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

Optimal. Leaf size=261 \[ \frac {\left (a^2 B+2 a b C-3 b^2 B\right ) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )}+\frac {b (b B-a C) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {x \left (a^2 B-4 a b C+6 b^2 B\right )}{2 a^4}-\frac {\left (a^3 (-C)+2 a^2 b B+2 a b^2 C-3 b^3 B\right ) \sin (c+d x)}{a^3 d \left (a^2-b^2\right )}-\frac {2 b^2 \left (-3 a^3 C+4 a^2 b B+2 a b^2 C-3 b^3 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}} \]

[Out]

1/2*(B*a^2+6*B*b^2-4*C*a*b)*x/a^4-2*b^2*(4*B*a^2*b-3*B*b^3-3*C*a^3+2*C*a*b^2)*arctanh((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-(2*B*a^2*b-3*B*b^3-C*a^3+2*C*a*b^2)*sin(d*x+c)/a^3/(a^2-b^2)
/d+1/2*(B*a^2-3*B*b^2+2*C*a*b)*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)/d+b*(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 = 0.93, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {4072, 4030, 4104, 3919, 3831, 2659, 208} \[ -\frac {\left (2 a^2 b B+a^3 (-C)+2 a b^2 C-3 b^3 B\right ) \sin (c+d x)}{a^3 d \left (a^2-b^2\right )}+\frac {\left (a^2 B+2 a b C-3 b^2 B\right ) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left (a^2-b^2\right )}-\frac {2 b^2 \left (4 a^2 b B-3 a^3 C+2 a b^2 C-3 b^3 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {b (b B-a C) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {x \left (a^2 B-4 a b C+6 b^2 B\right )}{2 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2*B + 6*b^2*B - 4*a*b*C)*x)/(2*a^4) - (2*b^2*(4*a^2*b*B - 3*b^3*B - 3*a^3*C + 2*a*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) - ((2*a^2*b*B - 3*b^3*B - a^3*C + 2*a*
b^2*C)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2*B - 3*b^2*B + 2*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2
- b^2)*d) + (b*(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 4030

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

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]

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 (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) (B+C \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx\\ &=\frac {b (b B-a C) \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 (-a^2 B+3 b^2 B-2 a b C+a (b B-a C) \sec (c+d x)-2 b (b B-a C) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac {\left (a^2 B-3 b^2 B+2 a b C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \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 (2 a^2 b B-3 b^3 B-a^3 C+2 a b^2 C\right )+a \left (a^2 B+b^2 B-2 a b C\right ) \sec (c+d x)+b \left (a^2 B-3 b^2 B+2 a b 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 (2 a^2 b B-3 b^3 B-a^3 C+2 a b^2 C\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 B-3 b^2 B+2 a b