3.10.21 \(\int \frac {\cos (c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [921]

Optimal. Leaf size=330 \[ -\frac {(3 A b-a B) x}{a^4}-\frac {\left (15 a^2 A b^4-6 A b^6+6 a^5 b B-5 a^3 b^3 B+2 a b^5 B-2 a^6 C-a^4 b^2 (12 A+C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {\left (11 a^2 A b^2-6 A b^4-5 a^3 b B+2 a b^3 B-a^4 (2 A-3 C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]

[Out]

-(3*A*b-B*a)*x/a^4-(15*a^2*A*b^4-6*A*b^6+6*a^5*b*B-5*a^3*b^3*B+2*a*b^5*B-2*a^6*C-a^4*b^2*(12*A+C))*arctanh((a-
b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^4/(a-b)^(5/2)/(a+b)^(5/2)/d-1/2*(11*a^2*A*b^2-6*A*b^4-5*a^3*b*B+2*a
*b^3*B-a^4*(2*A-3*C))*sin(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d
*x+c))^2-1/2*(3*A*b^4+4*a^3*b*B-a*b^3*B-2*a^4*C-a^2*b^2*(6*A+C))*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))

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Rubi [A]
time = 2.19, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4185, 4189, 4004, 3916, 2738, 214} \begin {gather*} -\frac {x (3 A b-a B)}{a^4}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\sin (c+d x) \left (-\left (a^4 (2 A-3 C)\right )-5 a^3 b B+11 a^2 A b^2+2 a b^3 B-6 A b^4\right )}{2 a^3 d \left (a^2-b^2\right )^2}-\frac {\sin (c+d x) \left (-2 a^4 C+4 a^3 b B-a^2 b^2 (6 A+C)-a b^3 B+3 A b^4\right )}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\left (-2 a^6 C+6 a^5 b B-a^4 b^2 (12 A+C)-5 a^3 b^3 B+15 a^2 A b^4+2 a b^5 B-6 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{5/2} (a+b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((3*A*b - a*B)*x)/a^4) - ((15*a^2*A*b^4 - 6*A*b^6 + 6*a^5*b*B - 5*a^3*b^3*B + 2*a*b^5*B - 2*a^6*C - a^4*b^2*
(12*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((11*a^
2*A*b^2 - 6*A*b^4 - 5*a^3*b*B + 2*a*b^3*B - a^4*(2*A - 3*C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 -
a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*A*b^4 + 4*a^3*b*B - a*b^3*B - 2
*a^4*C - a^2*b^2*(6*A + C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4185

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +
b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I
nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*
(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] &
& ILtQ[n, 0])

Rule 4189

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1
)*((d*Csc[e + f*x])^n/(a*f*n)), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (3 A b^2-a b B-a^2 (2 A-C)+2 a (A b-a B+b C) \sec (c+d x)-2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\cos (c+d x) \left (-11 a^2 A b^2+6 A b^4+5 a^3 b B-2 a b^3 B+a^4 (2 A-3 C)+a \left (A b^3+2 a^3 B+a b^2 B-a^2 b (4 A+3 C)\right ) \sec (c+d x)-\left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (11 a^2 A b^2-6 A b^4-5 a^3 b B+2 a b^3 B-a^4 (2 A-3 C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {2 \left (a^2-b^2\right )^2 (3 A b-a B)+a \left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {(3 A b-a B) x}{a^4}-\frac {\left (11 a^2 A b^2-6 A b^4-5 a^3 b B+2 a b^3 B-a^4 (2 A-3 C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (15 a^2 A b^4-6 A b^6+6 a^5 b B-5 a^3 b^3 B+2 a b^5 B-2 a^6 C-a^4 b^2 (12 A+C)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac {(3 A b-a B) x}{a^4}-\frac {\left (11 a^2 A b^2-6 A b^4-5 a^3 b B+2 a b^3 B-a^4 (2 A-3 C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (15 a^2 A b^4-6 A b^6+6 a^5 b B-5 a^3 b^3 B+2 a b^5 B-2 a^6 C-a^4 b^2 (12 A+C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^4 b \left (a^2-b^2\right )^2}\\ &=-\frac {(3 A b-a B) x}{a^4}-\frac {\left (11 a^2 A b^2-6 A b^4-5 a^3 b B+2 a b^3 B-a^4 (2 A-3 C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (15 a^2 A b^4-6 A b^6+6 a^5 b B-5 a^3 b^3 B+2 a b^5 B-2 a^6 C-a^4 b^2 (12 A+C)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 b \left (a^2-b^2\right )^2 d}\\ &=-\frac {(3 A b-a B) x}{a^4}+\frac {\left (12 a^4 A b^2-15 a^2 A b^4+6 A b^6-6 a^5 b B+5 a^3 b^3 B-2 a b^5 B+2 a^6 C+a^4 b^2 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {\left (11 a^2 A b^2-6 A b^4-5 a^3 b B+2 a b^3 B-a^4 (2 A-3 C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (3 A b^4+4 a^3 b B-a b^3 B-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.14, size = 1015, normalized size = 3.08 \begin {gather*} -\frac {2 (3 A b-a B) x (b+a \cos (c+d x))^3 \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {\left (12 a^4 A b^2-15 a^2 A b^4+6 A b^6-6 a^5 b B+5 a^3 b^3 B-2 a b^5 B+2 a^6 C+a^4 b^2 C\right ) (b+a \cos (c+d x))^3 \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {2 i \text {ArcTan}\left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \cos (c)}{a^4 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {2 \text {ArcTan}\left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \sin (c)}{a^4 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}\right )}{\left (-a^2+b^2\right )^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {(b+a \cos (c+d x)) \sec (c) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-A b^5 \sin (c)+a b^4 B \sin (c)-a^2 b^3 C \sin (c)+a A b^4 \sin (d x)-a^2 b^3 B \sin (d x)+a^3 b^2 C \sin (d x)\right )}{a^4 \left (a^2-b^2\right ) d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {(b+a \cos (c+d x))^2 \sec (c) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (9 a^2 A b^4 \sin (c)-6 A b^6 \sin (c)-7 a^3 b^3 B \sin (c)+4 a b^5 B \sin (c)+5 a^4 b^2 C \sin (c)-2 a^2 b^4 C \sin (c)-8 a^3 A b^3 \sin (d x)+5 a A b^5 \sin (d x)+6 a^4 b^2 B \sin (d x)-3 a^2 b^4 B \sin (d x)-4 a^5 b C \sin (d x)+a^3 b^3 C \sin (d x)\right )}{a^4 \left (a^2-b^2\right )^2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {2 A (b+a \cos (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \tan (c+d x)}{a^3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*(3*A*b - a*B)*x*(b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a^4*(A + 2*C
