3.914 \(\int \frac {\cos ^2(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^2} \, dx\)
Optimal. Leaf size=298 \[ -\frac {\sin (c+d x) \cos (c+d x) \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right )}{2 a^2 d \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {x \left (a^2 (A+2 C)-4 a b B+6 A b^2\right )}{2 a^4}+\frac {\sin (c+d x) \left (a^3 B-a^2 b (2 A-C)-2 a b^2 B+3 A b^3\right )}{a^3 d \left (a^2-b^2\right )}-\frac {2 b \left (2 a^4 C-3 a^3 b B+4 a^2 A b^2-a^2 b^2 C+2 a b^3 B-3 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}} \]
[Out]
1/2*(6*A*b^2-4*a*b*B+a^2*(A+2*C))*x/a^4-2*b*(4*A*a^2*b^2-3*A*b^4-3*B*a^3*b+2*B*a*b^3+2*C*a^4-C*a^2*b^2)*arctan
h((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^4/(a-b)^(3/2)/(a+b)^(3/2)/d+(3*A*b^3+a^3*B-2*a*b^2*B-a^2*b*(2*
A-C))*sin(d*x+c)/a^3/(a^2-b^2)/d-1/2*(3*A*b^2-2*a*b*B-a^2*(A-2*C))*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)/d+(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))
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Rubi [A] time = 1.24, antiderivative size = 298, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used =
{4100, 4104, 3919, 3831, 2659, 208} \[ \frac {\sin (c+d x) \left (-a^2 b (2 A-C)+a^3 B-2 a b^2 B+3 A b^3\right )}{a^3 d \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (a^2 (-(A-2 C))-2 a b B+3 A b^2\right )}{2 a^2 d \left (a^2-b^2\right )}-\frac {2 b \left (4 a^2 A b^2-a^2 b^2 C-3 a^3 b B+2 a^4 C+2 a b^3 B-3 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {x \left (a^2 (A+2 C)-4 a b B+6 A b^2\right )}{2 a^4} \]
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])^2,x]
[Out]
((6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*x)/(2*a^4) - (2*b*(4*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 2*a*b^3*B + 2*a^4*
C - a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((3*
A*b^3 + a^3*B - 2*a*b^2*B - a^2*b*(2*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - 2*a*b*B - a^2*(A
- 2*C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])
/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 2659
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 3831
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 3919
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 4100
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 4104
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))^2} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (3 A b^2-2 a b B-a^2 (A-2 C)+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)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {\cos (c+d x) \left (2 \left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right )+a \left (A b^2-2 a b B+a^2 (A+2 C)\right ) \sec (c+d x)-b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\int \frac {-\left (a^2-b^2\right ) \left (6 A b^2-4 a b B+a^2 (A+2 C)\right )+a b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}\\ &=\frac {\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) x}{2 a^4}+\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (b \left (3 A b^4+3 a^3 b B-2 a b^3 B-a^2 b^2 (4 A-C)-2 a^4 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^4 \left (a^2-b^2\right )}\\ &=\frac {\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) x}{2 