C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \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 (a^2 B+6 b^2 B-4 a b C\right )-a b \left (a^2 B-3 b^2 B+2 a b C\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}\\ &=\frac {\left (a^2 B+6 b^2 B-4 a b C\right ) x}{2 a^4}-\frac {\left (2 a^2 b B-3 b^3 B-a^3 C+2 a b^2 C\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 B-3 b^2 B+2 a b C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \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^2 \left (4 a^2 b B-3 b^3 B-3 a^3 C+2 a b^2 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 (a^2 B+6 b^2 B-4 a b C\right ) x}{2 a^4}-\frac {\left (2 a^2 b B-3 b^3 B-a^3 C+2 a b^2 C\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 B-3 b^2 B+2 a b C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \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 (4 a^2 b B-3 b^3 B-3 a^3 C+2 a b^2 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^4 \left (a^2-b^2\right )}\\ &=\frac {\left (a^2 B+6 b^2 B-4 a b C\right ) x}{2 a^4}-\frac {\left (2 a^2 b B-3 b^3 B-a^3 C+2 a b^2 C\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 B-3 b^2 B+2 a b C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \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 b \left (4 a^2 b B-3 b^3 B-3 a^3 C+2 a b^2 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 (a^2 B+6 b^2 B-4 a b C\right ) x}{2 a^4}-\frac {2 b^2 \left (4 a^2 b B-3 b^3 B-3 a^3 C+2 a 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 (2 a^2 b B-3 b^3 B-a^3 C+2 a b^2 C\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 B-3 b^2 B+2 a b C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (b B-a C) \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 = 3.29, size = 184, normalized size = 0.70 \[ \frac {2 (c+d x) \left (a^2 B-4 a b C+6 b^2 B\right )+a^2 B \sin (2 (c+d x))-\frac {8 b^2 \left (3 a^3 C-4 a^2 b B-2 a b^2 C+3 b^3 B\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {4 a b^3 (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}+4 a (a C-2 b B) \sin (c+d x)}{4 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a^2*B + 6*b^2*B - 4*a*b*C)*(c + d*x) - (8*b^2*(-4*a^2*b*B + 3*b^3*B + 3*a^3*C - 2*a*b^2*C)*ArcTanh[((-a +
b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 4*a*(-2*b*B + a*C)*Sin[c + d*x] - (4*a*b^3*(-(b*B)
+ a*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + a^2*B*Sin[2*(c + d*x)])/(4*a^4*d)

