3.690 \(\int \frac {\cos ^3(c+d x) (A+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=326 \[ -\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d \left (a^2-b^2\right )}+\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {b x \left (a^2 (A+2 C)+4 A b^2\right )}{a^5}-\frac {\left (-\left (a^4 (2 A+3 C)\right )-a^2 b^2 (7 A-6 C)+12 A b^4\right ) \sin (c+d x)}{3 a^4 d \left (a^2-b^2\right )}+\frac {b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a^3 d \left (a^2-b^2\right )}+\frac {2 b^2 \left (3 a^4 C+5 a^2 A b^2-2 a^2 b^2 C-4 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{3/2} (a+b)^{3/2}} \]

[Out]

-b*(4*A*b^2+a^2*(A+2*C))*x/a^5+2*b^2*(5*A*a^2*b^2-4*A*b^4+3*C*a^4-2*C*a^2*b^2)*arctanh((a-b)^(1/2)*tan(1/2*d*x
+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(3/2)/(a+b)^(3/2)/d-1/3*(12*A*b^4-a^2*b^2*(7*A-6*C)-a^4*(2*A+3*C))*sin(d*x+c)/a
^4/(a^2-b^2)/d+b*(2*A*b^2-a^2*(A-C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)/d-1/3*(4*A*b^2-a^2*(A-3*C))*cos(d*x+c
)^2*sin(d*x+c)/a^2/(a^2-b^2)/d+(A*b^2+C*a^2)*cos(d*x+c)^2*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))

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Rubi [A]  time = 1.26, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4101, 4104, 3919, 3831, 2659, 208} \[ -\frac {\left (-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+12 A b^4\right ) \sin (c+d x)}{3 a^4 d \left (a^2-b^2\right )}-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d \left (a^2-b^2\right )}+\frac {b \left (2 A b^2-a^2 (A-C)\right ) \sin (c+d x) \cos (c+d x)}{a^3 d \left (a^2-b^2\right )}+\frac {2 b^2 \left (5 a^2 A b^2-2 a^2 b^2 C+3 a^4 C-4 A b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {b x \left (a^2 (A+2 C)+4 A b^2\right )}{a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(4*A*b^2 + a^2*(A + 2*C))*x)/a^5) + (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C - 2*a^2*b^2*C)*ArcTanh[(Sqrt[
a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((12*A*b^4 - a^2*b^2*(7*A - 6*C)
- a^4*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) + (b*(2*A*b^2 - a^2*(A - C))*Cos[c + d*x]*Sin[c + d*x])
/(a^3*(a^2 - b^2)*d) - ((4*A*b^2 - a^2*(A - 3*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2
+ a^2*C)*Cos[c + d*x]^2*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 4101

Int[((A_.) + 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^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)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e
+ f*x])^n*Simp[a^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2
+ a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, 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 ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx &=\frac {\left (A b^2+a^2 C\right ) \cos ^2(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 ^3(c+d x) \left (4 A b^2-a^2 (A-3 C)+a b (A+C) \sec (c+d x)-3 \left (A b^2+a^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \cos ^2(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 (6 b \left (2 A b^2-a^2 (A-C)\right )+a \left (A b^2+a^2 (2 A+3 C)\right ) \sec (c+d x)-2 b \left (4 A b^2-a^2 (A-3 