+ 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((12*a^4*A*b^2 - 15*a^2*A*b^4 + 6*A*b^6 -
6*a^5*b*B + 5*a^3*b^3*B - 2*a*b^5*B + 2*a^6*C + a^4*b^2*C)*(b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c +
d*x] + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) -
(I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c]
)/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[C
os[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*
Sin[c + (d*x)/2])]*Sin[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^2*(A + 2*C + 2*
B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]*(A +
B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-(A*b^5*Sin[c]) + a*b^4*B*Sin[c] - a^2*b^3*C*Sin[c] + a*A*b^4*Sin[d*x] - a
^2*b^3*B*Sin[d*x] + a^3*b^2*C*Sin[d*x]))/(a^4*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*
(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*
(9*a^2*A*b^4*Sin[c] - 6*A*b^6*Sin[c] - 7*a^3*b^3*B*Sin[c] + 4*a*b^5*B*Sin[c] + 5*a^4*b^2*C*Sin[c] - 2*a^2*b^4*
C*Sin[c] - 8*a^3*A*b^3*Sin[d*x] + 5*a*A*b^5*Sin[d*x] + 6*a^4*b^2*B*Sin[d*x] - 3*a^2*b^4*B*Sin[d*x] - 4*a^5*b*C
*Sin[d*x] + a^3*b^3*C*Sin[d*x]))/(a^4*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b
*Sec[c + d*x])^3) + (2*A*(b + a*Cos[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Tan[c + d*x])/(a^3*d*(
A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)

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Maple [A]
time = 0.71, size = 398, normalized size = 1.21

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+\left (3 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{4}}-\frac {2 \left (\frac {-\frac {\left (8 a^{2} A \,b^{2}+a A \,b^{3}-4 A \,b^{4}-6 a^{3} b B -a^{2} b^{2} B +2 a \,b^{3} B +4 a^{4} C +a^{3} b C \right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (8 a^{2} A \,b^{2}-a A \,b^{3}-4 A \,b^{4}-6 a^{3} b B +a^{2} b^{2} B +2 a \,b^{3} B +4 a^{4} C -a^{3} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (12 A \,a^{4} b^{2}-15 a^{2} A \,b^{4}+6 A \,b^{6}-6 a^{5} b B +5 a^{3} b^{3} B -2 a \,b^{5} B +2 a^{6} C +a^{4} b^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4}}}{d}\)	\(398\)
default	\(\frac {-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+\left (3 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{4}}-\frac {2 \left (\frac {-\frac {\left (8 a^{2} A \,b^{2}+a A \,b^{3}-4 A \,b^{4}-6 a^{3} b B -a^{2} b^{2} B +2 a \,b^{3} B +4 a^{4} C +a^{3} b C \right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (8 a^{2} A \,b^{2}-a A \,b^{3}-4 A \,b^{4}-6 a^{3} b B +a^{2} b^{2} B +2 a \,b^{3} B +4 a^{4} C -a^{3} b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (12 A \,a^{4} b^{2}-15 a^{2} A \,b^{4}+6 A \,b^{6}-6 a^{5} b B +5 a^{3} b^{3} B -2 a \,b^{5} B +2 a^{6} C +a^{4} b^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4}}}{d}\)	\(398\)