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fricas [A]  time = 1.72, size = 970, normalized size = 3.72 \[ \left [\frac {{\left (B a^{7} - 4 \, C a^{6} b + 4 \, B a^{5} b^{2} + 8 \, C a^{4} b^{3} - 11 \, B a^{3} b^{4} - 4 \, C a^{2} b^{5} + 6 \, B a b^{6}\right )} d x \cos \left (d x + c\right ) + {\left (B a^{6} b - 4 \, C a^{5} b^{2} + 4 \, B a^{4} b^{3} + 8 \, C a^{3} b^{4} - 11 \, B a^{2} b^{5} - 4 \, C a b^{6} + 6 \, B b^{7}\right )} d x + {\left (3 \, C a^{3} b^{3} - 4 \, B a^{2} b^{4} - 2 \, C a b^{5} + 3 \, B b^{6} + {\left (3 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} - 2 \, C a^{2} b^{4} + 3 \, B a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + {\left (2 \, C a^{6} b - 4 \, B a^{5} b^{2} - 6 \, C a^{4} b^{3} + 10 \, B a^{3} b^{4} + 4 \, C a^{2} b^{5} - 6 \, B a b^{6} + {\left (B a^{7} - 2 \, B a^{5} b^{2} + B a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{7} - 3 \, B a^{6} b - 4 \, C a^{5} b^{2} + 6 \, B a^{4} b^{3} + 2 \, C a^{3} b^{4} - 3 \, B a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d\right )}}, \frac {{\left (B a^{7} - 4 \, C a^{6} b + 4 \, B a^{5} b^{2} + 8 \, C a^{4} b^{3} - 11 \, B a^{3} b^{4} - 4 \, C a^{2} b^{5} + 6 \, B a b^{6}\right )} d x \cos \left (d x + c\right ) + {\left (B a^{6} b - 4 \, C a^{5} b^{2} + 4 \, B a^{4} b^{3} + 8 \, C a^{3} b^{4} - 11 \, B a^{2} b^{5} - 4 \, C a b^{6} + 6 \, B b^{7}\right )} d x + 2 \, {\left (3 \, C a^{3} b^{3} - 4 \, B a^{2} b^{4} - 2 \, C a b^{5} + 3 \, B b^{6} + {\left (3 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} - 2 \, C a^{2} b^{4} + 3 \, B a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left (2 \, C a^{6} b - 4 \, B a^{5} b^{2} - 6 \, C a^{4} b^{3} + 10 \, B a^{3} b^{4} + 4 \, C a^{2} b^{5} - 6 \, B a b^{6} + {\left (B a^{7} - 2 \, B a^{5} b^{2} + B a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{7} - 3 \, B a^{6} b - 4 \, C a^{5} b^{2} + 6 \, B a^{4} b^{3} + 2 \, C a^{3} b^{4} - 3 \, B a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((B*a^7 - 4*C*a^6*b + 4*B*a^5*b^2 + 8*C*a^4*b^3 - 11*B*a^3*b^4 - 4*C*a^2*b^5 + 6*B*a*b^6)*d*x*cos(d*x + c
) + (B*a^6*b - 4*C*a^5*b^2 + 4*B*a^4*b^3 + 8*C*a^3*b^4 - 11*B*a^2*b^5 - 4*C*a*b^6 + 6*B*b^7)*d*x + (3*C*a^3*b^
3 - 4*B*a^2*b^4 - 2*C*a*b^5 + 3*B*b^6 + (3*C*a^4*b^2 - 4*B*a^3*b^3 - 2*C*a^2*b^4 + 3*B*a*b^5)*cos(d*x + c))*sq
rt(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*C*a^6*b - 4*B*a^5*b^2 - 6*C*
a^4*b^3 + 10*B*a^3*b^4 + 4*C*a^2*b^5 - 6*B*a*b^6 + (B*a^7 - 2*B*a^5*b^2 + B*a^3*b^4)*cos(d*x + c)^2 + (2*C*a^7
- 3*B*a^6*b - 4*C*a^5*b^2 + 6*B*a^4*b^3 + 2*C*a^3*b^4 - 3*B*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*((B*a^7 - 4*C*a^6*b + 4*B*a^5*b^2 + 8*
C*a^4*b^3 - 11*B*a^3*b^4 - 4*C*a^2*b^5 + 6*B*a*b^6)*d*x*cos(d*x + c) + (B*a^6*b - 4*C*a^5*b^2 + 4*B*a^4*b^3 +
8*C*a^3*b^4 - 11*B*a^2*b^5 - 4*C*a*b^6 + 6*B*b^7)*d*x + 2*(3*C*a^3*b^3 - 4*B*a^2*b^4 - 2*C*a*b^5 + 3*B*b^6 + (
3*C*a^4*b^2 - 4*B*a^3*b^3 - 2*C*a^2*b^4 + 3*B*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*C*a^6*b - 4*B*a^5*b^2 - 6*C*a^4*b^3 + 10*B*a^3*b^4 + 4*C*
a^2*b^5 - 6*B*a*b^6 + (B*a^7 - 2*B*a^5*b^2 + B*a^3*b^4)*cos(d*x + c)^2 + (2*C*a^7 - 3*B*a^6*b - 4*C*a^5*b^2 +