C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=\frac {b \left (2 A b^2-a^2 (A-C)\right ) \cos (c+d x) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \cos ^2(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 (12 A b^4-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right )+2 a b \left (2 A b^2+a^2 (A+3 C)\right ) \sec (c+d x)-6 b^2 \left (2 A b^2-a^2 (A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )}\\ &=-\frac {\left (12 A b^4-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {b \left (2 A b^2-a^2 (A-C)\right ) \cos (c+d x) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {-6 b \left (a^2-b^2\right ) \left (4 A b^2+a^2 (A+2 C)\right )+6 a b^2 \left (2 A b^2-a^2 (A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )}\\ &=-\frac {b \left (4 A b^2+a^2 (A+2 C)\right ) x}{a^5}-\frac {\left (12 A b^4-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {b \left (2 A b^2-a^2 (A-C)\right ) \cos (c+d x) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (b^2 \left (4 A b^4-a^2 b^2 (5 A-2 C)-3 a^4 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^5 \left (a^2-b^2\right )}\\ &=-\frac {b \left (4 A b^2+a^2 (A+2 C)\right ) x}{a^5}-\frac {\left (12 A b^4-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {b \left (2 A b^2-a^2 (A-C)\right ) \cos (c+d x) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \cos ^2(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 (4 A b^4-a^2 b^2 (5 A-2 C)-3 a^4 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^5 \left (a^2-b^2\right )}\\ &=-\frac {b \left (4 A b^2+a^2 (A+2 C)\right ) x}{a^5}-\frac {\left (12 A b^4-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {b \left (2 A b^2-a^2 (A-C)\right ) \cos (c+d x) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (2 b \left (4 A b^4-a^2 b^2 (5 A-2 C)-3 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^5 \left (a^2-b^2\right ) d}\\ &=-\frac {b \left (4 A b^2+a^2 (A+2 C)\right ) x}{a^5}+\frac {2 b^2 \left (5 a^2 A b^2-4 A b^4+3 a^4 C-2 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^5 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {\left (12 A b^4-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {b \left (2 A b^2-a^2 (A-C)\right ) \cos (c+d x) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2+a^2 C\right ) \cos ^2(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 = 1.29, size = 212, normalized size = 0.65 \[ \frac {a^3 A \sin (3 (c+d x))-12 b (c+d x) \left (a^2 (A+2 C)+4 A b^2\right )+3 a \left (a^2 (3 A+4 C)+12 A b^2\right ) \sin (c+d x)-\frac {12 a b^3 \left (a^2 C+A b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-6 a^2 A b \sin (2 (c+d x))+\frac {24 b^2 \left (-3 a^4 C+a^2 b^2 (2 C-5 A)+4 A b^4\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}}{12 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-12*b*(4*A*b^2 + a^2*(A + 2*C))*(c + d*x) + (24*b^2*(4*A*b^4 - 3*a^4*C + a^2*b^2*(-5*A + 2*C))*ArcTanh[((-a +
b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 3*a*(12*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] - (1
2*a*b^3*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) - 6*a^2*A*b*Sin[2*(c + d*x)] + a^
3*A*Sin[3*(c + d*x)])/(12*a^5*d)