risch	\(\text {Expression too large to display}\)	\(1891\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-2/a^4*(-A*a*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+(3*A*b-B*a)*arctan(tan(1/2*d*x+1/2*c)))-2/a^4*((
-1/2*(8*A*a^2*b^2+A*a*b^3-4*A*b^4-6*B*a^3*b-B*a^2*b^2+2*B*a*b^3+4*C*a^4+C*a^3*b)*a*b/(a-b)/(a^2+2*a*b+b^2)*tan
(1/2*d*x+1/2*c)^3+1/2*b*a*(8*A*a^2*b^2-A*a*b^3-4*A*b^4-6*B*a^3*b+B*a^2*b^2+2*B*a*b^3+4*C*a^4-C*a^3*b)/(a+b)/(a
-b)^2*tan(1/2*d*x+1/2*c))/(a*tan(1/2*d*x+1/2*c)^2-b*tan(1/2*d*x+1/2*c)^2-a-b)^2-1/2*(12*A*a^4*b^2-15*A*a^2*b^4
+6*A*b^6-6*B*a^5*b+5*B*a^3*b^3-2*B*a*b^5+2*C*a^6+C*a^4*b^2)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a
-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 811 vs. \(2 (307) = 614\).
time = 2.98, size = 1680, normalized size = 5.09 \begin {gather*} \left [\frac {4 \, {\left (B a^{9} - 3 \, A a^{8} b - 3 \, B a^{7} b^{2} + 9 \, A a^{6} b^{3} + 3 \, B a^{5} b^{4} - 9 \, A a^{4} b^{5} - B a^{3} b^{6} + 3 \, A a^{2} b^{7}\right )} d x \cos \left (d x + c\right )^{2} + 8 \, {\left (B a^{8} b - 3 \, A a^{7} b^{2} - 3 \, B a^{6} b^{3} + 9 \, A a^{5} b^{4} + 3 \, B a^{4} b^{5} - 9 \, A a^{3} b^{6} - B a^{2} b^{7} + 3 \, A a b^{8}\right )} d x \cos \left (d x + c\right ) + 4 \, {\left (B a^{7} b^{2} - 3 \, A a^{6} b^{3} - 3 \, B a^{5} b^{4} + 9 \, A a^{4} b^{5} + 3 \, B a^{3} b^{6} - 9 \, A a^{2} b^{7} - B a b^{8} + 3 \, A b^{9}\right )} d x + {\left (2 \, C a^{6} b^{2} - 6 \, B a^{5} b^{3} + {\left (12 \, A + C\right )} a^{4} b^{4} + 5 \, B a^{3} b^{5} - 15 \, A a^{2} b^{6} - 2 \, B a b^{7} + 6 \, A b^{8} + {\left (2 \, C a^{8} - 6 \, B a^{7} b + {\left (12 \, A + C\right )} a^{6} b^{2} + 5 \, B a^{5} b^{3} - 15 \, A a^{4} b^{4} - 2 \, B a^{3} b^{5} + 6 \, A a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, C a^{7} b - 6 \, B a^{6} b^{2} + {\left (12 \, A + C\right )} a^{5} b^{3} + 5 \, B a^{4} b^{4} - 15 \, A a^{3} b^{5} - 2 \, B a^{2} b^{6} + 6 \, A a b^{7}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + 2 \, {\left ({\left (2 \, A - 3 \, C\right )} a^{7} b^{2} + 5 \, B a^{6} b^{3} - {\left (13 \, A - 3 \, C\right )} a^{5} b^{4} - 7 \, B a^{4} b^{5} + 17 \, A a^{3} b^{6} + 2 \, B a^{2} b^{7} - 6 \, A a b^{8} + 2 \, {\left (A a^{9} - 3 \, A a^{7} b^{2} + 3 \, A a^{5} b^{4} - A a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, {\left (A - C\right )} a^{8} b + 6 \, B a^{7} b^{2} - 5 \, {\left (4 \, A - C\right )} a^{6} b^{3} - 9 \, B a^{5} b^{4} + {\left (25 \, A - C\right )} a^{4} b^{5} + 3 \, B a^{3} b^{6} - 9 \, A a^{2} b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{12} - 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} - a^{6} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{11} b - 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} - a^{5} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{10} b^{2} - 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} - a^{4} b^{8}\right )} d\right )}}, \frac {2 \, {\left (B a^{9} - 3 \, A a^{8} b - 3 \, B a^{7} b^{2} + 9 \, A a^{6} b^{3} + 3 \, B a^{5} b^{4} - 9 \, A a^{4} b^{5} - B a^{3} b^{6} + 3 \, A a^{2} b^{7}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (B a^{8} b - 3 \, A a^{7} b^{2} - 3 \, B a^{6} b^{3} + 9 \, A a^{5} b^{4} + 3 \, B a^{4} b^{5} - 9 \, A a^{3} b^{6} - B a^{2} b^{7} + 3 \, A a b^{8}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (B a^{7} b^{2} - 3 \, A a^{6} b^{3} - 3 \, B a^{5} b^{4} + 9 \, A a^{4} b^{5} + 3 \, B a^{3} b^{6} - 9 \, A a^{2} b^{7} - B a b^{8} + 3 \, A b^{9}\right )} d x + {\left (2 \, C a^{6} b^{2} - 6 \, B a^{5} b^{3} + {\left (12 \, A + C\right )} a^{4} b^{4} + 5 \, B a^{3} b^{5} - 15 \, A a^{2} b^{6} - 2 \, B a b^{7} + 6 \, A b^{8} + {\left (2 \, C a^{8} - 6 \, B a^{7} b + {\left (12 \, A + C\right )} a^{6} b^{2} + 5 \, B a^{5} b^{3} - 15 \, A a^{4} b^{4} - 2 \, B a^{3} b^{5} + 6 \, A a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, C a^{7} b - 6 \, B a^{6} b^{2} + {\left (12 \, A + C\right )} a^{5} b^{3} + 5 \, B a^{4} b^{4} - 15 \, A a^{3} b^{5} - 2 \, B a^{2} b^{6} + 6 \, A a b^{7}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left ({\left (2 \, A - 3 \, C\right )} a^{7} b^{2} + 5 \, B a^{6} b^{3} - {\left (13 \, A - 3 \, C\right )} a^{5} b^{4} - 7 \, B a^{4} b^{5} + 17 \, A a^{3} b^{6} + 2 \, B a^{2} b^{7} - 6 \, A a b^{8} + 2 \, {\left (A a^{9} - 3 \, A a^{7} b^{2} + 3 \, A a^{5} b^{4} - A a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, {\left (A - C\right )} a^{8} b + 6 \, B a^{7} b^{2} - 5 \, {\left (4 \, A - C\right )} a^{6} b^{3} - 9 \, B a^{5} b^{4} + {\left (25 \, A - C\right )} a^{4} b^{5} + 3 \, B a^{3} b^{6} - 9 \, A a^{2} b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{12} - 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} - a^{6} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{11} b - 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} - a^{5} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{10} b^{2} - 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} - a^{4} b^{8}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(B*a^9 - 3*A*a^8*b - 3*B*a^7*b^2 + 9*A*a^6*b^3 + 3*B*a^5*b^4 - 9*A*a^4*b^5 - B*a^3*b^6 + 3*A*a^2*b^7)*
d*x*cos(d*x + c)^2 + 8*(B*a^8*b - 3*A*a^7*b^2 - 3*B*a^6*b^3 + 9*A*a^5*b^4 + 3*B*a^4*b^5 - 9*A*a^3*b^6 - B*a^2*
b^7 + 3*A*a*b^8)*d*x*cos(d*x + c) + 4*(B*a^7*b^2 - 3*A*a^6*b^3 - 3*B*a^5*b^4 + 9*A*a^4*b^5 + 3*B*a^3*b^6 - 9*A
*a^2*b^7 - B*a*b^8 + 3*A*b^9)*d*x + (2*C*a^6*b^2 - 6*B*a^5*b^3 + (12*A + C)*a^4*b^4 + 5*B*a^3*b^5 - 15*A*a^2*b
^6 - 2*B*a*b^7 + 6*A*b^8 + (2*C*a^8 - 6*B*a^7*b + (12*A + C)*a^6*b^2 + 5*B*a^5*b^3 - 15*A*a^4*b^4 - 2*B*a^3*b^
5 + 6*A*a^2*b^6)*cos(d*x + c)^2 + 2*(2*C*a^7*b - 6*B*a^6*b^2 + (12*A + C)*a^5*b^3 + 5*B*a^4*b^4 - 15*A*a^3*b^5
- 2*B*a^2*b^6 + 6*A*a*b^7)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)
^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x +
