3.10.22 \(\int \frac {\cos ^2(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [922]
Optimal. Leaf size=453 \[ \frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
[Out]
1/2*(12*A*b^2-6*a*b*B+a^2*(A+2*C))*x/a^5-b*(12*A*b^6-12*a^5*b*B+15*a^3*b^3*B-6*a*b^5*B-a^2*b^4*(29*A-2*C)+5*a^
4*b^2*(4*A-C)+6*a^6*C)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(5/2)/(a+b)^(5/2)/d-1/2*(
12*A*b^5-2*a^5*B+11*a^3*b^2*B-6*a*b^4*B+a^4*b*(6*A-5*C)-a^2*b^3*(21*A-2*C))*sin(d*x+c)/a^4/(a^2-b^2)^2/d+1/2*(
6*A*b^4+6*a^3*b*B-3*a*b^3*B+a^4*(A-4*C)-a^2*b^2*(10*A-C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*(A*b^2-a
*(B*b-C*a))*cos(d*x+c)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+1/2*(7*A*a^2*b^2-4*A*b^4-5*B*a^3*b+2*B*a*b^
3+3*C*a^4)*cos(d*x+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 = 3.49, antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4185, 4189,
4004, 3916, 2738, 214} \begin {gather*} \frac {\sin (c+d x) \cos (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 {x \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{2 a^5}+\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 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) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{2 a^4 d \left (a^2-b^2\right )^2}-\frac {b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 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^5 d (a-b)^{5/2} (a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]
[Out]
((12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*
b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a
- b)^(5/2)*(a + b)^(5/2)*d) - ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(
21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b
^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[
c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4
*C)*Cos[c + d*x]*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 ^2(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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (2 A b^2-a b B-a^2 (A-C)\right )+2 a (A b-a B+b C) \sec (c+d x)-3 \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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \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 ^2(c+d x) \left (2 \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right )+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)+2 \left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 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 (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \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 (2 \left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right )+2 a \left (2 A b^4+4 a^3 b B-a b^3 B-a^2 b^2 (4 A+C)-a^4 (A+2 C)\right ) \sec (c+d x)-2 b \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \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 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right )+2 a b \left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 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^5 \left (a^2-b^2\right )^2 d}\\ &=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (20 a^4 A b^2-29 a^2 A b^4+12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B+6 a^6 C-5 a^4 b^2 C+2 a^2 b^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \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 ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \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 [A]