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fricas [A]  time = 0.58, size = 989, normalized size = 3.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(6*((A + 2*C)*a^7*b + 2*(A - 2*C)*a^5*b^3 - (7*A - 2*C)*a^3*b^5 + 4*A*a*b^7)*d*x*cos(d*x + c) + 6*((A +
2*C)*a^6*b^2 + 2*(A - 2*C)*a^4*b^4 - (7*A - 2*C)*a^2*b^6 + 4*A*b^8)*d*x - 3*(3*C*a^4*b^3 + (5*A - 2*C)*a^2*b^5
- 4*A*b^7 + (3*C*a^5*b^2 + (5*A - 2*C)*a^3*b^4 - 4*A*a*b^6)*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 + (5*A - 9*C)*a^5*b^3 - (19*A - 6*C)*a^3*b^
5 + 12*A*a*b^7 + (A*a^8 - 2*A*a^6*b^2 + A*a^4*b^4)*cos(d*x + c)^3 - 2*(A*a^7*b - 2*A*a^5*b^3 + A*a^3*b^5)*cos(
d*x + c)^2 + ((2*A + 3*C)*a^8 + 2*(A - 3*C)*a^6*b^2 - (10*A - 3*C)*a^4*b^4 + 6*A*a^2*b^6)*cos(d*x + c))*sin(d*
x + c))/((a^10 - 2*a^8*b^2 + a^6*b^4)*d*cos(d*x + c) + (a^9*b - 2*a^7*b^3 + a^5*b^5)*d), -1/3*(3*((A + 2*C)*a^
7*b + 2*(A - 2*C)*a^5*b^3 - (7*A - 2*C)*a^3*b^5 + 4*A*a*b^7)*d*x*cos(d*x + c) + 3*((A + 2*C)*a^6*b^2 + 2*(A -
2*C)*a^4*b^4 - (7*A - 2*C)*a^2*b^6 + 4*A*b^8)*d*x - 3*(3*C*a^4*b^3 + (5*A - 2*C)*a^2*b^5 - 4*A*b^7 + (3*C*a^5*
b^2 + (5*A - 2*C)*a^3*b^4 - 4*A*a*b^6)*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 + (5*A - 9*C)*a^5*b^3 - (19*A - 6*C)*a^3*b^5 + 12*A*a*b
^7 + (A*a^8 - 2*A*a^6*b^2 + A*a^4*b^4)*cos(d*x + c)^3 - 2*(A*a^7*b - 2*A*a^5*b^3 + A*a^3*b^5)*cos(d*x + c)^2 +
((2*A + 3*C)*a^8 + 2*(A - 3*C)*a^6*b^2 - (10*A - 3*C)*a^4*b^4 + 6*A*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^
10 - 2*a^8*b^2 + a^6*b^4)*d*cos(d*x + c) + (a^9*b - 2*a^7*b^3 + a^5*b^5)*d)]

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giac [A]  time = 0.27, size = 441, normalized size = 1.35 \[ \frac {\frac {6 \, {\left (3 \, C a^{4} b^{2} + 5 \, A a^{2} b^{4} - 2 \, C a^{2} b^{4} - 4 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{7} - a^{5} b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {6 \, {\left (C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} - \frac {3 \, {\left (A a^{2} b + 2 \, C a^{2} b + 4 \, A b^{3}\right )} {\left (d x + c\right )}}{a^{5}} + \frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{4}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(6*(3*C*a^4*b^2 + 5*A*a^2*b^4 - 2*C*a^2*b^4 - 4*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^7 - a^5*b^2)*sqrt(-a^2 + b^2
)) + 6*(C*a^2*b^3*tan(1/2*d*x + 1/2*c) + A*b^5*tan(1/2*d*x + 1/2*c))/((a^6 - a^4*b^2)*(a*tan(1/2*d*x + 1/2*c)^
2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)) - 3*(A*a^2*b + 2*C*a^2*b + 4*A*b^3)*(d*x + c)/a^5 + 2*(3*A*a^2*tan(1/2*
d*x + 1/2*c)^5 + 3*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 3*A*a*b*tan(1/2*d*x + 1/2*c)^5 + 9*A*b^2*tan(1/2*d*x + 1/2*c
)^5 + 2*A*a^2*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*tan(1/2*d*x + 1/2*c)^3 + 18*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 3*A*
a^2*tan(1/2*d*x + 1/2*c) + 3*C*a^2*tan(1/2*d*x + 1/2*c) - 3*A*a*b*tan(1/2*d*x + 1/2*c) + 9*A*b^2*tan(1/2*d*x +
1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^4))/d