a^4}+\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (3 A b^4+3 a^3 b B-2 a b^3 B-a^2 b^2 (4 A-C)-2 a^4 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^4 \left (a^2-b^2\right )}\\ &=\frac {\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) x}{2 a^4}+\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (2 \left (3 A b^4+3 a^3 b B-2 a b^3 B-a^2 b^2 (4 A-C)-2 a^4 C\right )\right ) \operatorname {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 \left (a^2-b^2\right ) d}\\ &=\frac {\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) x}{2 a^4}-\frac {2 b \left (4 a^2 A b^2-3 A b^4-3 a^3 b B+2 a b^3 B+2 a^4 C-a^2 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)^{3/2} (a+b)^{3/2} d}+\frac {\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.44, size = 206, normalized size = 0.69 \[ \frac {2 (c+d x) \left (a^2 (A+2 C)-4 a b B+6 A b^2\right )+a^2 A \sin (2 (c+d x))-\frac {8 b \left (-2 a^4 C+3 a^3 b B+a^2 b^2 (C-4 A)-2 a b^3 B+3 A b^4\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {4 a b^2 \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{(a-b) (a+b) (a \cos (c+d x)+b)}+4 a (a B-2 A b) \sin (c+d x)}{4 a^4 d} \]
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])^2,x]
[Out]
(2*(6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*(c + d*x) - (8*b*(3*A*b^4 + 3*a^3*b*B - 2*a*b^3*B - 2*a^4*C + a^2*b^2*(
-4*A + C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 4*a*(-2*A*b + a*B)*Sin[c
+ d*x] + (4*a*b^2*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + a^2*A*Sin[
2*(c + d*x)])/(4*a^4*d)
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fricas [A] time = 0.61, size = 1102, normalized size = 3.70 \[ \text {result too large to display} \]
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))^2,x, algorithm="fricas")
[Out]
[1/2*(((A + 2*C)*a^7 - 4*B*a^6*b + 4*(A - C)*a^5*b^2 + 8*B*a^4*b^3 - (11*A - 2*C)*a^3*b^4 - 4*B*a^2*b^5 + 6*A*
a*b^6)*d*x*cos(d*x + c) + ((A + 2*C)*a^6*b - 4*B*a^5*b^2 + 4*(A - C)*a^4*b^3 + 8*B*a^3*b^4 - (11*A - 2*C)*a^2*
b^5 - 4*B*a*b^6 + 6*A*b^7)*d*x + (2*C*a^4*b^2 - 3*B*a^3*b^3 + (4*A - C)*a^2*b^4 + 2*B*a*b^5 - 3*A*b^6 + (2*C*a
^5*b - 3*B*a^4*b^2 + (4*A - C)*a^3*b^3 + 2*B*a^2*b^4 - 3*A*a*b^5)*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*B*a^6*b - 2*(2*A - C)*a^5*b^2 - 6*B*a^4*b^3 + 2*(5*A - C
)*a^3*b^4 + 4*B*a^2*b^5 - 6*A*a*b^6 + (A*a^7 - 2*A*a^5*b^2 + A*a^3*b^4)*cos(d*x + c)^2 + (2*B*a^7 - 3*A*a^6*b
- 4*B*a^5*b^2 + 6*A*a^4*b^3 + 2*B*a^3*b^4 - 3*A*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/((a^9 - 2*a^7*b^2 + a^5*b
^4)*d*cos(d*x + c) + (a^8*b - 2*a^6*b^3 + a^4*b^5)*d), 1/2*(((A + 2*C)*a^7 - 4*B*a^6*b + 4*(A - C)*a^5*b^2 + 8
*B*a^4*b^3 - (11*A - 2*C)*a^3*b^4 - 4*B*a^2*b^5 + 6*A*a*b^6)*d*x*cos(d*x + c) + ((A + 2*C)*a^6*b - 4*B*a^5*b^2
+ 4*(A - C)*a^4*b^3 + 8*B*a^3*b^4 - (11*A - 2*C)*a^2*b^5 - 4*B*a*b^6 + 6*A*b^7)*d*x - 2*(2*C*a^4*b^2 - 3*B*a^
3*b^3 + (4*A - C)*a^2*b^4 + 2*B*a*b^5 - 3*A*b^6 + (2*C*a^5*b - 3*B*a^4*b^2 + (4*A - C)*a^3*b^3 + 2*B*a^2*b^4 -
3*A*a*b^5)*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^6*b - 2*(2*A - C)*a^5*b^2 - 6*B*a^4*b^3 + 2*(5*A - C)*a^3*b^4 + 4*B*a^2*b^5 - 6*A*a*b^6 + (A*a
^7 - 2*A*a^5*b^2 + A*a^3*b^4)*cos(d*x + c)^2 + (2*B*a^7 - 3*A*a^6*b - 4*B*a^5*b^2 + 6*A*a^4*b^3 + 2*B*a^3*b^4