time = 4.97, size = 881, normalized size = 1.94 \begin {gather*} \frac {\frac {16 b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B+5 a^4 b^2 (4 A-C)+6 a^6 C+a^2 b^4 (-29 A+2 C)\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {4 a^8 A c+48 a^6 A b^2 c-12 a^4 A b^4 c-136 a^2 A b^6 c+96 A b^8 c-24 a^7 b B c+72 a^3 b^5 B c-48 a b^7 B c+8 a^8 c C-24 a^4 b^4 c C+16 a^2 b^6 c C+4 a^8 A d x+48 a^6 A b^2 d x-12 a^4 A b^4 d x-136 a^2 A b^6 d x+96 A b^8 d x-24 a^7 b B d x+72 a^3 b^5 B d x-48 a b^7 B d x+8 a^8 C d x-24 a^4 b^4 C d x+16 a^2 b^6 C d x+16 a b \left (a^2-b^2\right )^2 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) (c+d x) \cos (c+d x)+4 \left (a^3-a b^2\right )^2 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) (c+d x) \cos (2 (c+d x))-8 a^7 A b \sin (c+d x)-32 a^5 A b^3 \sin (c+d x)+160 a^3 A b^5 \sin (c+d x)-96 a A b^7 \sin (c+d x)+4 a^8 B \sin (c+d x)+8 a^6 b^2 B \sin (c+d x)-84 a^4 b^4 B \sin (c+d x)+48 a^2 b^6 B \sin (c+d x)+40 a^5 b^3 C \sin (c+d x)-16 a^3 b^5 C \sin (c+d x)+2 a^8 A \sin (2 (c+d x))-48 a^6 A b^2 \sin (2 (c+d x))+130 a^4 A b^4 \sin (2 (c+d x))-72 a^2 A b^6 \sin (2 (c+d x))+16 a^7 b B \sin (2 (c+d x))-64 a^5 b^3 B \sin (2 (c+d x))+36 a^3 b^5 B \sin (2 (c+d x))+24 a^6 b^2 C \sin (2 (c+d x))-12 a^4 b^4 C \sin (2 (c+d x))-8 a^7 A b \sin (3 (c+d x))+16 a^5 A b^3 \sin (3 (c+d x))-8 a^3 A b^5 \sin (3 (c+d x))+4 a^8 B \sin (3 (c+d x))-8 a^6 b^2 B \sin (3 (c+d x))+4 a^4 b^4 B \sin (3 (c+d x))+a^8 A \sin (4 (c+d x))-2 a^6 A b^2 \sin (4 (c+d x))+a^4 A b^4 \sin (4 (c+d x))}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{16 a^5 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]
[Out]
((16*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B + 5*a^4*b^2*(4*A - C) + 6*a^6*C + a^2*b^4*(-29*A + 2*
C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^8*A*c + 48*a^6*A*b^2*c - 12
*a^4*A*b^4*c - 136*a^2*A*b^6*c + 96*A*b^8*c - 24*a^7*b*B*c + 72*a^3*b^5*B*c - 48*a*b^7*B*c + 8*a^8*c*C - 24*a^
4*b^4*c*C + 16*a^2*b^6*c*C + 4*a^8*A*d*x + 48*a^6*A*b^2*d*x - 12*a^4*A*b^4*d*x - 136*a^2*A*b^6*d*x + 96*A*b^8*
d*x - 24*a^7*b*B*d*x + 72*a^3*b^5*B*d*x - 48*a*b^7*B*d*x + 8*a^8*C*d*x - 24*a^4*b^4*C*d*x + 16*a^2*b^6*C*d*x +
16*a*b*(a^2 - b^2)^2*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[c + d*x] + 4*(a^3 - a*b^2)^2*(12*A*b^
2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[2*(c + d*x)] - 8*a^7*A*b*Sin[c + d*x] - 32*a^5*A*b^3*Sin[c + d*x] +
160*a^3*A*b^5*Sin[c + d*x] - 96*a*A*b^7*Sin[c + d*x] + 4*a^8*B*Sin[c + d*x] + 8*a^6*b^2*B*Sin[c + d*x] - 84*a
^4*b^4*B*Sin[c + d*x] + 48*a^2*b^6*B*Sin[c + d*x] + 40*a^5*b^3*C*Sin[c + d*x] - 16*a^3*b^5*C*Sin[c + d*x] + 2*
a^8*A*Sin[2*(c + d*x)] - 48*a^6*A*b^2*Sin[2*(c + d*x)] + 130*a^4*A*b^4*Sin[2*(c + d*x)] - 72*a^2*A*b^6*Sin[2*(
c + d*x)] + 16*a^7*b*B*Sin[2*(c + d*x)] - 64*a^5*b^3*B*Sin[2*(c + d*x)] + 36*a^3*b^5*B*Sin[2*(c + d*x)] + 24*a
^6*b^2*C*Sin[2*(c + d*x)] - 12*a^4*b^4*C*Sin[2*(c + d*x)] - 8*a^7*A*b*Sin[3*(c + d*x)] + 16*a^5*A*b^3*Sin[3*(c
+ d*x)] - 8*a^3*A*b^5*Sin[3*(c + d*x)] + 4*a^8*B*Sin[3*(c + d*x)] - 8*a^6*b^2*B*Sin[3*(c + d*x)] + 4*a^4*b^4*
B*Sin[3*(c + d*x)] + a^8*A*Sin[4*(c + d*x)] - 2*a^6*A*b^2*Sin[4*(c + d*x)] + a^4*A*b^4*Sin[4*(c + d*x)])/((a^2
- b^2)^2*(b + a*Cos[c + d*x])^2))/(16*a^5*d)
________________________________________________________________________________________
Maple [A]
