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

Optimal. Leaf size=205 \[ \frac {B x}{a^3}+\frac {b (b B-a C) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {b \left (-3 a^3 C+5 a^2 b B-2 b^3 B\right ) \tan (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\left (-2 a^5 C+6 a^4 b B-a^3 b^2 C-5 a^2 b^3 B+2 b^5 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.58, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {4072, 3923, 4060, 3919, 3831, 2659, 208} \[ -\frac {\left (-5 a^2 b^3 B-a^3 b^2 C+6 a^4 b B-2 a^5 C+2 b^5 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {b \left (5 a^2 b B-3 a^3 C-2 b^3 B\right ) \tan (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {b (b B-a C) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {B x}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*x)/a^3 - ((6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 2*a^5*C - a^3*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/
Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b*(b*B - a*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec
[c + d*x])^2) + (b*(5*a^2*b*B - 2*b^3*B - 3*a^3*C)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^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 3923

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[(b*(
b*c - a*d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(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)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))*Csc[e + f*x] + b
*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m,
-1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rule 4060

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(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))/(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)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rule 4072

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

Rubi steps

\begin {align*} \int \frac {\cos (c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=\int \frac {B+C \sec (c+d x)}{(a+b \sec (c+d x))^3} \, dx\\ &=\frac {b (b B-a C) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {-2 \left (a^2-b^2\right ) B+2 a (b B-a C) \sec (c+d x)-b (b B-a C) \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {b (b B-a C) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \tan (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 B-a \left (4 a^2 b B-b^3 B-2 a^3 C-a b^2 C\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {B x}{a^3}+\frac {b (b B-a C) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (6 a^4 b B-5 a^2 b^3 B+2 b^5 B-2 a^5 C-a^3 b^2 C\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {B x}{a^3}+\frac {b (b B-a C) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (6 a^4 b B-5 a^2 b^3 B+2 b^5 B-2 a^5 C-a^3 b^2 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^3 b \left (a^2-b^2\right )^2}\\ &=\frac {B x}{a^3}+\frac {b (b B-a C) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (6 a^4 b B-5 a^2 b^3 B+2 b^5 B-2 a^5 C-a^3 b^2 C\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^3 b \left (a^2-b^2\right )^2 d}\\ &=\frac {B x}{a^3}-\frac {\left (6 a^4 b B-5 a^2 b^3 B+2 b^5 B-2 a^5 C-a^3 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^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {b (b B-a C) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {b \left (5 a^2 b B-2 b^3 B-3 a^3 C\right ) \tan (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 = 1.86, size = 203, normalized size = 0.99 \[ \frac {\frac {a b \left (-4 a^3 C+6 a^2 b B+a b^2 C-3 b^3 B\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}-\frac {2 \left (2 a^5 C-6 a^4 b B+a^3 b^2 C+5 a^2 b^3 B-2 b^5 B\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a b^2 (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)^2}+2 B (c+d x)}{2 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.56, size = 1152, normalized size = 5.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(B*a^8 - 3*B*a^6*b^2 + 3*B*a^4*b^4 - B*a^2*b^6)*d*x*cos(d*x + c)^2 + 8*(B*a^7*b - 3*B*a^5*b^3 + 3*B*a^
3*b^5 - B*a*b^7)*d*x*cos(d*x + c) + 4*(B*a^6*b^2 - 3*B*a^4*b^4 + 3*B*a^2*b^6 - B*b^8)*d*x - (2*C*a^5*b^2 - 6*B
*a^4*b^3 + C*a^3*b^4 + 5*B*a^2*b^5 - 2*B*b^7 + (2*C*a^7 - 6*B*a^6*b + C*a^5*b^2 + 5*B*a^4*b^3 - 2*B*a^2*b^5)*c
os(d*x + c)^2 + 2*(2*C*a^6*b - 6*B*a^5*b^2 + C*a^4*b^3 + 5*B*a^3*b^4 - 2*B*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*(3*C*a^6*b^2 - 5*B*a^5*b^3 - 3*C*a^4*b^4
+ 7*B*a^3*b^5 - 2*B*a*b^7 + (4*C*a^7*b - 6*B*a^6*b^2 - 5*C*a^5*b^3 + 9*B*a^4*b^4 + C*a^3*b^5 - 3*B*a^2*b^6)*co
s(d*x + c))*sin(d*x + c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cos(d*x + c)^2 + 2*(a^10*b - 3*a^8*b^3 +
3*a^6*b^5 - a^4*b^7)*d*cos(d*x + c) + (a^9*b^2 - 3*a^7*b^4 + 3*a^5*b^6 - a^3*b^8)*d), 1/2*(2*(B*a^8 - 3*B*a^6
*b^2 + 3*B*a^4*b^4 - B*a^2*b^6)*d*x*cos(d*x + c)^2 + 4*(B*a^7*b - 3*B*a^5*b^3 + 3*B*a^3*b^5 - B*a*b^7)*d*x*cos
(d*x + c) + 2*(B*a^6*b^2 - 3*B*a^4*b^4 + 3*B*a^2*b^6 - B*b^8)*d*x + (2*C*a^5*b^2 - 6*B*a^4*b^3 + C*a^3*b^4 + 5
*B*a^2*b^5 - 2*B*b^7 + (2*C*a^7 - 6*B*a^6*b + C*a^5*b^2 + 5*B*a^4*b^3 - 2*B*a^2*b^5)*cos(d*x + c)^2 + 2*(2*C*a
^6*b - 6*B*a^5*b^2 + C*a^4*b^3 + 5*B*a^3*b^4 - 2*B*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))) - (3*C*a^6*b^2 - 5*B*a^5*b^3 - 3*C*a^4*b^4 + 7*B*a^3*b^5
- 2*B*a*b^7 + (4*C*a^7*b - 6*B*a^6*b^2 - 5*C*a^5*b^3 + 9*B*a^4*b^4 + C*a^3*b^5 - 3*B*a^2*b^6)*cos(d*x + c))*si
n(d*x + c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cos(d*x + c)^2 + 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a
^4*b^7)*d*cos(d*x + c) + (a^9*b^2 - 3*a^7*b^4 + 3*a^5*b^6 - a^3*b^8)*d)]