c) + b^2)) + 2*((2*A - 3*C)*a^7*b^2 + 5*B*a^6*b^3 - (13*A - 3*C)*a^5*b^4 - 7*B*a^4*b^5 + 17*A*a^3*b^6 + 2*B*a^
2*b^7 - 6*A*a*b^8 + 2*(A*a^9 - 3*A*a^7*b^2 + 3*A*a^5*b^4 - A*a^3*b^6)*cos(d*x + c)^2 + (4*(A - C)*a^8*b + 6*B*
a^7*b^2 - 5*(4*A - C)*a^6*b^3 - 9*B*a^5*b^4 + (25*A - C)*a^4*b^5 + 3*B*a^3*b^6 - 9*A*a^2*b^7)*cos(d*x + c))*si
n(d*x + c))/((a^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*d*cos(d*x + c)^2 + 2*(a^11*b - 3*a^9*b^3 + 3*a^7*b^5 -
a^5*b^7)*d*cos(d*x + c) + (a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*d), 1/2*(2*(B*a^9 - 3*A*a^8*b - 3*B*a^7
*b^2 + 9*A*a^6*b^3 + 3*B*a^5*b^4 - 9*A*a^4*b^5 - B*a^3*b^6 + 3*A*a^2*b^7)*d*x*cos(d*x + c)^2 + 4*(B*a^8*b - 3*
A*a^7*b^2 - 3*B*a^6*b^3 + 9*A*a^5*b^4 + 3*B*a^4*b^5 - 9*A*a^3*b^6 - B*a^2*b^7 + 3*A*a*b^8)*d*x*cos(d*x + c) +
2*(B*a^7*b^2 - 3*A*a^6*b^3 - 3*B*a^5*b^4 + 9*A*a^4*b^5 + 3*B*a^3*b^6 - 9*A*a^2*b^7 - B*a*b^8 + 3*A*b^9)*d*x +
(2*C*a^6*b^2 - 6*B*a^5*b^3 + (12*A + C)*a^4*b^4 + 5*B*a^3*b^5 - 15*A*a^2*b^6 - 2*B*a*b^7 + 6*A*b^8 + (2*C*a^8
- 6*B*a^7*b + (12*A + C)*a^6*b^2 + 5*B*a^5*b^3 - 15*A*a^4*b^4 - 2*B*a^3*b^5 + 6*A*a^2*b^6)*cos(d*x + c)^2 + 2*
(2*C*a^7*b - 6*B*a^6*b^2 + (12*A + C)*a^5*b^3 + 5*B*a^4*b^4 - 15*A*a^3*b^5 - 2*B*a^2*b^6 + 6*A*a*b^7)*cos(d*x
+ c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + ((2*A - 3*C
)*a^7*b^2 + 5*B*a^6*b^3 - (13*A - 3*C)*a^5*b^4 - 7*B*a^4*b^5 + 17*A*a^3*b^6 + 2*B*a^2*b^7 - 6*A*a*b^8 + 2*(A*a
^9 - 3*A*a^7*b^2 + 3*A*a^5*b^4 - A*a^3*b^6)*cos(d*x + c)^2 + (4*(A - C)*a^8*b + 6*B*a^7*b^2 - 5*(4*A - C)*a^6*
b^3 - 9*B*a^5*b^4 + (25*A - C)*a^4*b^5 + 3*B*a^3*b^6 - 9*A*a^2*b^7)*cos(d*x + c))*sin(d*x + c))/((a^12 - 3*a^1
0*b^2 + 3*a^8*b^4 - a^6*b^6)*d*cos(d*x + c)^2 + 2*(a^11*b - 3*a^9*b^3 + 3*a^7*b^5 - a^5*b^7)*d*cos(d*x + c) +
(a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (307) = 614\).
time = 0.56, size = 667, normalized size = 2.02 \begin {gather*} \frac {\frac {{\left (2 \, C a^{6} - 6 \, B a^{5} b + 12 \, A a^{4} b^{2} + C a^{4} b^{2} + 5 \, B a^{3} b^{3} - 15 \, A a^{2} b^{4} - 2 \, B a b^{5} + 6 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {4 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}} + \frac {{\left (B a - 3 \, A b\right )} {\left (d x + c\right )}}{a^{4}} + \frac {2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*C*a^6 - 6*B*a^5*b + 12*A*a^4*b^2 + C*a^4*b^2 + 5*B*a^3*b^3 - 15*A*a^2*b^4 - 2*B*a*b^5 + 6*A*b^6)*(pi*floor
(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2
+ b^2)))/((a^8 - 2*a^6*b^2 + a^4*b^4)*sqrt(-a^2 + b^2)) + (4*C*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 6*B*a^4*b^2*tan
(1/2*d*x + 1/2*c)^3 - 3*C*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 + 8*A*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^3*b^3*ta
n(1/2*d*x + 1/2*c)^3 - C*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 7*A*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 + 3*B*a^2*b^4*tan