time = 0.49, size = 487, normalized size = 1.08
| | |
method |
result |
size |
| | |
derivativedivides |
\(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-3 a A b +a^{2} B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-3 a A b +a^{2} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+12 A \,b^{2}-6 a b B +2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}+\frac {2 b \left (\frac {-\frac {\left (10 a^{2} A \,b^{2}+a A \,b^{3}-6 A \,b^{4}-8 a^{3} b B -a^{2} b^{2} B +4 a \,b^{3} B +6 a^{4} C +a^{3} b C -2 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (10 a^{2} A \,b^{2}-a A \,b^{3}-6 A \,b^{4}-8 a^{3} b B +a^{2} b^{2} B +4 a \,b^{3} B +6 a^{4} C -a^{3} b C -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (20 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+12 A \,b^{6}-12 a^{5} b B +15 a^{3} b^{3} B -6 a \,b^{5} B +6 a^{6} C -5 a^{4} b^{2} C +2 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\) |
\(487\) |
default |
\(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-3 a A b +a^{2} B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2}-3 a A b +a^{2} B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (A \,a^{2}+12 A \,b^{2}-6 a b B +2 a^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}+\frac {2 b \left (\frac {-\frac {\left (10 a^{2} A \,b^{2}+a A \,b^{3}-6 A \,b^{4}-8 a^{3} b B -a^{2} b^{2} B +4 a \,b^{3} B +6 a^{4} C +a^{3} b C -2 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (10 a^{2} A \,b^{2}-a A \,b^{3}-6 A \,b^{4}-8 a^{3} b B +a^{2} b^{2} B +4 a \,b^{3} B +6 a^{4} C -a^{3} b C -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (20 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+12 A \,b^{6}-12 a^{5} b B +15 a^{3} b^{3} B -6 a \,b^{5} B +6 a^{6} C -5 a^{4} b^{2} C +2 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\) |
\(487\) |
risch |
\(\text {Expression too large to display}\) |
\(2160\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^2*(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^5*(((-1/2*A*a^2-3*a*A*b+a^2*B)*tan(1/2*d*x+1/2*c)^3+(1/2*A*a^2-3*a*A*b+a^2*B)*tan(1/2*d*x+1/2*c))/(1+
tan(1/2*d*x+1/2*c)^2)^2+1/2*(A*a^2+12*A*b^2-6*B*a*b+2*C*a^2)*arctan(tan(1/2*d*x+1/2*c)))+2*b/a^5*((-1/2*(10*A*
a^2*b^2+A*a*b^3-6*A*b^4-8*B*a^3*b-B*a^2*b^2+4*B*a*b^3+6*C*a^4+C*a^3*b-2*C*a^2*b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)*t
an(1/2*d*x+1/2*c)^3+1/2*b*a*(10*A*a^2*b^2-A*a*b^3-6*A*b^4-8*B*a^3*b+B*a^2*b^2+4*B*a*b^3+6*C*a^4-C*a^3*b-2*C*a^
2*b^2)/(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*(20*A*a^4*b
^2-29*A*a^2*b^4+12*A*b^6-12*B*a^5*b+15*B*a^3*b^3-6*B*a*b^5+6*C*a^6-5*C*a^4*b^2+2*C*a^2*b^4)/(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))))
________________________________________________________________________________________
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)^2*(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. 1034 vs.
\(2 (433) = 866\).
time = 4.30, size = 2127, normalized size = 4.70 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^2*(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*(2*((A + 2*C)*a^10 - 6*B*a^9*b + 3*(3*A - 2*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(11*A - 2*C)*a^6*b^4 - 18*B*a^5
*b^5 + (35*A - 2*C)*a^4*b^6 + 6*B*a^3*b^7 - 12*A*a^2*b^8)*d*x*cos(d*x + c)^2 + 4*((A + 2*C)*a^9*b - 6*B*a^8*b^