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maple [B]  time = 1.27, size = 836, normalized size = 2.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*b^5/a^4/(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)*C+10/d/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))*A*b^4-8/d*b^6/a^5/(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))*C-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))*C+2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^3*A*tan(1/2*d*x+1/2*c)^5+2/d/a^3/(1+ta
n(1/2*d*x+1/2*c)^2)^3*A*tan(1/2*d*x+1/2*c)^5*b+6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*A*tan(1/2*d*x+1/2*c)^5*b^2+2
/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^3*C*tan(1/2*d*x+1/2*c)^5+4/3/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*
c)^3*A+12/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3*A*b^2+4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1
/2*d*x+1/2*c)^3*C+2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^3*A*tan(1/2*d*x+1/2*c)-2/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^3*A
*tan(1/2*d*x+1/2*c)*b+6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^3*A*tan(1/2*d*x+1/2*c)*b^2+2/d/a^2/(1+tan(1/2*d*x+1/2*c
)^2)^3*C*tan(1/2*d*x+1/2*c)-2/d*A/a^3*b*arctan(tan(1/2*d*x+1/2*c))-8/d/a^5*A*arctan(tan(1/2*d*x+1/2*c))*b^3-4/
d/a^3*C*arctan(tan(1/2*d*x+1/2*c))*b

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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)^3*(A+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 = 13.70, size = 6978, normalized size = 21.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*tan(c/2 + (d*x)/2)^3*(A*a^5 + 36*A*b^5 - 3*C*a^5 - 19*A*a^2*b^3 - 7*A*a^3*b^2 + 18*C*a^2*b^3 + 3*C*a^3*b
^2 + 6*A*a*b^4 - 8*A*a^4*b - 9*C*a^4*b))/(3*a^4*(a + b)*(a - b)) - (2*tan(c/2 + (d*x)/2)^5*(A*a^5 - 36*A*b^5 -
3*C*a^5 + 19*A*a^2*b^3 - 7*A*a^3*b^2 - 18*C*a^2*b^3 + 3*C*a^3*b^2 + 6*A*a*b^4 + 8*A*a^4*b + 9*C*a^4*b))/(3*a^
4*(a + b)*(a - b)) + (2*tan(c/2 + (d*x)/2)^7*(A*a^5 + 4*A*b^5 + C*a^5 - 3*A*a^2*b^3 + A*a^3*b^2 + 2*C*a^2*b^3
- C*a^3*b^2 - 2*A*a*b^4 - C*a^4*b))/(a^4*(a + b)*(a - b)) - (2*tan(c/2 + (d*x)/2)*(A*a^5 - 4*A*b^5 + C*a^5 + 3
*A*a^2*b^3 + A*a^3*b^2 - 2*C*a^2*b^3 - C*a^3*b^2 - 2*A*a*b^4 + C*a^4*b))/(a^4*(a + b)*(a - b)))/(d*(a + b - ta
n(c/2 + (d*x)/2)^8*(a - b) + tan(c/2 + (d*x)/2)^2*(2*a + 4*b) - tan(c/2 + (d*x)/2)^6*(2*a - 4*b) + 6*b*tan(c/2
+ (d*x)/2)^4)) - (atan(-((((((32*(2*A*a^11*b^7 - 4*A*a^10*b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A*a^14*b^4 +
A*a^15*b^3 - 2*C*a^12*b^6 + C*a^13*b^5 + 5*C*a^14*b^4 - 3*C*a^15*b^3 - 3*C*a^16*b^2 + A*a^17*b + 2*C*a^17*b))/
(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (32*tan(c/2 + (d*x)/2)*(A*b^3*4i + a^2*b*(A + 2*C)*1i)*(2*a^15*b - 2*a
^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(A*b^
3*4i + a^2*b*(A + 2*C)*1i))/a^5 + (32*tan(c/2 + (d*x)/2)*(32*A^2*b^12 - 32*A^2*a*b^11 - 48*A^2*a^2*b^10 + 48*A
^2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6 - 12*A^2*a^7*b^5 + 7*A^2*a^8*b^4 - 2*A^2*a^9*b^3 +
A^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - 16*C^2*a^6*b^6 + 16*C^2*a^7*b^5 + 5*C^2*a^8*b^4 - 8*C^2*a^9*b^3
+ 4*C^2*a^10*b^2 + 32*A*C*a^2*b^10 - 32*A*C*a^3*b^9 - 56*A*C*a^4*b^8 + 56*A*C*a^5*b^7 + 10*A*C*a^6*b^6 - 16*A