(1/2*d*x + 1/2*c)^3 - 5*A*a*b^5*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 4*A*b^6*tan(1/2*d*
x + 1/2*c)^3 - 4*C*a^5*b*tan(1/2*d*x + 1/2*c) + 6*B*a^4*b^2*tan(1/2*d*x + 1/2*c) - 3*C*a^4*b^2*tan(1/2*d*x + 1
/2*c) - 8*A*a^3*b^3*tan(1/2*d*x + 1/2*c) + 5*B*a^3*b^3*tan(1/2*d*x + 1/2*c) + C*a^3*b^3*tan(1/2*d*x + 1/2*c) -
7*A*a^2*b^4*tan(1/2*d*x + 1/2*c) - 3*B*a^2*b^4*tan(1/2*d*x + 1/2*c) + 5*A*a*b^5*tan(1/2*d*x + 1/2*c) - 2*B*a*
b^5*tan(1/2*d*x + 1/2*c) + 4*A*b^6*tan(1/2*d*x + 1/2*c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*(a*tan(1/2*d*x + 1/2*c)^
2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^2) + (B*a - 3*A*b)*(d*x + c)/a^4 + 2*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*
x + 1/2*c)^2 + 1)*a^3))/d

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Mupad [B]
time = 11.07, size = 2500, normalized size = 7.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(2*A*a^5 + 6*A*b^5 - 12*A*a^2*b^3 - 4*A*a^3*b^2 - B*a^2*b^3 + 6*B*a^3*b^2 + C*a^3*b^2 + 3
*A*a*b^4 + 2*A*a^4*b - 2*B*a*b^4 - 4*C*a^4*b))/((a + b)*(a^5 - 2*a^4*b + a^3*b^2)) - (tan(c/2 + (d*x)/2)^5*(2*
A*a^5 - 6*A*b^5 + 12*A*a^2*b^3 - 4*A*a^3*b^2 - B*a^2*b^3 - 6*B*a^3*b^2 + C*a^3*b^2 + 3*A*a*b^4 - 2*A*a^4*b + 2
*B*a*b^4 + 4*C*a^4*b))/((a^3*b - a^4)*(a + b)^2) + (2*tan(c/2 + (d*x)/2)^3*(2*A*a^6 - 6*A*b^6 + 13*A*a^2*b^4 -
6*A*a^4*b^2 - 5*B*a^3*b^3 + 3*C*a^4*b^2 + 2*B*a*b^5))/(a*(a^2*b - a^3)*(a + b)^2*(a - b)))/(d*(2*a*b + tan(c/
2 + (d*x)/2)^2*(2*a*b - a^2 + 3*b^2) + tan(c/2 + (d*x)/2)^6*(a^2 - 2*a*b + b^2) + a^2 + b^2 - tan(c/2 + (d*x)/
2)^4*(2*a*b + a^2 - 3*b^2))) + (log(tan(c/2 + (d*x)/2) - 1i)*(3*A*b - B*a)*1i)/(a^4*d) - (log(tan(c/2 + (d*x)/
2) + 1i)*(A*b*3i - B*a*1i))/(a^4*d) - (atan(((((a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(72*A^2*b^12
+ 4*B^2*a^12 + 4*C^2*a^12 - 72*A^2*a*b^11 - 8*B^2*a^11*b - 288*A^2*a^2*b^10 + 288*A^2*a^3*b^9 + 441*A^2*a^4*b^
8 - 432*A^2*a^5*b^7 - 288*A^2*a^6*b^6 + 288*A^2*a^7*b^5 + 36*A^2*a^8*b^4 - 72*A^2*a^9*b^3 + 36*A^2*a^10*b^2 +
8*B^2*a^2*b^10 - 8*B^2*a^3*b^9 - 32*B^2*a^4*b^8 + 32*B^2*a^5*b^7 + 57*B^2*a^6*b^6 - 48*B^2*a^7*b^5 - 52*B^2*a^
8*b^4 + 32*B^2*a^9*b^3 + 24*B^2*a^10*b^2 + C^2*a^8*b^4 + 4*C^2*a^10*b^2 - 48*A*B*a*b^11 - 24*A*B*a^11*b - 24*B
*C*a^11*b + 48*A*B*a^2*b^10 + 192*A*B*a^3*b^9 - 192*A*B*a^4*b^8 - 318*A*B*a^5*b^7 + 288*A*B*a^6*b^6 + 252*A*B*
a^7*b^5 - 192*A*B*a^8*b^4 - 72*A*B*a^9*b^3 + 48*A*B*a^10*b^2 + 12*A*C*a^4*b^8 - 6*A*C*a^6*b^6 - 36*A*C*a^8*b^4
+ 48*A*C*a^10*b^2 - 4*B*C*a^5*b^7 + 2*B*C*a^7*b^5 + 8*B*C*a^9*b^3))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^
8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (((a + b)^5*(a - b)^5)^(1/2)*((8*(4*B*a^18 + 4*C*a^18 + 12*A*a^
8*b^10 - 6*A*a^9*b^9 - 54*A*a^10*b^8 + 24*A*a^11*b^7 + 96*A*a^12*b^6 - 42*A*a^13*b^5 - 78*A*a^14*b^4 + 36*A*a^
15*b^3 + 24*A*a^16*b^2 - 4*B*a^9*b^9 + 2*B*a^10*b^8 + 18*B*a^11*b^7 - 4*B*a^12*b^6 - 36*B*a^13*b^5 + 6*B*a^14*