- 3*A*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/((a^9 - 2*a^7*b^2 + a^5*b^4)*d*cos(d*x + c) + (a^8*b - 2*a^6*b^3 +
a^4*b^5)*d)]
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giac [A] time = 0.67, size = 380, normalized size = 1.28 \[ -\frac {\frac {4 \, {\left (2 \, C a^{4} b - 3 \, B a^{3} b^{2} + 4 \, A a^{2} b^{3} - C a^{2} b^{3} + 2 \, B a b^{4} - 3 \, A b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {4 \, {\left (C a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} - \frac {{\left (A a^{2} + 2 \, C a^{2} - 4 \, B a b + 6 \, A b^{2}\right )} {\left (d x + c\right )}}{a^{4}} + \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}}{2 \, d} \]
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))^2,x, algorithm="giac")
[Out]
-1/2*(4*(2*C*a^4*b - 3*B*a^3*b^2 + 4*A*a^2*b^3 - C*a^2*b^3 + 2*B*a*b^4 - 3*A*b^5)*(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^6 - a
^4*b^2)*sqrt(-a^2 + b^2)) + 4*(C*a^2*b^2*tan(1/2*d*x + 1/2*c) - B*a*b^3*tan(1/2*d*x + 1/2*c) + A*b^4*tan(1/2*d
*x + 1/2*c))/((a^5 - a^3*b^2)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)) - (A*a^2 + 2*C*a^
2 - 4*B*a*b + 6*A*b^2)*(d*x + c)/a^4 + 2*(A*a*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*tan(1/2*d*x + 1/2*c)^3 + 4*A*b*ta
n(1/2*d*x + 1/2*c)^3 - A*a*tan(1/2*d*x + 1/2*c) - 2*B*a*tan(1/2*d*x + 1/2*c) + 4*A*b*tan(1/2*d*x + 1/2*c))/((t
an(1/2*d*x + 1/2*c)^2 + 1)^2*a^3))/d
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maple [B] time = 0.99, size = 857, normalized size = 2.88 \[ \text {result too large to display} \]
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))^2,x)
[Out]
-2/d*b^4/a^3/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)*A+2/d*b^3/a^2/(a
^2-b^2)*tan(1/2*d*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)*B-2/d*b^2/a/(a^2-b^2)*tan(1/2*d
*x+1/2*c)/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)*C-8/d/a^2/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctan
h(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^3+6/d*b^5/a^4/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh(tan(
1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+6/d*b^2/a/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*
c)*(a-b)/((a-b)*(a+b))^(1/2))*B-4/d*b^4/a^3/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/(
(a-b)*(a+b))^(1/2))*B-4/d/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2)
)*C*b+2/d*b^3/a^2/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*C-1/d/
a^2/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*A-4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*
A*b+2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^2*B*tan(1/2*d*x+1/2*c)^3+1/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^2*A*tan(1/2*d*x
+1/2*c)-4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*A*b+2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^2*B*tan(1/2
*d*x+1/2*c)+1/d/a^2*arctan(tan(1/2*d*x+1/2*c))*A+6/d/a^4*arctan(tan(1/2*d*x+1/2*c))*A*b^2-4/d/a^3*arctan(tan(1
/2*d*x+1/2*c))*B*b+2/d/a^2*arctan(tan(1/2*d*x+1/2*c))*C
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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))^2,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 details)Is 4*a^2-4*b^2 positive or negative?
