/2*c)^3 - 9*C*b^3*tan(1/2*d*x + 1/2*c)^3 - 24*C*a^3*tan
(1/2*d*x + 1/2*c) - 72*B*a^2*b*tan(1/2*d*x + 1/2*c) - 36*C*a^2*b*tan(1/2*d*x + 1/2*c) - 72*A*a*b^2*tan(1/2*d*x
+ 1/2*c) - 36*B*a*b^2*tan(1/2*d*x + 1/2*c) - 72*C*a*b^2*tan(1/2*d*x + 1/2*c) - 12*A*b^3*tan(1/2*d*x + 1/2*c)
- 24*B*b^3*tan(1/2*d*x + 1/2*c) - 15*C*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d

________________________________________________________________________________________

Mupad [B]
time = 6.91, size = 2500, normalized size = 12.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(atan(((((A*b^3)/2 + B*a^3 + (3*C*b^3)/8 + 3*A*a^2*b + (3*B*a*b^2)/2 + (3*C*a^2*b)/2)*(32*A*a^3 + 16*A*b^3 + 3
2*B*a^3 + 12*C*b^3 + 96*A*a^2*b + 48*B*a*b^2 + 48*C*a^2*b) + tan(c/2 + (d*x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B
^2*a^6 + (9*C^2*b^6)/2 + 96*A^2*a^2*b^4 + 288*A^2*a^4*b^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 36*C^2*a^2*b^4 +
72*C^2*a^4*b^2 + 12*A*C*b^6 + 48*A*B*a*b^5 + 192*A*B*a^5*b + 36*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 +
120*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 168*B*C*a^3*b^3))*((A*b^3)/2 + B*a^3 + (3*C*b^3)/8 + 3*A*a^2*b + (3*B*a*b^
2)/2 + (3*C*a^2*b)/2)*1i - (((A*b^3)/2 + B*a^3 + (3*C*b^3)/8 + 3*A*a^2*b + (3*B*a*b^2)/2 + (3*C*a^2*b)/2)*(32*
A*a^3 + 16*A*b^3 + 32*B*a^3 + 12*C*b^3 + 96*A*a^2*b + 48*B*a*b^2 + 48*C*a^2*b) - tan(c/2 + (d*x)/2)*(32*A^2*a^
6 + 8*A^2*b^6 + 32*B^2*a^6 + (9*C^2*b^6)/2 + 96*A^2*a^2*b^4 + 288*A^2*a^4*b^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^
2 + 36*C^2*a^2*b^4 + 72*C^2*a^4*b^2 + 12*A*C*b^6 + 48*A*B*a*b^5 + 192*A*B*a^5*b + 36*B*C*a*b^5 + 96*B*C*a^5*b
+ 320*A*B*a^3*b^3 + 120*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 168*B*C*a^3*b^3))*((A*b^3)/2 + B*a^3 + (3*C*b^3)/8 + 3
*A*a^2*b + (3*B*a*b^2)/2 + (3*C*a^2*b)/2)*1i)/(64*A*B^2*a^9 - (((A*b^3)/2 + B*a^3 + (3*C*b^3)/8 + 3*A*a^2*b +
(3*B*a*b^2)/2 + (3*C*a^2*b)/2)*(32*A*a^3 + 16*A*b^3 + 32*B*a^3 + 12*C*b^3 + 96*A*a^2*b + 48*B*a*b^2 + 48*C*a^2
*b) - tan(c/2 + (d*x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B^2*a^6 + (9*C^2*b^6)/2 + 96*A^2*a^2*b^4 + 288*A^2*a^4*b
^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 36*C^2*a^2*b^4 + 72*C^2*a^4*b^2 + 12*A*C*b^6 + 48*A*B*a*b^5 + 192*A*B*a
^5*b + 36*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 120*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 168*B*C*a^3*b^3))*(
(A*b^3)/2 + B*a^3 + (3*C*b^3)/8 + 3*A*a^2*b + (3*B*a*b^2)/2 + (3*C*a^2*b)/2) - (((A*b^3)/2 + B*a^3 + (3*C*b^3)
/8 + 3*A*a^2*b + (3*B*a*b^2)/2 + (3*C*a^2*b)/2)*(32*A*a^3 + 16*A*b^3 + 32*B*a^3 + 12*C*b^3 + 96*A*a^2*b + 48*B
*a*b^2 + 48*C*a^2*b) + tan(c/2 + (d*x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B^2*a^6 + (9*C^2*b^6)/2 + 96*A^2*a^2*b^
4 + 288*A^2*a^4*b^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 36*C^2*a^2*b^4 + 72*C^2*a^4*b^2 + 12*A*C*b^6 + 48*A*B*
a*b^5 + 192*A*B*a^5*b + 36*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 120*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 16