2 + 3*(3*A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4*b^6 + (35*A - 2*C)*a^3*b^7 + 6*B*
a^2*b^8 - 12*A*a*b^9)*d*x*cos(d*x + c) + 2*((A + 2*C)*a^8*b^2 - 6*B*a^7*b^3 + 3*(3*A - 2*C)*a^6*b^4 + 18*B*a^5
*b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18*B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*d*x + (6*C*a^6*b^
3 - 12*B*a^5*b^4 + 5*(4*A - C)*a^4*b^5 + 15*B*a^3*b^6 - (29*A - 2*C)*a^2*b^7 - 6*B*a*b^8 + 12*A*b^9 + (6*C*a^8
*b - 12*B*a^7*b^2 + 5*(4*A - C)*a^6*b^3 + 15*B*a^5*b^4 - (29*A - 2*C)*a^4*b^5 - 6*B*a^3*b^6 + 12*A*a^2*b^7)*co
s(d*x + c)^2 + 2*(6*C*a^7*b^2 - 12*B*a^6*b^3 + 5*(4*A - C)*a^5*b^4 + 15*B*a^4*b^5 - (29*A - 2*C)*a^3*b^6 - 6*B
*a^2*b^7 + 12*A*a*b^8)*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*B*a^8*b^2 - (6*A - 5*C)*a^7*b^3 - 13*B*a^6*b^4 + (27*A - 7*C)*a^5*b^5 + 17*B*a^4*b^6 - (33*A - 2*
C)*a^3*b^7 - 6*B*a^2*b^8 + 12*A*a*b^9 + (A*a^10 - 3*A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b^6)*cos(d*x + c)^3 + 2*(B
*a^10 - 2*A*a^9*b - 3*B*a^8*b^2 + 6*A*a^7*b^3 + 3*B*a^6*b^4 - 6*A*a^5*b^5 - B*a^4*b^6 + 2*A*a^3*b^7)*cos(d*x +
c)^2 + (4*B*a^9*b - (11*A - 6*C)*a^8*b^2 - 20*B*a^7*b^3 + (43*A - 9*C)*a^6*b^4 + 25*B*a^5*b^5 - (50*A - 3*C)*
a^4*b^6 - 9*B*a^3*b^7 + 18*A*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^13 - 3*a^11*b^2 + 3*a^9*b^4 - a^7*b^6)*d
*cos(d*x + c)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*d*cos(d*x + c) + (a^11*b^2 - 3*a^9*b^4 + 3*a^7
*b^6 - a^5*b^8)*d), 1/2*(((A + 2*C)*a^10 - 6*B*a^9*b + 3*(3*A - 2*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(11*A - 2*C)*a
^6*b^4 - 18*B*a^5*b^5 + (35*A - 2*C)*a^4*b^6 + 6*B*a^3*b^7 - 12*A*a^2*b^8)*d*x*cos(d*x + c)^2 + 2*((A + 2*C)*a
^9*b - 6*B*a^8*b^2 + 3*(3*A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4*b^6 + (35*A - 2*
C)*a^3*b^7 + 6*B*a^2*b^8 - 12*A*a*b^9)*d*x*cos(d*x + c) + ((A + 2*C)*a^8*b^2 - 6*B*a^7*b^3 + 3*(3*A - 2*C)*a^6
*b^4 + 18*B*a^5*b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18*B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*d*
x - (6*C*a^6*b^3 - 12*B*a^5*b^4 + 5*(4*A - C)*a^4*b^5 + 15*B*a^3*b^6 - (29*A - 2*C)*a^2*b^7 - 6*B*a*b^8 + 12*A
*b^9 + (6*C*a^8*b - 12*B*a^7*b^2 + 5*(4*A - C)*a^6*b^3 + 15*B*a^5*b^4 - (29*A - 2*C)*a^4*b^5 - 6*B*a^3*b^6 + 1
2*A*a^2*b^7)*cos(d*x + c)^2 + 2*(6*C*a^7*b^2 - 12*B*a^6*b^3 + 5*(4*A - C)*a^5*b^4 + 15*B*a^4*b^5 - (29*A - 2*C
)*a^3*b^6 - 6*B*a^2*b^7 + 12*A*a*b^8)*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*B*a^8*b^2 - (6*A - 5*C)*a^7*b^3 - 13*B*a^6*b^4 + (27*A - 7*C)*a^5*b^5 +
17*B*a^4*b^6 - (33*A - 2*C)*a^3*b^7 - 6*B*a^2*b^8 + 12*A*a*b^9 + (A*a^10 - 3*A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b
^6)*cos(d*x + c)^3 + 2*(B*a^10 - 2*A*a^9*b - 3*B*a^8*b^2 + 6*A*a^7*b^3 + 3*B*a^6*b^4 - 6*A*a^5*b^5 - B*a^4*b^6
+ 2*A*a^3*b^7)*cos(d*x + c)^2 + (4*B*a^9*b - (11*A - 6*C)*a^8*b^2 - 20*B*a^7*b^3 + (43*A - 9*C)*a^6*b^4 + 25*
B*a^5*b^5 - (50*A - 3*C)*a^4*b^6 - 9*B*a^3*b^7 + 18*A*a^2*b^8)*cos(d*x + c))*sin(d*x + c))/((a^13 - 3*a^11*b^2
+ 3*a^9*b^4 - a^7*b^6)*d*cos(d*x + c)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*d*cos(d*x + c) + (a^1
1*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*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 ^{2}{\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)**2*(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)**2/(a + b*sec(c + d*x))**3, x)
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3408 vs.