*C*a^7*b^5 + 12*A*C*a^8*b^4 - 8*A*C*a^9*b^3 + 4*A*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(A*b^3*4i
+ a^2*b*(A + 2*C)*1i)*1i)/a^5 - (((((32*(2*A*a^11*b^7 - 4*A*a^10*b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A*a^14*
b^4 + A*a^15*b^3 - 2*C*a^12*b^6 + C*a^13*b^5 + 5*C*a^14*b^4 - 3*C*a^15*b^3 - 3*C*a^16*b^2 + A*a^17*b + 2*C*a^1
7*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (32*tan(c/2 + (d*x)/2)*(A*b^3*4i + a^2*b*(A + 2*C)*1i)*(2*a^15*b
- 2*a^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))
*(A*b^3*4i + a^2*b*(A + 2*C)*1i))/a^5 - (32*tan(c/2 + (d*x)/2)*(32*A^2*b^12 - 32*A^2*a*b^11 - 48*A^2*a^2*b^10
+ 48*A^2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6 - 12*A^2*a^7*b^5 + 7*A^2*a^8*b^4 - 2*A^2*a^9*
b^3 + A^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - 16*C^2*a^6*b^6 + 16*C^2*a^7*b^5 + 5*C^2*a^8*b^4 - 8*C^2*a
^9*b^3 + 4*C^2*a^10*b^2 + 32*A*C*a^2*b^10 - 32*A*C*a^3*b^9 - 56*A*C*a^4*b^8 + 56*A*C*a^5*b^7 + 10*A*C*a^6*b^6
- 16*A*C*a^7*b^5 + 12*A*C*a^8*b^4 - 8*A*C*a^9*b^3 + 4*A*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(A*b
^3*4i + a^2*b*(A + 2*C)*1i)*1i)/a^5)/((((((32*(2*A*a^11*b^7 - 4*A*a^10*b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A
*a^14*b^4 + A*a^15*b^3 - 2*C*a^12*b^6 + C*a^13*b^5 + 5*C*a^14*b^4 - 3*C*a^15*b^3 - 3*C*a^16*b^2 + A*a^17*b + 2
*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) - (32*tan(c/2 + (d*x)/2)*(A*b^3*4i + a^2*b*(A + 2*C)*1i)*(2*
a^15*b - 2*a^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*
b^2)))*(A*b^3*4i + a^2*b*(A + 2*C)*1i))/a^5 + (32*tan(c/2 + (d*x)/2)*(32*A^2*b^12 - 32*A^2*a*b^11 - 48*A^2*a^2
*b^10 + 48*A^2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6 - 12*A^2*a^7*b^5 + 7*A^2*a^8*b^4 - 2*A^
2*a^9*b^3 + A^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - 16*C^2*a^6*b^6 + 16*C^2*a^7*b^5 + 5*C^2*a^8*b^4 - 8
*C^2*a^9*b^3 + 4*C^2*a^10*b^2 + 32*A*C*a^2*b^10 - 32*A*C*a^3*b^9 - 56*A*C*a^4*b^8 + 56*A*C*a^5*b^7 + 10*A*C*a^
6*b^6 - 16*A*C*a^7*b^5 + 12*A*C*a^8*b^4 - 8*A*C*a^9*b^3 + 4*A*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2)
)*(A*b^3*4i + a^2*b*(A + 2*C)*1i))/a^5 - (64*(64*A^3*b^14 - 32*A^3*a*b^13 - 112*A^3*a^2*b^12 + 48*A^3*a^3*b^11
+ 12*A^3*a^4*b^10 - 6*A^3*a^5*b^9 + 31*A^3*a^6*b^8 - 5*A^3*a^7*b^7 + 5*A^3*a^8*b^6 + 8*C^3*a^6*b^8 - 4*C^3*a^
7*b^7 - 20*C^3*a^8*b^6 + 6*C^3*a^9*b^5 + 12*C^3*a^10*b^4 + 48*A*C^2*a^4*b^10 - 24*A*C^2*a^5*b^9 - 108*A*C^2*a^
6*b^8 + 36*A*C^2*a^7*b^7 + 48*A*C^2*a^8*b^6 - 3*A*C^2*a^9*b^5 + 12*A*C^2*a^10*b^4 + 96*A^2*C*a^2*b^12 - 48*A^2
*C*a^3*b^11 - 192*A^2*C*a^4*b^10 + 72*A^2*C*a^5*b^9 + 54*A^2*C*a^6*b^8 - 9*A^2*C*a^7*b^7 + 39*A^2*C*a^8*b^6 -
3*A^2*C*a^9*b^5 + 3*A^2*C*a^10*b^4))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (((((32*(2*A*a^11*b^7 - 4*A*a^10*
b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A*a^14*b^4 + A*a^15*b^3 - 2*C*a^12*b^6 + C*a^13*b^5 + 5*C*a^14*b^4 - 3*C
*a^15*b^3 - 3*C*a^16*b^2 + A*a^17*b + 2*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (32*tan(c/2 + (d*x)
/2)*(A*b^3*4i + a^2*b*(A + 2*C)*1i)*(2*a^15*b - 2*a^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2*a^14*b^2
))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(A*b^3*4i + a^2*b*(A + 2*C)*1i))/a^5 - (32*tan(c/2 + (d*x)/2)*(3