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mupad [B] time = 15.97, size = 9997, normalized size = 33.55 \[ \text {result too large to display} \]
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))^2,x)
[Out]
(atan(-((((((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b
^3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5
+ 12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*tan(c/2 +
(d*x)/2)*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a
^11*b^3 - 8*a^12*b^2))/(a^4*(a^8*b + a^9 - a^6*b^3 - a^7*b^2)))*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i))
/a^4 + (8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b -
120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^
2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^
3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*
a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^
4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 -
100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*
a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2
))*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i)*1i)/a^4 - (((((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^
9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*
b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/
(a^11*b + a^12 - a^9*b^3 - a^10*b^2) + (8*tan(c/2 + (d*x)/2)*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i)*(8*
a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a^12*b^2))/(a^4*(a^8*b + a^9 - a^6*b^3 - a^7*b^
2)))*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i))/a^4 - (8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^
2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^
2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^
6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2
*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^
9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 +
16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*
a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3
+ 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2))*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i)*1i)/a^4)/(
(((((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A
*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a
^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*tan(c/2 + (d*x)/2
)*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3
- 8*a^12*b^2))/(a^4*(a^8*b + a^9 - a^6*b^3 - a^7*b^2)))*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i))/a^4 + (
8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*A^2
*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^
2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B
^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10 - 9
6*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 -
40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A*C*
a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b^6
+ 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2))*(A*b^
2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i))/a^4 - (16*(108*A^3*b^11 - 54*A^3*a*b^10 + 8*C^3*a^10*b - 216*A^3*a^2
*b^9 + 81*A^3*a^3*b^8 + 63*A^3*a^4*b^7 - 9*A^3*a^5*b^6 + 41*A^3*a^6*b^5 - 4*A^3*a^7*b^4 + 4*A^3*a^8*b^3 - 32*B
^3*a^3*b^8 + 16*B^3*a^4*b^7 + 80*B^3*a^5*b^6 - 24*B^3*a^6*b^5 - 48*B^3*a^7*b^4 + 4*C^3*a^6*b^5 - 4*C^3*a^7*b^4
- 12*C^3*a^8*b^3 + 8*C^3*a^9*b^2 - 216*A^2*B*a*b^10 + 8*A*C^2*a^10*b + 2*A^2*C*a^10*b + 144*A*B^2*a^2*b^9 - 7
2*A*B^2*a^3*b^8 - 336*A*B^2*a^4*b^7 + 108*A*B^2*a^5*b^6 + 168*A*B^2*a^6*b^5 - 6*A*B^2*a^7*b^4 + 24*A*B^2*a^8*b
^3 + 108*A^2*B*a^2*b^9 + 468*A^2*B*a^3*b^8 - 162*A^2*B*a^4*b^7 - 186*A^2*B*a^5*b^6 + 15*A^2*B*a^6*b^5 - 63*A^2
*B*a^7*b^4 + 3*A^2*B*a^8*b^3 - 3*A^2*B*a^9*b^2 + 36*A*C^2*a^4*b^7 - 30*A*C^2*a^5*b^6 - 96*A*C^2*a^6*b^5 + 52*A
*C^2*a^7*b^4 + 52*A*C^2*a^8*b^3 + 108*A^2*C*a^2*b^9 - 72*A^2*C*a^3*b^8 - 252*A^2*C*a^4*b^7 + 111*A^2*C*a^5*b^6
+ 105*A^2*C*a^6*b^5 - 5*A^2*C*a^7*b^4 + 37*A^2*C*a^8*b^3 - 2*A^2*C*a^9*b^2 - 24*B*C^2*a^5*b^6 + 20*B*C^2*a^6*
b^5 + 68*B*C^2*a^7*b^4 - 36*B*C^2*a^8*b^3 - 44*B*C^2*a^9*b^2 + 48*B^2*C*a^4*b^7 - 32*B^2*C*a^5*b^6 - 128*B^2*C
*a^6*b^5 + 52*B^2*C*a^7*b^4 + 80*B^2*C*a^8*b^3 - 144*A*B*C*a^3*b^8 + 96*A*B*C*a^4*b^7 + 360*A*B*C*a^5*b^6 - 15