\(2 (433) = 866\).
time = 0.93, size = 3408, normalized size = 7.52 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="giac")
[Out]
-1/2*(((a^6 - a^5*b + 10*a^4*b^2 + 10*a^3*b^3 - 23*a^2*b^4 - 6*a*b^5 + 12*b^6)*sqrt(-a^2 + b^2)*A*abs(a^9 - 2*
a^7*b^2 + a^5*b^4)*abs(-a + b) - 3*(2*a^5*b + 2*a^4*b^2 - 4*a^3*b^3 - a^2*b^4 + 2*a*b^5)*sqrt(-a^2 + b^2)*B*ab
s(a^9 - 2*a^7*b^2 + a^5*b^4)*abs(-a + b) + (2*a^6 + 4*a^5*b - 4*a^4*b^2 - a^3*b^3 + 2*a^2*b^4)*sqrt(-a^2 + b^2
)*C*abs(a^9 - 2*a^7*b^2 + a^5*b^4)*abs(-a + b) - (a^15 - a^14*b + 8*a^13*b^2 - 28*a^12*b^3 - 42*a^11*b^4 + 111
*a^10*b^5 + 68*a^9*b^6 - 158*a^8*b^7 - 47*a^7*b^8 + 100*a^6*b^9 + 12*a^5*b^10 - 24*a^4*b^11)*sqrt(-a^2 + b^2)*
A*abs(-a + b) + 3*(2*a^14*b - 6*a^13*b^2 - 8*a^12*b^3 + 21*a^11*b^4 + 12*a^10*b^5 - 28*a^9*b^6 - 8*a^8*b^7 + 1
7*a^7*b^8 + 2*a^6*b^9 - 4*a^5*b^10)*sqrt(-a^2 + b^2)*B*abs(-a + b) - (2*a^15 - 8*a^14*b - 8*a^13*b^2 + 25*a^12
*b^3 + 12*a^11*b^4 - 30*a^10*b^5 - 8*a^9*b^6 + 17*a^8*b^7 + 2*a^7*b^8 - 4*a^6*b^9)*sqrt(-a^2 + b^2)*C*abs(-a +
b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^8*b - 2*a^6*b^3 + a^4*b^5 + sqrt
((a^9 + a^8*b - 2*a^7*b^2 - 2*a^6*b^3 + a^5*b^4 + a^4*b^5)*(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^
4*b^5) + (a^8*b - 2*a^6*b^3 + a^4*b^5)^2))/(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5))))/((a^9
- 2*a^7*b^2 + a^5*b^4)^2*(a^2 - 2*a*b + b^2) + (a^10*b - 2*a^9*b^2 - a^8*b^3 + 4*a^7*b^4 - a^6*b^5 - 2*a^5*b^6
+ a^4*b^7)*abs(a^9 - 2*a^7*b^2 + a^5*b^4)) + (A*a^15 + 2*C*a^15 - A*a^14*b - 6*B*a^14*b - 8*C*a^14*b + 8*A*a^
13*b^2 + 18*B*a^13*b^2 - 8*C*a^13*b^2 - 28*A*a^12*b^3 + 24*B*a^12*b^3 + 25*C*a^12*b^3 - 42*A*a^11*b^4 - 63*B*a
^11*b^4 + 12*C*a^11*b^4 + 111*A*a^10*b^5 - 36*B*a^10*b^5 - 30*C*a^10*b^5 + 68*A*a^9*b^6 + 84*B*a^9*b^6 - 8*C*a
^9*b^6 - 158*A*a^8*b^7 + 24*B*a^8*b^7 + 17*C*a^8*b^7 - 47*A*a^7*b^8 - 51*B*a^7*b^8 + 2*C*a^7*b^8 + 100*A*a^6*b
^9 - 6*B*a^6*b^9 - 4*C*a^6*b^9 + 12*A*a^5*b^10 + 12*B*a^5*b^10 - 24*A*a^4*b^11 + A*a^6*abs(a^9 - 2*a^7*b^2 + a
^5*b^4) + 2*C*a^6*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - A*a^5*b*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 6*B*a^5*b*abs(a^9
- 2*a^7*b^2 + a^5*b^4) + 4*C*a^5*b*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 10*A*a^4*b^2*abs(a^9 - 2*a^7*b^2 + a^5*b^4
) - 6*B*a^4*b^2*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 4*C*a^4*b^2*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 10*A*a^3*b^3*abs
(a^9 - 2*a^7*b^2 + a^5*b^4) + 12*B*a^3*b^3*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - C*a^3*b^3*abs(a^9 - 2*a^7*b^2 + a^
5*b^4) - 23*A*a^2*b^4*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 3*B*a^2*b^4*abs(a^9 - 2*a^7*b^2 + a^5*b^4) + 2*C*a^2*b^