b^4 + 34*B*a^15*b^3 - 8*B*a^16*b^2 - 2*C*a^11*b^7 + 2*C*a^12*b^6 + 6*C*a^15*b^3 - 6*C*a^16*b^2 - 12*A*a^17*b -
12*B*a^17*b - 4*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^1
4*b^2) - (4*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^6 + 2*C*a^6 - 15*A*a^2*b^4 + 12*A*a^4*b^2 +
5*B*a^3*b^3 + C*a^4*b^2 - 2*B*a*b^5 - 6*B*a^5*b)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^
7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/((a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^
8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*
a^11*b^2)))*(6*A*b^6 + 2*C*a^6 - 15*A*a^2*b^4 + 12*A*a^4*b^2 + 5*B*a^3*b^3 + C*a^4*b^2 - 2*B*a*b^5 - 6*B*a^5*b
))/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)))*(6*A*b^6 + 2*C*a^6 - 15*A*a^2*b^
4 + 12*A*a^4*b^2 + 5*B*a^3*b^3 + C*a^4*b^2 - 2*B*a*b^5 - 6*B*a^5*b)*1i)/(2*(a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a
^8*b^6 + 10*a^10*b^4 - 5*a^12*b^2)) + (((a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(72*A^2*b^12 + 4*B^2
*a^12 + 4*C^2*a^12 - 72*A^2*a*b^11 - 8*B^2*a^11*b - 288*A^2*a^2*b^10 + 288*A^2*a^3*b^9 + 441*A^2*a^4*b^8 - 432
*A^2*a^5*b^7 - 288*A^2*a^6*b^6 + 288*A^2*a^7*b^5 + 36*A^2*a^8*b^4 - 72*A^2*a^9*b^3 + 36*A^2*a^10*b^2 + 8*B^2*a
^2*b^10 - 8*B^2*a^3*b^9 - 32*B^2*a^4*b^8 + 32*B^2*a^5*b^7 + 57*B^2*a^6*b^6 - 48*B^2*a^7*b^5 - 52*B^2*a^8*b^4 +
32*B^2*a^9*b^3 + 24*B^2*a^10*b^2 + C^2*a^8*b^4 + 4*C^2*a^10*b^2 - 48*A*B*a*b^11 - 24*A*B*a^11*b - 24*B*C*a^11
*b + 48*A*B*a^2*b^10 + 192*A*B*a^3*b^9 - 192*A*B*a^4*b^8 - 318*A*B*a^5*b^7 + 288*A*B*a^6*b^6 + 252*A*B*a^7*b^5
- 192*A*B*a^8*b^4 - 72*A*B*a^9*b^3 + 48*A*B*a^10*b^2 + 12*A*C*a^4*b^8 - 6*A*C*a^6*b^6 - 36*A*C*a^8*b^4 + 48*A
*C*a^10*b^2 - 4*B*C*a^5*b^7 + 2*B*C*a^7*b^5 + 8*B*C*a^9*b^3))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 +
3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (((a + b)^5*(a - b)^5)^(1/2)*((8*(4*B*a^18 + 4*C*a^18 + 12*A*a^8*b^10
- 6*A*a^9*b^9 - 54*A*a^10*b^8 + 24*A*a^11*b^7 + 96*A*a^12*b^6 - 42*A*a^13*b^5 - 78*A*a^14*b^4 + 36*A*a^15*b^3
+ 24*A*a^16*b^2 - 4*B*a^9*b^9 + 2*B*a^10*b^8 + 18*B*a^11*b^7 - 4*B*a^12*b^6 - 36*B*a^13*b^5 + 6*B*a^14*b^4 + 3
4*B*a^15*b^3 - 8*B*a^16*b^2 - 2*C*a^11*b^7 + 2*C*a^12*b^6 + 6*C*a^15*b^3 - 6*C*a^16*b^2 - 12*A*a^17*b - 12*B*a
^17*b - 4*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2)
+ (4*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^6 + 2*C*a^6 - 15*A*a^2*b^4 + 12*A*a^4*b^2 + 5*B*a^3
*b^3 + C*a^4*b^2 - 2*B*a*b^5 - 6*B*a^5*b)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^7 - 48*
a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2))/((a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 +
10*a^10*b^4 - 5*a^12*b^2)*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^
2)))*(6*A*b^6 + 2*C*a^6 - 15*A*a^2*b^4 + 12*A*a^4*b^2 + 5*B*a^3*b^3 + C*a^4*b^2 - 2*B*a*b^5 - 6*B*a^5*b))/(2*(
a^14 - a^4*b^10 + 5*a^6*b^8 - 10*a^8*b^6 + 10*a...

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