6*B*a^4*b^3 + 2*C*a^3*b^4 - 3*B*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 = 0.25, size = 340, normalized size = 1.30 \[ \frac {\frac {4 \, {\left (3 \, C a^{3} b^{2} - 4 \, B a^{2} b^{3} - 2 \, C a b^{4} + 3 \, B b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {4 \, {\left (C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} + \frac {{\left (B a^{2} - 4 \, C a b + 6 \, B b^{2}\right )} {\left (d x + c\right )}}{a^{4}} - \frac {2 \, {\left (B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(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*(3*C*a^3*b^2 - 4*B*a^2*b^3 - 2*C*a*b^4 + 3*B*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + a
rctan(-(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*b^3*tan(1/2*d*x + 1/2*c) - B*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)) + (B*a^2 - 4*C*a*b + 6*B*b^2)*(d*x + c)/a^4 - 2*(B*a*tan(1/2*d*x + 1/2*c)^3
- 2*C*a*tan(1/2*d*x + 1/2*c)^3 + 4*B*b*tan(1/2*d*x + 1/2*c)^3 - B*a*tan(1/2*d*x + 1/2*c) - 2*C*a*tan(1/2*d*x
+ 1/2*c) + 4*B*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^3))/d

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maple [B]  time = 1.08, size = 651, normalized size = 2.49 \[ -\frac {2 b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B}{d \,a^{3} \left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b \right )}+\frac {2 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{d \,a^{2} \left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b \right )}-\frac {8 \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) b^{3} B}{d \,a^{2} \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {6 b^{5} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) B}{d \,a^{4} \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {6 b^{2} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) C}{d a \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {4 b^{4} \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) C}{d \,a^{3} \left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B b}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B b}{d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{2}}+\frac {6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2} B}{d \,a^{4}}-\frac {4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C b}{d \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(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)*B+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-8/d/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))*b^3*B+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))*B+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-1/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^2*B*tan(1/2*d*x+1/2*c)^3-4/d/a^3/(1+ta