4*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 6*A*a*b^5*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 6*B*a*b^5*abs(a^9 - 2*a^7*b^2 +
a^5*b^4) + 12*A*b^6*abs(a^9 - 2*a^7*b^2 + a^5*b^4))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1
/2*c)/sqrt(-(a^8*b - 2*a^6*b^3 + a^4*b^5 - sqrt((a^9 + a^8*b - 2*a^7*b^2 - 2*a^6*b^3 + a^5*b^4 + a^4*b^5)*(a^9
- a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5) + (a^8*b - 2*a^6*b^3 + a^4*b^5)^2))/(a^9 - a^8*b - 2*a^7
*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5))))/(a^8*b*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - 2*a^6*b^3*abs(a^9 - 2*a^7*b^2
+ a^5*b^4) + a^4*b^5*abs(a^9 - 2*a^7*b^2 + a^5*b^4) - (a^9 - 2*a^7*b^2 + a^5*b^4)^2) + 2*(A*a^7*tan(1/2*d*x +
1/2*c)^7 - 2*B*a^7*tan(1/2*d*x + 1/2*c)^7 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 4*B*a^6*b*tan(1/2*d*x + 1/2*c)
^7 - 13*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 2*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 6*C*a^5*b^2*tan(1/2*d*x + 1/2*
c)^7 - 2*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 5*C*a^4*b^3*tan(1/2*d*x + 1/
2*c)^7 + 33*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 3*C*a^3*b^4*tan(1/2*d*x +
1/2*c)^7 - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 2*C*a^2*b^5*tan(1/2*d*x
+ 1/2*c)^7 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 6*B*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^7*tan(1/2*d*x + 1/2
*c)^7 - 3*A*a^7*tan(1/2*d*x + 1/2*c)^5 + 2*B*a^7*tan(1/2*d*x + 1/2*c)^5 - 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 4
*B*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 10*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 6
*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5
- 15*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 35*B*a^3*b^4*tan(1/2*d*x + 1/2*c
)^5 - 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 9*B*a^2*b^5*tan(1/2*d*x + 1/2
*c)^5 + 6*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 18*B*a*b^6*tan(1/2*d*x + 1/2*
c)^5 + 36*A*b^7*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^3 + 2*B*a^7*tan(1/2*d*x + 1/2*c)^3 - 4*A
*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 4*B*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 10*B*a
^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 16*
B*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 +
35*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 - 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^
3 - 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^...
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Mupad [B]
