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

[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]))

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Rubi [A]  time = 1.25529, antiderivative size = 326, normalized size of antiderivative = 1., 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 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]

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 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 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 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]

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*}

Mathematica [A]  time = 1.16316, size = 212, normalized size = 0.65 \[ \frac{-12 b (c+d x) \left (a^2 (A+2 C)+4 A b^2\right )+3 a \left (a^2 (3 A+4 C)+12 A b^2\right ) \sin (c+d x)-\frac{12 a b^3 \left (a^2 C+A b^2\right ) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}+\frac{24 b^2 \left (a^2 b^2 (2 C-5 A)-3 a^4 C+4 A b^4\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-6 a^2 A b \sin (2 (c+d x))+a^3 A \sin (3 (c+d x))}{12 a^5 d} \]

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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Maple [B]  time = 0.128, size = 836, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

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

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

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

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Fricas [A]  time = 0.785335, size = 2187, normalized size = 6.71 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 1.25442, size = 595, normalized size = 1.83 \begin{align*} \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} \end{align*}

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