n(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*B*b+2/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*C+1/d/a
^2/(1+tan(1/2*d*x+1/2*c)^2)^2*B*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)*B*b+2
/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*C+1/d/a^2*arctan(tan(1/2*d*x+1/2*c))*B+6/d/a^4*arctan(tan
(1/2*d*x+1/2*c))*b^2*B-4/d/a^3*arctan(tan(1/2*d*x+1/2*c))*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*(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 = 13.93, size = 6730, normalized size = 25.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((8*tan(c/2 + (d*x)/2)*(B^2*a^10 + 72*B^2*b^10 - 72*B^2*a*b^9 - 2*B^2*a^9*b - 120*B^2*a^2*b^8 + 120*B^
2*a^3*b^7 + 17*B^2*a^4*b^6 - 26*B^2*a^5*b^5 + 23*B^2*a^6*b^4 - 20*B^2*a^7*b^3 + 11*B^2*a^8*b^2 + 32*C^2*a^2*b^
8 - 32*C^2*a^3*b^7 - 64*C^2*a^4*b^6 + 64*C^2*a^5*b^5 + 20*C^2*a^6*b^4 - 32*C^2*a^7*b^3 + 16*C^2*a^8*b^2 - 96*B
*C*a*b^9 - 8*B*C*a^9*b + 96*B*C*a^2*b^8 + 176*B*C*a^3*b^7 - 176*B*C*a^4*b^6 - 40*B*C*a^5*b^5 + 64*B*C*a^6*b^4
- 40*B*C*a^7*b^3 + 16*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (((8*(2*B*a^15 - 12*B*a^8*b^7 + 6*B*a^
9*b^6 + 28*B*a^10*b^5 - 14*B*a^11*b^4 - 16*B*a^12*b^3 + 6*B*a^13*b^2 + 8*C*a^9*b^6 - 4*C*a^10*b^5 - 20*C*a^11*
b^4 + 12*C*a^12*b^3 + 12*C*a^13*b^2 - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (4*tan(c/2 + (d*x)/2
)*(B*a^2*1i + B*b^2*6i - C*a*b*4i)*(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)))*(B*a^2*1i + B*b^2*6i - C*a*b*4i))/(2*a^4))*(B*a^2*1i + B*b^2*6i - C
*a*b*4i)*1i)/(2*a^4) + (((8*tan(c/2 + (d*x)/2)*(B^2*a^10 + 72*B^2*b^10 - 72*B^2*a*b^9 - 2*B^2*a^9*b - 120*B^2*
a^2*b^8 + 120*B^2*a^3*b^7 + 17*B^2*a^4*b^6 - 26*B^2*a^5*b^5 + 23*B^2*a^6*b^4 - 20*B^2*a^7*b^3 + 11*B^2*a^8*b^2
+ 32*C^2*a^2*b^8 - 32*C^2*a^3*b^7 - 64*C^2*a^4*b^6 + 64*C^2*a^5*b^5 + 20*C^2*a^6*b^4 - 32*C^2*a^7*b^3 + 16*C^
2*a^8*b^2 - 96*B*C*a*b^9 - 8*B*C*a^9*b + 96*B*C*a^2*b^8 + 176*B*C*a^3*b^7 - 176*B*C*a^4*b^6 - 40*B*C*a^5*b^5 +
64*B*C*a^6*b^4 - 40*B*C*a^7*b^3 + 16*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (((8*(2*B*a^15 - 12*B*
a^8*b^7 + 6*B*a^9*b^6 + 28*B*a^10*b^5 - 14*B*a^11*b^4 - 16*B*a^12*b^3 + 6*B*a^13*b^2 + 8*C*a^9*b^6 - 4*C*a^10*
b^5 - 20*C*a^11*b^4 + 12*C*a^12*b^3 + 12*C*a^13*b^2 - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) + (4*t
an(c/2 + (d*x)/2)*(B*a^2*1i + B*b^2*6i - C*a*b*4i)*(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)))*(B*a^2*1i + B*b^2*6i - C*a*b*4i))/(2*a^4))*(B*a^2*1
i + B*b^2*6i - C*a*b*4i)*1i)/(2*a^4))/((16*(108*B^3*b^11 - 54*B^3*a*b^10 - 216*B^3*a^2*b^9 + 81*B^3*a^3*b^8 +
63*B^3*a^4*b^7 - 9*B^3*a^5*b^6 + 41*B^3*a^6*b^5 - 4*B^3*a^7*b^4 + 4*B^3*a^8*b^3 - 32*C^3*a^3*b^8 + 16*C^3*a^4*
b^7 + 80*C^3*a^5*b^6 - 24*C^3*a^6*b^5 - 48*C^3*a^7*b^4 - 216*B^2*C*a*b^10 + 144*B*C^2*a^2*b^9 - 72*B*C^2*a^3*b