2*A*B*C*a^6*b^5 - 188*A*B*C*a^7*b^4 + 4*A*B*C*a^8*b^3 - 28*A*B*C*a^9*b^2))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2
) + (((((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 +
6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12
*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) + (8*tan(c/2 + (d*
x)/2)*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*
b^3 - 8*a^12*b^2))/(a^4*(a^8*b + a^9 - a^6*b^3 - a^7*b^2)))*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i))/a^4
- (8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120
*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^
8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 +
16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10
- 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^
6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*
A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*
b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2))*(
A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i))/a^4))*(A*b^2*3i + a^2*((A*1i)/2 + C*1i) - B*a*b*2i)*2i)/(a^4*d)
- ((tan(c/2 + (d*x)/2)*(A*a^4 + 6*A*b^4 + 2*B*a^4 - 5*A*a^2*b^2 - 2*B*a^2*b^2 + 2*C*a^2*b^2 + 3*A*a*b^3 - 3*A*
a^3*b - 4*B*a*b^3 + 2*B*a^3*b))/((a^3*b - a^4)*(a + b)) + (tan(c/2 + (d*x)/2)^5*(A*a^4 + 6*A*b^4 - 2*B*a^4 - 5
*A*a^2*b^2 + 2*B*a^2*b^2 + 2*C*a^2*b^2 - 3*A*a*b^3 + 3*A*a^3*b - 4*B*a*b^3 + 2*B*a^3*b))/((a^3*b - a^4)*(a + b
)) - (2*tan(c/2 + (d*x)/2)^3*(A*a^4 - 6*A*b^4 + 3*A*a^2*b^2 - 2*C*a^2*b^2 + 4*B*a*b^3 - 2*B*a^3*b))/(a*(a^2*b
- a^3)*(a + b)))/(d*(a + b + tan(c/2 + (d*x)/2)^2*(a + 3*b) - tan(c/2 + (d*x)/2)^4*(a - 3*b) - tan(c/2 + (d*x)
/2)^6*(a - b))) + (b*atan(((b*((8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A
^2*a^9*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4
- 20*A^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2
*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 +
12*C^2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*
A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2
*b^8 - 48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 -
32*B*C*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b +
a^9 - a^6*b^3 - a^7*b^2) + (b*((a + b)^3*(a - b)^3)^(1/2)*((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b
^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4
+ 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^
11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*b*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(8*a^13*b - 8*a^8*b^6
+ 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a^12*b^2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3
+ 3*B*a^3*b))/((a^8*b + a^9 - a^6*b^3 - a^7*b^2)*(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2)))*(3*A*b^4 - 2*C*a^
4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*((a + b)^3*(a
- b)^3)^(1/2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b)*1i)/(a^10 - a^4*b^6 + 3*a^
6*b^4 - 3*a^8*b^2) + (b*((8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9
*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A
^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b
^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^
2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^
3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 -
48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C
*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 -
a^6*b^3 - a^7*b^2) - (b*((a + b)^3*(a - b)^3)^(1/2)*((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 2
8*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*
B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b +
a^12 - a^9*b^3 - a^10*b^2) + (8*b*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(8*a^13*b - 8*a^8*b^6 + 8*a^
9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a^12*b^2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B
*a^3*b))/((a^8*b + a^9 - a^6*b^3 - a^7*b^2)*(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2)))*(3*A*b^4 - 2*C*a^4 - 4*
A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*((a + b)^3*(a - b)^3
)^(1/2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b)*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4
- 3*a^8*b^2))/((16*(108*A^3*b^11 - 54*A^3*a*b^10 + 8*C^3*a^10*b - 216*A^3*a^2*b^9 + 81*A^3*a^3*b^8 + 63*A^3*a^