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giac [B]  time = 0.39, size = 457, normalized size = 2.23 \[ \frac {\frac {{\left (2 \, C a^{5} - 6 \, B a^{4} b + C a^{3} b^{2} + 5 \, B a^{2} b^{3} - 2 \, B b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {{\left (d x + c\right )} B}{a^{3}} + \frac {4 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*C*a^5 - 6*B*a^4*b + C*a^3*b^2 + 5*B*a^2*b^3 - 2*B*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^7 - 2*a^5*b^2 + a^3*b^4)*sqr
t(-a^2 + b^2)) + (d*x + c)*B/a^3 + (4*C*a^4*b*tan(1/2*d*x + 1/2*c)^3 - 6*B*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 3*
C*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - C*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 3*B
*a*b^4*tan(1/2*d*x + 1/2*c)^3 - 2*B*b^5*tan(1/2*d*x + 1/2*c)^3 - 4*C*a^4*b*tan(1/2*d*x + 1/2*c) + 6*B*a^3*b^2*
tan(1/2*d*x + 1/2*c) - 3*C*a^3*b^2*tan(1/2*d*x + 1/2*c) + 5*B*a^2*b^3*tan(1/2*d*x + 1/2*c) + C*a^2*b^3*tan(1/2
*d*x + 1/2*c) - 3*B*a*b^4*tan(1/2*d*x + 1/2*c) - 2*B*b^5*tan(1/2*d*x + 1/2*c))/((a^6 - 2*a^4*b^2 + a^2*b^4)*(a
*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^2))/d