2*A^2*b^12 - 32*A^2*a*b^11 - 48*A^2*a^2*b^10 + 48*A^2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6
- 12*A^2*a^7*b^5 + 7*A^2*a^8*b^4 - 2*A^2*a^9*b^3 + A^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - 16*C^2*a^6*b
^6 + 16*C^2*a^7*b^5 + 5*C^2*a^8*b^4 - 8*C^2*a^9*b^3 + 4*C^2*a^10*b^2 + 32*A*C*a^2*b^10 - 32*A*C*a^3*b^9 - 56*A
*C*a^4*b^8 + 56*A*C*a^5*b^7 + 10*A*C*a^6*b^6 - 16*A*C*a^7*b^5 + 12*A*C*a^8*b^4 - 8*A*C*a^9*b^3 + 4*A*C*a^10*b^
2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2))*(A*b^3*4i + a^2*b*(A + 2*C)*1i))/a^5))*(A*b^3*4i + a^2*b*(A + 2*C)*1i
)*2i)/(a^5*d) - (b^2*atan(((b^2*((32*tan(c/2 + (d*x)/2)*(32*A^2*b^12 - 32*A^2*a*b^11 - 48*A^2*a^2*b^10 + 48*A^
2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6 - 12*A^2*a^7*b^5 + 7*A^2*a^8*b^4 - 2*A^2*a^9*b^3 + A
^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - 16*C^2*a^6*b^6 + 16*C^2*a^7*b^5 + 5*C^2*a^8*b^4 - 8*C^2*a^9*b^3
+ 4*C^2*a^10*b^2 + 32*A*C*a^2*b^10 - 32*A*C*a^3*b^9 - 56*A*C*a^4*b^8 + 56*A*C*a^5*b^7 + 10*A*C*a^6*b^6 - 16*A*
C*a^7*b^5 + 12*A*C*a^8*b^4 - 8*A*C*a^9*b^3 + 4*A*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2) + (b^2*((32*
(2*A*a^11*b^7 - 4*A*a^10*b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A*a^14*b^4 + A*a^15*b^3 - 2*C*a^12*b^6 + C*a^13
*b^5 + 5*C*a^14*b^4 - 3*C*a^15*b^3 - 3*C*a^16*b^2 + A*a^17*b + 2*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b
^2) - (32*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2)*(
2*a^15*b - 2*a^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2*a^14*b^2))/((a^10*b + a^11 - a^8*b^3 - a^9*b^
2)*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2
*C*a^2*b^2))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^
2*b^2 + 2*C*a^2*b^2)*1i)/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2) + (b^2*((32*tan(c/2 + (d*x)/2)*(32*A^2*b^12
- 32*A^2*a*b^11 - 48*A^2*a^2*b^10 + 48*A^2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6 - 12*A^2*a^
7*b^5 + 7*A^2*a^8*b^4 - 2*A^2*a^9*b^3 + A^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - 16*C^2*a^6*b^6 + 16*C^2
*a^7*b^5 + 5*C^2*a^8*b^4 - 8*C^2*a^9*b^3 + 4*C^2*a^10*b^2 + 32*A*C*a^2*b^10 - 32*A*C*a^3*b^9 - 56*A*C*a^4*b^8
+ 56*A*C*a^5*b^7 + 10*A*C*a^6*b^6 - 16*A*C*a^7*b^5 + 12*A*C*a^8*b^4 - 8*A*C*a^9*b^3 + 4*A*C*a^10*b^2))/(a^10*b
+ a^11 - a^8*b^3 - a^9*b^2) - (b^2*((32*(2*A*a^11*b^7 - 4*A*a^10*b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A*a^14
*b^4 + A*a^15*b^3 - 2*C*a^12*b^6 + C*a^13*b^5 + 5*C*a^14*b^4 - 3*C*a^15*b^3 - 3*C*a^16*b^2 + A*a^17*b + 2*C*a^
17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (32*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4
- 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2)*(2*a^15*b - 2*a^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2*a^14
*b^2))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(
1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*((a + b)^3*(a
- b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2)*1i)/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))/((
64*(64*A^3*b^14 - 32*A^3*a*b^13 - 112*A^3*a^2*b^12 + 48*A^3*a^3*b^11 + 12*A^3*a^4*b^10 - 6*A^3*a^5*b^9 + 31*A^