^8 - 336*B*C^2*a^4*b^7 + 108*B*C^2*a^5*b^6 + 168*B*C^2*a^6*b^5 - 6*B*C^2*a^7*b^4 + 24*B*C^2*a^8*b^3 + 108*B^2*
C*a^2*b^9 + 468*B^2*C*a^3*b^8 - 162*B^2*C*a^4*b^7 - 186*B^2*C*a^5*b^6 + 15*B^2*C*a^6*b^5 - 63*B^2*C*a^7*b^4 +
3*B^2*C*a^8*b^3 - 3*B^2*C*a^9*b^2))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (((8*tan(c/2 + (d*x)/2)*(B^2*a^10 +
72*B^2*b^10 - 72*B^2*a*b^9 - 2*B^2*a^9*b - 120*B^2*a^2*b^8 + 120*B^2*a^3*b^7 + 17*B^2*a^4*b^6 - 26*B^2*a^5*b^
5 + 23*B^2*a^6*b^4 - 20*B^2*a^7*b^3 + 11*B^2*a^8*b^2 + 32*C^2*a^2*b^8 - 32*C^2*a^3*b^7 - 64*C^2*a^4*b^6 + 64*C
^2*a^5*b^5 + 20*C^2*a^6*b^4 - 32*C^2*a^7*b^3 + 16*C^2*a^8*b^2 - 96*B*C*a*b^9 - 8*B*C*a^9*b + 96*B*C*a^2*b^8 +
176*B*C*a^3*b^7 - 176*B*C*a^4*b^6 - 40*B*C*a^5*b^5 + 64*B*C*a^6*b^4 - 40*B*C*a^7*b^3 + 16*B*C*a^8*b^2))/(a^8*b
+ a^9 - a^6*b^3 - a^7*b^2) + (((8*(2*B*a^15 - 12*B*a^8*b^7 + 6*B*a^9*b^6 + 28*B*a^10*b^5 - 14*B*a^11*b^4 - 16
*B*a^12*b^3 + 6*B*a^13*b^2 + 8*C*a^9*b^6 - 4*C*a^10*b^5 - 20*C*a^11*b^4 + 12*C*a^12*b^3 + 12*C*a^13*b^2 - 8*C*
a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (4*tan(c/2 + (d*x)/2)*(B*a^2*1i + B*b^2*6i - C*a*b*4i)*(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)))
*(B*a^2*1i + B*b^2*6i - C*a*b*4i))/(2*a^4))*(B*a^2*1i + B*b^2*6i - C*a*b*4i))/(2*a^4) + (((8*tan(c/2 + (d*x)/2
)*(B^2*a^10 + 72*B^2*b^10 - 72*B^2*a*b^9 - 2*B^2*a^9*b - 120*B^2*a^2*b^8 + 120*B^2*a^3*b^7 + 17*B^2*a^4*b^6 -
26*B^2*a^5*b^5 + 23*B^2*a^6*b^4 - 20*B^2*a^7*b^3 + 11*B^2*a^8*b^2 + 32*C^2*a^2*b^8 - 32*C^2*a^3*b^7 - 64*C^2*a
^4*b^6 + 64*C^2*a^5*b^5 + 20*C^2*a^6*b^4 - 32*C^2*a^7*b^3 + 16*C^2*a^8*b^2 - 96*B*C*a*b^9 - 8*B*C*a^9*b + 96*B
*C*a^2*b^8 + 176*B*C*a^3*b^7 - 176*B*C*a^4*b^6 - 40*B*C*a^5*b^5 + 64*B*C*a^6*b^4 - 40*B*C*a^7*b^3 + 16*B*C*a^8
*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (((8*(2*B*a^15 - 12*B*a^8*b^7 + 6*B*a^9*b^6 + 28*B*a^10*b^5 - 14*B*
a^11*b^4 - 16*B*a^12*b^3 + 6*B*a^13*b^2 + 8*C*a^9*b^6 - 4*C*a^10*b^5 - 20*C*a^11*b^4 + 12*C*a^12*b^3 + 12*C*a^
13*b^2 - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) + (4*tan(c/2 + (d*x)/2)*(B*a^2*1i + B*b^2*6i - C*a*
b*4i)*(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)))*(B*a^2*1i + B*b^2*6i - C*a*b*4i))/(2*a^4))*(B*a^2*1i + B*b^2*6i - C*a*b*4i))/(2*a^4)))*(B*a^2*1i
+ B*b^2*6i - C*a*b*4i)*1i)/(a^4*d) - ((tan(c/2 + (d*x)/2)^5*(B*a^4 + 6*B*b^4 - 2*C*a^4 - 5*B*a^2*b^2 + 2*C*a^
2*b^2 - 3*B*a*b^3 + 3*B*a^3*b - 4*C*a*b^3 + 2*C*a^3*b))/((a^3*b - a^4)*(a + b)) + (tan(c/2 + (d*x)/2)*(B*a^4 +
6*B*b^4 + 2*C*a^4 - 5*B*a^2*b^2 - 2*C*a^2*b^2 + 3*B*a*b^3 - 3*B*a^3*b - 4*C*a*b^3 + 2*C*a^3*b))/((a^3*b - a^4
)*(a + b)) - (2*tan(c/2 + (d*x)/2)^3*(B*a^4 - 6*B*b^4 + 3*B*a^2*b^2 + 4*C*a*b^3 - 2*C*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^2*atan(((b^2*((8*tan(c/2 + (d*x)/2)*(B^2*a^10 + 72*B^2*b^10 - 72*B^2*a*b^9 - 2*B^2*a^9*b - 120*