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maple [B]  time = 1.23, size = 1063, normalized size = 5.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-1/
d/a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^3/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+2/d
/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^4/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+4/
d*a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*b*C+1/d/(
a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C*b^2+6/d/(a*t
an(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-1/d/a/(a*tan(1/2*d*x+
1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^3/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-2/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-
tan(1/2*d*x+1/2*c)^2*b-a-b)^2*b^4/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-4/d*a/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x
+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*b*C+1/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b
)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C*b^2-6/d*a*b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x
+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+5/d/a/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*
(a-b)/((a-b)*(a+b))^(1/2))*B*b^3-2/d/a^3/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a
-b)/((a-b)*(a+b))^(1/2))*B*b^5+2/d*a^2/(a^4-2*a^2*b^2+b^4)/((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^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(
a+b))^(1/2))*b^2*C+2/d/a^3*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)*(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 details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 15.43, size = 6911, normalized size = 33.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*atan(-((B*((B*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^
10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2
- 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11
*b^2) - (B*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*
a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^3*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3
*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*1i)/a^3 + (8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*
B^2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 +
32*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5
+ 8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))/a^3 -
(B*((B*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*B
*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^14
*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (B*t
an(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 3
2*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^3*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3
*a^8*b^3 - 3*a^9*b^2)))*1i)/a^3 - (8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 -
8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*
b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*
b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))/a^3)/((B*((B*((8*(
4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*B*a^11*b^4 +
34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^14*b - 4*C*a^1
4*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (B*tan(c/2 + (d*
x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 -
32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^3*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3
*a^9*b^2)))*1i)/a^3 + (8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B^2*a^9*b
- 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3 + 24*B^2
*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3))/(a^10*
b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2))*1i)/a^3 - (16*(4*B^3*b^9 + 4*B*
C^2*a^9 - 4*B^2*C*a^9 - 2*B^3*a*b^8 + 12*B^3*a^8*b - 18*B^3*a^2*b^7 + 13*B^3*a^3*b^6 + 36*B^3*a^4*b^5 - 26*B^3
*a^5*b^4 - 34*B^3*a^6*b^3 + 24*B^3*a^7*b^2 - 20*B^2*C*a^8*b + B*C^2*a^5*b^4 + 4*B*C^2*a^7*b^2 - 2*B^2*C*a^2*b^
7 - 2*B^2*C*a^3*b^6 + 2*B^2*C*a^4*b^5 + 2*B^2*C*a^6*b^3 + 6*B^2*C*a^7*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6
+ 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (B*((B*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*
b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 +
2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3
*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (B*tan(c/2 + (d*x)/2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 3
2*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^3*(a^10*b
+ a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*1i)/a^3 - (8*tan(c/2 + (d*x)/2)*
(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a
^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*
B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3
*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2))*1i)/a^3)))/(a^3*d) - ((tan(c/2 + (d*x)/2)^3*(2*B*b^4 - 6*B*a^2*b^2 + C*a^2*
b^2 - B*a*b^3 + 4*C*a^3*b))/((a^2*b - a^3)*(a + b)^2) + (tan(c/2 + (d*x)/2)*(2*B*b^4 - 6*B*a^2*b^2 - C*a^2*b^2
+ B*a*b^3 + 4*C*a^3*b))/((a + b)*(a^4 - 2*a^3*b + a^2*b^2)))/(d*(2*a*b - tan(c/2 + (d*x)/2)^2*(2*a^2 - 2*b^2)
+ tan(c/2 + (d*x)/2)^4*(a^2 - 2*a*b + b^2) + a^2 + b^2)) + (atan(((((8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2
*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5
*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a
^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^
3 - 3*a^9*b^2) + (((a + b)^5*(a - b)^5)^(1/2)*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*
b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 +
6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^
9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (4*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*
a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 +
48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4
- 5*a^11*b^2)*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*(2*C*a^5
- 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 -
5*a^11*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)*1i)/(2*(a^
13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)) + (((8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B
^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a
^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C
*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*
b^3 - 3*a^9*b^2) - (((a + b)^5*(a - b)^5)^(1/2)*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^
8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6
+ 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*
a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (4*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*
B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6
+ 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b
^4 - 5*a^11*b^2)*(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*(2*C*a^
5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4
- 5*a^11*b^2)))*((a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)*1i)/(2*(
a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))/((16*(4*B^3*b^9 + 4*B*C^2*a^9 - 4*B^2*C*
a^9 - 2*B^3*a*b^8 + 12*B^3*a^8*b - 18*B^3*a^2*b^7 + 13*B^3*a^3*b^6 + 36*B^3*a^4*b^5 - 26*B^3*a^5*b^4 - 34*B^3*
a^6*b^3 + 24*B^3*a^7*b^2 - 20*B^2*C*a^8*b + B*C^2*a^5*b^4 + 4*B*C^2*a^7*b^2 - 2*B^2*C*a^2*b^7 - 2*B^2*C*a^3*b^
6 + 2*B^2*C*a^4*b^5 + 2*B^2*C*a^6*b^3 + 6*B^2*C*a^7*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a
^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (((8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^
9 - 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*
a^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*
a^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) + (((a + b)^5*(a
- b)^5)^(1/2)*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b
^5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 1
2*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2
) - (4*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b
)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*
a^13*b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)*(a^10*b + a^11 -
a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3
*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*((a + b)^5*(a - b
)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7
*b^6 + 10*a^9*b^4 - 5*a^11*b^2)) + (((8*tan(c/2 + (d*x)/2)*(4*B^2*a^10 + 8*B^2*b^10 + 4*C^2*a^10 - 8*B^2*a*b^9
- 8*B^2*a^9*b - 32*B^2*a^2*b^8 + 32*B^2*a^3*b^7 + 57*B^2*a^4*b^6 - 48*B^2*a^5*b^5 - 52*B^2*a^6*b^4 + 32*B^2*a
^7*b^3 + 24*B^2*a^8*b^2 + C^2*a^6*b^4 + 4*C^2*a^8*b^2 - 24*B*C*a^9*b - 4*B*C*a^3*b^7 + 2*B*C*a^5*b^5 + 8*B*C*a
^7*b^3))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) - (((a + b)^5*(a
- b)^5)^(1/2)*((8*(4*B*a^15 + 4*C*a^15 - 4*B*a^6*b^9 + 2*B*a^7*b^8 + 18*B*a^8*b^7 - 4*B*a^9*b^6 - 36*B*a^10*b^
5 + 6*B*a^11*b^4 + 34*B*a^12*b^3 - 8*B*a^13*b^2 - 2*C*a^8*b^7 + 2*C*a^9*b^6 + 6*C*a^12*b^3 - 6*C*a^13*b^2 - 12
*B*a^14*b - 4*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2)
+ (4*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b)
*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10*b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a
^13*b^3 - 8*a^14*b^2))/((a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)*(a^10*b + a^11 -
a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2)))*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*
b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2)))*((a + b)^5*(a - b)
^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6*B*a^4*b))/(2*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*
b^6 + 10*a^9*b^4 - 5*a^11*b^2))))*((a + b)^5*(a - b)^5)^(1/2)*(2*C*a^5 - 2*B*b^5 + 5*B*a^2*b^3 + C*a^3*b^2 - 6
*B*a^4*b)*1i)/(d*(a^13 - a^3*b^10 + 5*a^5*b^8 - 10*a^7*b^6 + 10*a^9*b^4 - 5*a^11*b^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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