3*a^6*b^8 - 5*A^3*a^7*b^7 + 5*A^3*a^8*b^6 + 8*C^3*a^6*b^8 - 4*C^3*a^7*b^7 - 20*C^3*a^8*b^6 + 6*C^3*a^9*b^5 + 1
2*C^3*a^10*b^4 + 48*A*C^2*a^4*b^10 - 24*A*C^2*a^5*b^9 - 108*A*C^2*a^6*b^8 + 36*A*C^2*a^7*b^7 + 48*A*C^2*a^8*b^
6 - 3*A*C^2*a^9*b^5 + 12*A*C^2*a^10*b^4 + 96*A^2*C*a^2*b^12 - 48*A^2*C*a^3*b^11 - 192*A^2*C*a^4*b^10 + 72*A^2*
C*a^5*b^9 + 54*A^2*C*a^6*b^8 - 9*A^2*C*a^7*b^7 + 39*A^2*C*a^8*b^6 - 3*A^2*C*a^9*b^5 + 3*A^2*C*a^10*b^4))/(a^14
*b + a^15 - a^12*b^3 - a^13*b^2) - (b^2*((32*tan(c/2 + (d*x)/2)*(32*A^2*b^12 - 32*A^2*a*b^11 - 48*A^2*a^2*b^10
+ 48*A^2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6 - 12*A^2*a^7*b^5 + 7*A^2*a^8*b^4 - 2*A^2*a^9
*b^3 + A^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - 16*C^2*a^6*b^6 + 16*C^2*a^7*b^5 + 5*C^2*a^8*b^4 - 8*C^2*
a^9*b^3 + 4*C^2*a^10*b^2 + 32*A*C*a^2*b^10 - 32*A*C*a^3*b^9 - 56*A*C*a^4*b^8 + 56*A*C*a^5*b^7 + 10*A*C*a^6*b^6
- 16*A*C*a^7*b^5 + 12*A*C*a^8*b^4 - 8*A*C*a^9*b^3 + 4*A*C*a^10*b^2))/(a^10*b + a^11 - a^8*b^3 - a^9*b^2) + (b
^2*((32*(2*A*a^11*b^7 - 4*A*a^10*b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A*a^14*b^4 + A*a^15*b^3 - 2*C*a^12*b^6
+ C*a^13*b^5 + 5*C*a^14*b^4 - 3*C*a^15*b^3 - 3*C*a^16*b^2 + A*a^17*b + 2*C*a^17*b))/(a^14*b + a^15 - a^12*b^3
- a^13*b^2) - (32*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^
2*b^2)*(2*a^15*b - 2*a^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2*a^14*b^2))/((a^10*b + a^11 - a^8*b^3
- a^9*b^2)*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2
*b^2 + 2*C*a^2*b^2))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4
- 5*A*a^2*b^2 + 2*C*a^2*b^2))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2) + (b^2*((32*tan(c/2 + (d*x)/2)*(32*A^2*
b^12 - 32*A^2*a*b^11 - 48*A^2*a^2*b^10 + 48*A^2*a^3*b^9 + 2*A^2*a^4*b^8 - 2*A^2*a^5*b^7 + 7*A^2*a^6*b^6 - 12*A
^2*a^7*b^5 + 7*A^2*a^8*b^4 - 2*A^2*a^9*b^3 + A^2*a^10*b^2 + 8*C^2*a^4*b^8 - 8*C^2*a^5*b^7 - 16*C^2*a^6*b^6 + 1
6*C^2*a^7*b^5 + 5*C^2*a^8*b^4 - 8*C^2*a^9*b^3 + 4*C^2*a^10*b^2 + 32*A*C*a^2*b^10 - 32*A*C*a^3*b^9 - 56*A*C*a^4
*b^8 + 56*A*C*a^5*b^7 + 10*A*C*a^6*b^6 - 16*A*C*a^7*b^5 + 12*A*C*a^8*b^4 - 8*A*C*a^9*b^3 + 4*A*C*a^10*b^2))/(a
^10*b + a^11 - a^8*b^3 - a^9*b^2) - (b^2*((32*(2*A*a^11*b^7 - 4*A*a^10*b^8 + 9*A*a^12*b^6 - 4*A*a^13*b^5 - 5*A
*a^14*b^4 + A*a^15*b^3 - 2*C*a^12*b^6 + C*a^13*b^5 + 5*C*a^14*b^4 - 3*C*a^15*b^3 - 3*C*a^16*b^2 + A*a^17*b + 2
*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^13*b^2) + (32*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(4*
A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2)*(2*a^15*b - 2*a^10*b^6 + 2*a^11*b^5 + 4*a^12*b^4 - 4*a^13*b^3 - 2
*a^14*b^2))/((a^10*b + a^11 - a^8*b^3 - a^9*b^2)*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))*((a + b)^3*(a - b)
^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))*((a + b)^
3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2))/(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))
*((a + b)^3*(a - b)^3)^(1/2)*(4*A*b^4 - 3*C*a^4 - 5*A*a^2*b^2 + 2*C*a^2*b^2)*2i)/(d*(a^11 - a^5*b^6 + 3*a^7*b^
4 - 3*a^9*b^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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