4*b^7 - 9*A^3*a^5*b^6 + 41*A^3*a^6*b^5 - 4*A^3*a^7*b^4 + 4*A^3*a^8*b^3 - 32*B^3*a^3*b^8 + 16*B^3*a^4*b^7 + 80*
B^3*a^5*b^6 - 24*B^3*a^6*b^5 - 48*B^3*a^7*b^4 + 4*C^3*a^6*b^5 - 4*C^3*a^7*b^4 - 12*C^3*a^8*b^3 + 8*C^3*a^9*b^2
- 216*A^2*B*a*b^10 + 8*A*C^2*a^10*b + 2*A^2*C*a^10*b + 144*A*B^2*a^2*b^9 - 72*A*B^2*a^3*b^8 - 336*A*B^2*a^4*b
^7 + 108*A*B^2*a^5*b^6 + 168*A*B^2*a^6*b^5 - 6*A*B^2*a^7*b^4 + 24*A*B^2*a^8*b^3 + 108*A^2*B*a^2*b^9 + 468*A^2*
B*a^3*b^8 - 162*A^2*B*a^4*b^7 - 186*A^2*B*a^5*b^6 + 15*A^2*B*a^6*b^5 - 63*A^2*B*a^7*b^4 + 3*A^2*B*a^8*b^3 - 3*
A^2*B*a^9*b^2 + 36*A*C^2*a^4*b^7 - 30*A*C^2*a^5*b^6 - 96*A*C^2*a^6*b^5 + 52*A*C^2*a^7*b^4 + 52*A*C^2*a^8*b^3 +
108*A^2*C*a^2*b^9 - 72*A^2*C*a^3*b^8 - 252*A^2*C*a^4*b^7 + 111*A^2*C*a^5*b^6 + 105*A^2*C*a^6*b^5 - 5*A^2*C*a^
7*b^4 + 37*A^2*C*a^8*b^3 - 2*A^2*C*a^9*b^2 - 24*B*C^2*a^5*b^6 + 20*B*C^2*a^6*b^5 + 68*B*C^2*a^7*b^4 - 36*B*C^2
*a^8*b^3 - 44*B*C^2*a^9*b^2 + 48*B^2*C*a^4*b^7 - 32*B^2*C*a^5*b^6 - 128*B^2*C*a^6*b^5 + 52*B^2*C*a^7*b^4 + 80*
B^2*C*a^8*b^3 - 144*A*B*C*a^3*b^8 + 96*A*B*C*a^4*b^7 + 360*A*B*C*a^5*b^6 - 152*A*B*C*a^6*b^5 - 188*A*B*C*a^7*b
^4 + 4*A*B*C*a^8*b^3 - 28*A*B*C*a^9*b^2))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (b*((8*tan(c/2 + (d*x)/2)*(A^
2*a^10 + 72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b
^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*
B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b
^6 - 8*C^2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*
b - 8*A*C*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*
a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*A*C*a^5*b^5
+ 36*A*C*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*
C*a^6*b^4 - 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (b*((a + b)^3*(a - b)^3)^(1/
2)*((8*(2*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A
*a^13*b^2 + 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a
^12*b^3 - 4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*b*tan(c/2 + (d*x)
/2)*((a + b)^3*(a - b)^3)^(1/2)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a^12*b^2)*(3
*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/((a^8*b + a^9 - a^6*b^3 - a^7*b^2)*(a^10
- a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2)))*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a
^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b
^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2) + (b*((8*tan(c/2 + (d*x)/2)*(A^2*a^10 +
72*A^2*b^10 + 4*C^2*a^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 8*C^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A
^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b
^7 - 64*B^2*a^4*b^6 + 64*B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 + 8*C^2*a^4*b^6 - 8*C^
2*a^5*b^5 - 20*C^2*a^6*b^4 + 16*C^2*a^7*b^3 + 12*C^2*a^8*b^2 + 4*A*C*a^10 - 96*A*B*a*b^9 - 8*A*B*a^9*b - 8*A*C
*a^9*b - 16*B*C*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 -
40*A*B*a^7*b^3 + 16*A*B*a^8*b^2 + 48*A*C*a^2*b^8 - 48*A*C*a^3*b^7 - 100*A*C*a^4*b^6 + 88*A*C*a^5*b^5 + 36*A*C
*a^6*b^4 - 32*A*C*a^7*b^3 + 20*A*C*a^8*b^2 - 32*B*C*a^3*b^7 + 32*B*C*a^4*b^6 + 72*B*C*a^5*b^5 - 64*B*C*a^6*b^4
- 32*B*C*a^7*b^3 + 32*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (b*((a + b)^3*(a - b)^3)^(1/2)*((8*(2
*A*a^15 + 4*C*a^15 - 12*A*a^8*b^7 + 6*A*a^9*b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2
+ 8*B*a^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 4*C*a^10*b^5 + 12*C*a^12*b^3 -
4*C*a^13*b^2 - 8*B*a^14*b - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) + (8*b*tan(c/2 + (d*x)/2)*((a +
b)^3*(a - b)^3)^(1/2)*(8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a^12*b^2)*(3*A*b^4 -
2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/((a^8*b + a^9 - a^6*b^3 - a^7*b^2)*(a^10 - a^4*b^6
+ 3*a^6*b^4 - 3*a^8*b^2)))*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b))/(a^10 - a^4
*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 2*C*a^4 - 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*
a*b^3 + 3*B*a^3*b))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^4 - 2*C*a^4
- 4*A*a^2*b^2 + C*a^2*b^2 - 2*B*a*b^3 + 3*B*a^3*b)*2i)/(d*(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
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))**2,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))**2, x)
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