time = 21.93, size = 2500, normalized size = 5.52 \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)^2*(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)^5*(3*A*a^7 - 36*A*b^7 - 2*B*a^7 + 67*A*a^2*b^5 - 29*A*a^3*b^4 - 26*A*a^4*b^3 + 5*A*a^5*b^
2 - 9*B*a^2*b^5 - 35*B*a^3*b^4 + 16*B*a^4*b^3 + 10*B*a^5*b^2 - 6*C*a^2*b^5 + 3*C*a^3*b^4 + 15*C*a^4*b^3 - 6*C*
a^5*b^2 + 18*A*a*b^6 + 4*A*a^6*b + 18*B*a*b^6 - 4*B*a^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) - (tan(c/2 +
(d*x)/2)^3*(3*A*a^7 + 36*A*b^7 + 2*B*a^7 - 67*A*a^2*b^5 - 29*A*a^3*b^4 + 26*A*a^4*b^3 + 5*A*a^5*b^2 - 9*B*a^2
*b^5 + 35*B*a^3*b^4 + 16*B*a^4*b^3 - 10*B*a^5*b^2 + 6*C*a^2*b^5 + 3*C*a^3*b^4 - 15*C*a^4*b^3 - 6*C*a^5*b^2 + 1
8*A*a*b^6 - 4*A*a^6*b - 18*B*a*b^6 - 4*B*a^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) + (tan(c/2 + (d*x)/2)^7
*(A*a^6 - 12*A*b^6 - 2*B*a^6 + 23*A*a^2*b^4 - 10*A*a^3*b^3 - 8*A*a^4*b^2 - 3*B*a^2*b^4 - 12*B*a^3*b^3 + 4*B*a^
4*b^2 - 2*C*a^2*b^4 + C*a^3*b^3 + 6*C*a^4*b^2 + 6*A*a*b^5 + 5*A*a^5*b + 6*B*a*b^5 + 2*B*a^5*b))/((a^4*b - a^5)
*(a + b)^2) + (tan(c/2 + (d*x)/2)*(A*a^6 - 12*A*b^6 + 2*B*a^6 + 23*A*a^2*b^4 + 10*A*a^3*b^3 - 8*A*a^4*b^2 + 3*
B*a^2*b^4 - 12*B*a^3*b^3 - 4*B*a^4*b^2 - 2*C*a^2*b^4 - C*a^3*b^3 + 6*C*a^4*b^2 - 6*A*a*b^5 - 5*A*a^5*b + 6*B*a
*b^5 + 2*B*a^5*b))/((a + b)*(a^6 - 2*a^5*b + a^4*b^2)))/(d*(2*a*b - tan(c/2 + (d*x)/2)^4*(2*a^2 - 6*b^2) + tan
(c/2 + (d*x)/2)^2*(4*a*b + 4*b^2) - tan(c/2 + (d*x)/2)^6*(4*a*b - 4*b^2) + tan(c/2 + (d*x)/2)^8*(a^2 - 2*a*b +
b^2) + a^2 + b^2)) - (atan(((((((4*(4*A*a^21 + 8*C*a^21 - 48*A*a^10*b^11 + 24*A*a^11*b^10 + 212*A*a^12*b^9 -
100*A*a^13*b^8 - 360*A*a^14*b^7 + 164*A*a^15*b^6 + 276*A*a^16*b^5 - 120*A*a^17*b^4 - 80*A*a^18*b^3 + 28*A*a^19
*b^2 + 24*B*a^11*b^10 - 12*B*a^12*b^9 - 108*B*a^13*b^8 + 48*B*a^14*b^7 + 192*B*a^15*b^6 - 84*B*a^16*b^5 - 156*
B*a^17*b^4 + 72*B*a^18*b^3 + 48*B*a^19*b^2 - 8*C*a^12*b^9 + 4*C*a^13*b^8 + 36*C*a^14*b^7 - 8*C*a^15*b^6 - 72*C
*a^16*b^5 + 12*C*a^17*b^4 + 68*C*a^18*b^3 - 16*C*a^19*b^2 - 24*B*a^20*b - 24*C*a^20*b))/(a^18*b + a^19 - a^12*
b^7 - a^13*b^6 + 3*a^14*b^5 + 3*a^15*b^4 - 3*a^16*b^3 - 3*a^17*b^2) - (8*tan(c/2 + (d*x)/2)*(A*b^2*6i + a^2*((
A*1i)/2 + C*1i) - B*a*b*3i)*(8*a^19*b - 8*a^10*b^10 + 8*a^11*b^9 + 32*a^12*b^8 - 32*a^13*b^7 - 48*a^14*b^6 + 4
8*a^15*b^5 + 32*a^16*b^4 - 32*a^17*b^3 - 8*a^18*b^2))/(a^5*(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3
*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2)))*(A*b^2*6i + a^2*((A*1i)/2 + C*1i) - B*a*b*3i))/a^5 + (8*tan(c/2 + (d*x)
/2)*(A^2*a^14 + 288*A^2*b^14 + 4*C^2*a^14 - 288*A^2*a*b^13 - 2*A^2*a^13*b - 8*C^2*a^13*b - 1104*A^2*a^2*b^12 +