B^2*a^2*b^8 + 120*B^2*a^3*b^7 + 17*B^2*a^4*b^6 - 26*B^2*a^5*b^5 + 23*B^2*a^6*b^4 - 20*B^2*a^7*b^3 + 11*B^2*a^8
*b^2 + 32*C^2*a^2*b^8 - 32*C^2*a^3*b^7 - 64*C^2*a^4*b^6 + 64*C^2*a^5*b^5 + 20*C^2*a^6*b^4 - 32*C^2*a^7*b^3 + 1
6*C^2*a^8*b^2 - 96*B*C*a*b^9 - 8*B*C*a^9*b + 96*B*C*a^2*b^8 + 176*B*C*a^3*b^7 - 176*B*C*a^4*b^6 - 40*B*C*a^5*b
^5 + 64*B*C*a^6*b^4 - 40*B*C*a^7*b^3 + 16*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (b^2*((8*(2*B*a^15
- 12*B*a^8*b^7 + 6*B*a^9*b^6 + 28*B*a^10*b^5 - 14*B*a^11*b^4 - 16*B*a^12*b^3 + 6*B*a^13*b^2 + 8*C*a^9*b^6 - 4
*C*a^10*b^5 - 20*C*a^11*b^4 + 12*C*a^12*b^3 + 12*C*a^13*b^2 - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2
) - (8*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 4*B*a^2*b - 2*C*a*b^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))/((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)))*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 4*B*a^2*b - 2*C*a*b^2))/
(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*B*b^3 + 3*C*a^3 - 4*B*a^2*b - 2*C*a*b
^2)*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2) + (b^2*((8*tan(c/2 + (d*x)/2)*(B^2*a^10 + 72*B^2*b^10 - 72*B^
2*a*b^9 - 2*B^2*a^9*b - 120*B^2*a^2*b^8 + 120*B^2*a^3*b^7 + 17*B^2*a^4*b^6 - 26*B^2*a^5*b^5 + 23*B^2*a^6*b^4 -
20*B^2*a^7*b^3 + 11*B^2*a^8*b^2 + 32*C^2*a^2*b^8 - 32*C^2*a^3*b^7 - 64*C^2*a^4*b^6 + 64*C^2*a^5*b^5 + 20*C^2*
a^6*b^4 - 32*C^2*a^7*b^3 + 16*C^2*a^8*b^2 - 96*B*C*a*b^9 - 8*B*C*a^9*b + 96*B*C*a^2*b^8 + 176*B*C*a^3*b^7 - 17
6*B*C*a^4*b^6 - 40*B*C*a^5*b^5 + 64*B*C*a^6*b^4 - 40*B*C*a^7*b^3 + 16*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a
^7*b^2) - (b^2*((8*(2*B*a^15 - 12*B*a^8*b^7 + 6*B*a^9*b^6 + 28*B*a^10*b^5 - 14*B*a^11*b^4 - 16*B*a^12*b^3 + 6*
B*a^13*b^2 + 8*C*a^9*b^6 - 4*C*a^10*b^5 - 20*C*a^11*b^4 + 12*C*a^12*b^3 + 12*C*a^13*b^2 - 8*C*a^14*b))/(a^11*b
+ a^12 - a^9*b^3 - a^10*b^2) + (8*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 4*B
*a^2*b - 2*C*a*b^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))/((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)))*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^
3 - 4*B*a^2*b - 2*C*a*b^2))/(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*B*b^3 + 3
*C*a^3 - 4*B*a^2*b - 2*C*a*b^2)*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))/((16*(108*B^3*b^11 - 54*B^3*a*b^
10 - 216*B^3*a^2*b^9 + 81*B^3*a^3*b^8 + 63*B^3*a^4*b^7 - 9*B^3*a^5*b^6 + 41*B^3*a^6*b^5 - 4*B^3*a^7*b^4 + 4*B^
3*a^8*b^3 - 32*C^3*a^3*b^8 + 16*C^3*a^4*b^7 + 80*C^3*a^5*b^6 - 24*C^3*a^6*b^5 - 48*C^3*a^7*b^4 - 216*B^2*C*a*b
^10 + 144*B*C^2*a^2*b^9 - 72*B*C^2*a^3*b^8 - 336*B*C^2*a^4*b^7 + 108*B*C^2*a^5*b^6 + 168*B*C^2*a^6*b^5 - 6*B*C
^2*a^7*b^4 + 24*B*C^2*a^8*b^3 + 108*B^2*C*a^2*b^9 + 468*B^2*C*a^3*b^8 - 162*B^2*C*a^4*b^7 - 186*B^2*C*a^5*b^6