8*B*C*a^3*b^3))*((A*b^3)/2 + B*a^3 + (3*C*b^3)/8 + 3*A*a^2*b + (3*B*a*b^2)/2 + (3*C*a^2*b)/2) - 64*A^2*B*a^9 -
192*A^3*a^8*b + 16*A^3*a^3*b^6 + 192*A^3*a^5*b^4 - 32*A^3*a^6*b^3 + 576*A^3*a^7*b^2 + 384*A^2*B*a^8*b - 96*A^
2*C*a^8*b + 144*A*B^2*a^5*b^4 + 192*A*B^2*a^7*b^2 + 96*A^2*B*a^4*b^5 + 640*A^2*B*a^6*b^3 - 96*A^2*B*a^7*b^2 +
9*A*C^2*a^3*b^6 + 72*A*C^2*a^5*b^4 + 144*A*C^2*a^7*b^2 + 24*A^2*C*a^3*b^6 + 240*A^2*C*a^5*b^4 - 24*A^2*C*a^6*b
^3 + 576*A^2*C*a^7*b^2 + 192*A*B*C*a^8*b + 72*A*B*C*a^4*b^5 + 336*A*B*C*a^6*b^3))*(A*b^3*1i + B*a^3*2i + (C*b^
3*3i)/4 + A*a^2*b*6i + B*a*b^2*3i + C*a^2*b*3i))/d - (tan(c/2 + (d*x)/2)^7*(2*B*b^3 - A*b^3 + 2*C*a^3 - (5*C*b
^3)/4 + 6*A*a*b^2 - 3*B*a*b^2 + 6*B*a^2*b + 6*C*a*b^2 - 3*C*a^2*b) + tan(c/2 + (d*x)/2)^3*(A*b^3 + (10*B*b^3)/
3 + 6*C*a^3 - (3*C*b^3)/4 + 18*A*a*b^2 + 3*B*a*b^2 + 18*B*a^2*b + 10*C*a*b^2 + 3*C*a^2*b) - tan(c/2 + (d*x)/2)
^5*((10*B*b^3)/3 - A*b^3 + 6*C*a^3 + (3*C*b^3)/4 + 18*A*a*b^2 - 3*B*a*b^2 + 18*B*a^2*b + 10*C*a*b^2 - 3*C*a^2*
b) - tan(c/2 + (d*x)/2)*(A*b^3 + 2*B*b^3 + 2*C*a^3 + (5*C*b^3)/4 + 6*A*a*b^2 + 3*B*a*b^2 + 6*B*a^2*b + 6*C*a*b
^2 + 3*C*a^2*b))/(d*(6*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x
)/2)^8 + 1)) + (2*A*a^3*atan((A*a^3*(tan(c/2 + (d*x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B^2*a^6 + (9*C^2*b^6)/2 +
96*A^2*a^2*b^4 + 288*A^2*a^4*b^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 36*C^2*a^2*b^4 + 72*C^2*a^4*b^2 + 12*A*C
*b^6 + 48*A*B*a*b^5 + 192*A*B*a^5*b + 36*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 120*A*C*a^2*b^4 + 288*A*
C*a^4*b^2 + 168*B*C*a^3*b^3) - A*a^3*(32*A*a^3 + 16*A*b^3 + 32*B*a^3 + 12*C*b^3 + 96*A*a^2*b + 48*B*a*b^2 + 48
*C*a^2*b)*1i) + A*a^3*(tan(c/2 + (d*x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B^2*a^6 + (9*C^2*b^6)/2 + 96*A^2*a^2*b^
4 + 288*A^2*a^4*b^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 36*C^2*a^2*b^4 + 72*C^2*a^4*b^2 + 12*A*C*b^6 + 48*A*B*
a*b^5 + 192*A*B*a^5*b + 36*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 120*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 16
8*B*C*a^3*b^3) + A*a^3*(32*A*a^3 + 16*A*b^3 + 32*B*a^3 + 12*C*b^3 + 96*A*a^2*b + 48*B*a*b^2 + 48*C*a^2*b)*1i))
/(64*A*B^2*a^9 - 64*A^2*B*a^9 - 192*A^3*a^8*b + A*a^3*(tan(c/2 + (d*x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B^2*a^6
+ (9*C^2*b^6)/2 + 96*A^2*a^2*b^4 + 288*A^2*a^4*b^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 36*C^2*a^2*b^4 + 72*C^
2*a^4*b^2 + 12*A*C*b^6 + 48*A*B*a*b^5 + 192*A*B*a^5*b + 36*B*C*a*b^5 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 120*A*
C*a^2*b^4 + 288*A*C*a^4*b^2 + 168*B*C*a^3*b^3) - A*a^3*(32*A*a^3 + 16*A*b^3 + 32*B*a^3 + 12*C*b^3 + 96*A*a^2*b
+ 48*B*a*b^2 + 48*C*a^2*b)*1i)*1i - A*a^3*(tan(c/2 + (d*x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B^2*a^6 + (9*C^2*b
^6)/2 + 96*A^2*a^2*b^4 + 288*A^2*a^4*b^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 36*C^2*a^2*b^4 + 72*C^2*a^4*b^2 +
12*A*C*b^6 + 48*A*B*a*b^5 + 192*A*B*a^5*b + 36...

________________________________________________________________________________________