1104*A^2*a^3*b^11 + 1538*A^2*a^4*b^10 - 1538*A^2*a^5*b^9 - 827*A^2*a^6*b^8 + 872*A^2*a^7*b^7 + 18*A^2*a^8*b^6
- 108*A^2*a^9*b^5 + 74*A^2*a^10*b^4 - 40*A^2*a^11*b^3 + 21*A^2*a^12*b^2 + 72*B^2*a^2*b^12 - 72*B^2*a^3*b^11 -
288*B^2*a^4*b^10 + 288*B^2*a^5*b^9 + 441*B^2*a^6*b^8 - 432*B^2*a^7*b^7 - 288*B^2*a^8*b^6 + 288*B^2*a^9*b^5 +
36*B^2*a^10*b^4 - 72*B^2*a^11*b^3 + 36*B^2*a^12*b^2 + 8*C^2*a^4*b^10 - 8*C^2*a^5*b^9 - 32*C^2*a^6*b^8 + 32*C^2
*a^7*b^7 + 57*C^2*a^8*b^6 - 48*C^2*a^9*b^5 - 52*C^2*a^10*b^4 + 32*C^2*a^11*b^3 + 24*C^2*a^12*b^2 + 4*A*C*a^14
- 288*A*B*a*b^13 - 12*A*B*a^13*b - 8*A*C*a^13*b - 24*B*C*a^13*b + 288*A*B*a^2*b^12 + 1128*A*B*a^3*b^11 - 1128*
A*B*a^4*b^10 - 1650*A*B*a^5*b^9 + 1632*A*B*a^6*b^8 + 984*A*B*a^7*b^7 - 1008*A*B*a^8*b^6 - 72*A*B*a^9*b^5 + 192
*A*B*a^10*b^4 - 108*A*B*a^11*b^3 + 24*A*B*a^12*b^2 + 96*A*C*a^2*b^12 - 96*A*C*a^3*b^11 - 376*A*C*a^4*b^10 + 37
6*A*C*a^5*b^9 + 598*A*C*a^6*b^8 - 544*A*C*a^7*b^7 - 444*A*C*a^8*b^6 + 336*A*C*a^9*b^5 + 104*A*C*a^10*b^4 - 64*
A*C*a^11*b^3 + 36*A*C*a^12*b^2 - 48*B*C*a^3*b^11 + 48*B*C*a^4*b^10 + 192*B*C*a^5*b^9 - 192*B*C*a^6*b^8 - 318*B
*C*a^7*b^7 + 288*B*C*a^8*b^6 + 252*B*C*a^9*b^5 - 192*B*C*a^10*b^4 - 72*B*C*a^11*b^3 + 48*B*C*a^12*b^2))/(a^14*
b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2))*(A*b^2*6i + a^2*((A*1i)/2 +
C*1i) - B*a*b*3i)*1i)/a^5 - (((((4*(4*A*a^21 + 8*C*a^21 - 48*A*a^10*b^11 + 24*A*a^11*b^10 + 212*A*a^12*b^9 -
100*A*a^13*b^8 - 360*A*a^14*b^7 + 164*A*a^15*b^6 + 276*A*a^16*b^5 - 120*A*a^17*b^4 - 80*A*a^18*b^3 + 28*A*a^19
*b^2 + 24*B*a^11*b^10 - 12*B*a^12*b^9 - 108*B*a^13*b^8 + 48*B*a^14*b^7 + 192*B*a^15*b^6 - 84*B*a^16*b^5 - 156*
B*a^17*b^4 + 72*B*a^18*b^3 + 48*B*a^19*b^2 - 8*C*a^12*b^9 + 4*C*a^13*b^8 + 36*C*a^14*b^7 - 8*C*a^15*b^6 - 72*C
*a^16*b^5 + 12*C*a^17*b^4 + 68*C*a^18*b^3 - 16*C*a^19*b^2 - 24*B*a^20*b - 24*C*a^20*b))/(a^18*b + a^19 - a^12*
b^7 - a^13*b^6 + 3*a^14*b^5 + 3*a^15*b^4 - 3*a^16*b^3 - 3*a^17*b^2) + (8*tan(c/2 + (d*x)/2)*(A*b^2*6i + a^2*((
A*1i)/2 + C*1i) - B*a*b*3i)*(8*a^19*b - 8*a^10*b^10 + 8*a^11*b^9 + 32*a^12*b^8 - 32*a^13*b^7 - 48*a^14*b^6 + 4
8*a^15*b^5 + 32*a^16*b^4 - 32*a^17*b^3 - 8*a^18*b^2))/(a^5*(a^14*b + a^15 - a^8*b^7 - a^9*b^6 + 3*a^10*b^5 + 3
*a^11*b^4 - 3*a^12*b^3 - 3*a^13*b^2)))*(A*b^2*6i + a^2*((A*1i)/2 + C*1i) - B*a*b*3i))/a^5 - (8*tan(c/2 + (d*x)
/2)*(A^2*a^14 + 288*A^2*b^14 + 4*C^2*a^14 - 288*A^2*a*b^13 - 2*A^2*a^13*b - 8*C^2*a^13*b - 1104*A^2*a^2*b^12 +
1104*A^2*a^3*b^11 + 1538*A^2*a^4*b^10 - 1538*A^2*a^5*b^9 - 827*A^2*a^6*b^8 + 872*A^2*a^7*b^7 + 18*A^2*a^8*b^6
- 108*A^2*a^9*b^5 + 74*A^2*a^10*b^4 - 40*A^2*a...
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