+ 15*B^2*C*a^6*b^5 - 63*B^2*C*a^7*b^4 + 3*B^2*C*a^8*b^3 - 3*B^2*C*a^9*b^2))/(a^11*b + a^12 - a^9*b^3 - a^10*b^
2) - (b^2*((8*tan(c/2 + (d*x)/2)*(B^2*a^10 + 72*B^2*b^10 - 72*B^2*a*b^9 - 2*B^2*a^9*b - 120*B^2*a^2*b^8 + 120*
B^2*a^3*b^7 + 17*B^2*a^4*b^6 - 26*B^2*a^5*b^5 + 23*B^2*a^6*b^4 - 20*B^2*a^7*b^3 + 11*B^2*a^8*b^2 + 32*C^2*a^2*
b^8 - 32*C^2*a^3*b^7 - 64*C^2*a^4*b^6 + 64*C^2*a^5*b^5 + 20*C^2*a^6*b^4 - 32*C^2*a^7*b^3 + 16*C^2*a^8*b^2 - 96
*B*C*a*b^9 - 8*B*C*a^9*b + 96*B*C*a^2*b^8 + 176*B*C*a^3*b^7 - 176*B*C*a^4*b^6 - 40*B*C*a^5*b^5 + 64*B*C*a^6*b^
4 - 40*B*C*a^7*b^3 + 16*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (b^2*((8*(2*B*a^15 - 12*B*a^8*b^7 +
6*B*a^9*b^6 + 28*B*a^10*b^5 - 14*B*a^11*b^4 - 16*B*a^12*b^3 + 6*B*a^13*b^2 + 8*C*a^9*b^6 - 4*C*a^10*b^5 - 20*C
*a^11*b^4 + 12*C*a^12*b^3 + 12*C*a^13*b^2 - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*b^2*tan(c/2
+ (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 4*B*a^2*b - 2*C*a*b^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))/((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)))*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 4*B*a^2*b - 2*C*a*b^2))/(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*B*b^3 + 3*C*a^3 - 4*B*a^2*b - 2*C*a*b^2))/(a^10 - a^4*b
^6 + 3*a^6*b^4 - 3*a^8*b^2) + (b^2*((8*tan(c/2 + (d*x)/2)*(B^2*a^10 + 72*B^2*b^10 - 72*B^2*a*b^9 - 2*B^2*a^9*b
- 120*B^2*a^2*b^8 + 120*B^2*a^3*b^7 + 17*B^2*a^4*b^6 - 26*B^2*a^5*b^5 + 23*B^2*a^6*b^4 - 20*B^2*a^7*b^3 + 11*
B^2*a^8*b^2 + 32*C^2*a^2*b^8 - 32*C^2*a^3*b^7 - 64*C^2*a^4*b^6 + 64*C^2*a^5*b^5 + 20*C^2*a^6*b^4 - 32*C^2*a^7*
b^3 + 16*C^2*a^8*b^2 - 96*B*C*a*b^9 - 8*B*C*a^9*b + 96*B*C*a^2*b^8 + 176*B*C*a^3*b^7 - 176*B*C*a^4*b^6 - 40*B*
C*a^5*b^5 + 64*B*C*a^6*b^4 - 40*B*C*a^7*b^3 + 16*B*C*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (b^2*((8*(2
*B*a^15 - 12*B*a^8*b^7 + 6*B*a^9*b^6 + 28*B*a^10*b^5 - 14*B*a^11*b^4 - 16*B*a^12*b^3 + 6*B*a^13*b^2 + 8*C*a^9*
b^6 - 4*C*a^10*b^5 - 20*C*a^11*b^4 + 12*C*a^12*b^3 + 12*C*a^13*b^2 - 8*C*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a
^10*b^2) + (8*b^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 4*B*a^2*b - 2*C*a*b^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))/((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)))*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 4*B*a^2*b - 2*C*a
*b^2))/(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*B*b^3 + 3*C*a^3 - 4*B*a^2*b -
2*C*a*b^2))/(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*B*b^3 + 3*C*a^3 - 4*B*a^
2*b - 2*C*a*b^2)*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 \[ \int \frac {\left (B + C \sec {\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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