3.8.3 \(\int \frac {\cos (c+d x) (A+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [703]

Optimal. Leaf size=367 \[ -\frac {4 A b x}{a^5}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (68 a^2 A b^4-24 A b^6+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]

[Out]

-4*A*b*x/a^5-(35*a^4*A*b^4-28*a^2*A*b^6+8*A*b^8-2*a^8*C-a^6*b^2*(20*A+3*C))*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/
2*c)/(a+b)^(1/2))/a^5/(a-b)^(7/2)/(a+b)^(7/2)/d+1/6*(68*a^2*A*b^4-24*A*b^6+a^6*(6*A-11*C)-a^4*b^2*(65*A+4*C))*
sin(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*(A*b^2+C*a^2)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3-1/6*(4*A*b^4-3*a^4*
C-a^2*b^2*(9*A+2*C))*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2-1/2*(11*a^2*A*b^4-4*A*b^6-2*a^6*C-3*a^4*b
^2*(4*A+C))*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))

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Rubi [A]
time = 1.23, antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4186, 4185, 4189, 4004, 3916, 2738, 214} \begin {gather*} -\frac {4 A b x}{a^5}+\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}-\frac {\left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {\left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\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)^{7/2} (a+b)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]

[Out]

(-4*A*b*x)/a^5 - ((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTanh[(Sqrt[a - b
]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((68*a^2*A*b^4 - 24*A*b^6 + a^6*(6*A -
11*C) - a^4*b^2*(65*A + 4*C))*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^
2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((4*A*b^4 - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^
2)^2*d*(a + b*Sec[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Sin[c + d*x])/(2*a^
3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2738

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 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

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 4185

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

Rule 4186

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 4189

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 (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (4 A b^2-a^2 (3 A-C)+3 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))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) \left (-23 a^2 A b^2+12 A b^4+a^4 (6 A-5 C)+2 a b \left (A b^2-a^2 (6 A+5 C)\right ) \sec (c+d x)-2 \left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-68 a^2 A b^4+24 A b^6-a^6 (6 A-11 C)+a^4 b^2 (65 A+4 C)+a b \left (4 A b^4-a^2 b^2 (7 A-4 C)+a^4 (18 A+11 C)\right ) \sec (c+d x)+3 \left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 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 )^3}\\ &=\frac {\left (68 a^2 A b^4-24 A b^6+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {-24 A b \left (a^2-b^2\right )^3-3 a \left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3}\\ &=-\frac {4 A b x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac {4 A b x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 b \left (a^2-b^2\right )^3}\\ &=-\frac {4 A b x}{a^5}+\frac {\left (68 a^2 A b^4-24 A b^6+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \text {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 b \left (a^2-b^2\right )^3 d}\\ &=-\frac {4 A b x}{a^5}+\frac {\left (20 a^6 A b^2-35 a^4 A b^4+28 a^2 A b^6-8 A b^8+2 a^8 C+3 a^6 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)^{7/2} (a+b)^{7/2} d}+\frac {\left (68 a^2 A b^4-24 A b^6+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.49, size = 1089, normalized size = 2.97 \begin {gather*} -\frac {8 A b x (b+a \cos (c+d x))^4 \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a^5 (A+2 C+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {\left (-20 a^6 A b^2+35 a^4 A b^4-28 a^2 A b^6+8 A b^8-2 a^8 C-3 a^6 b^2 C\right ) (b+a \cos (c+d x))^4 \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (-\frac {2 i \text {ArcTan}\left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \cos (c)}{a^5 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {2 \text {ArcTan}\left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \sin (c)}{a^5 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}\right )}{\left (-a^2+b^2\right )^3 (A+2 C+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {2 (b+a \cos (c+d x)) \sec (c) \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (A b^6 \sin (c)+a^2 b^4 C \sin (c)-a A b^5 \sin (d x)-a^3 b^3 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right ) d (A+2 C+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {(b+a \cos (c+d x))^2 \sec (c) \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (-17 a^2 A b^5 \sin (c)+12 A b^7 \sin (c)-11 a^4 b^3 C \sin (c)+6 a^2 b^5 C \sin (c)+15 a^3 A b^4 \sin (d x)-10 a A b^6 \sin (d x)+9 a^5 b^2 C \sin (d x)-4 a^3 b^4 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right )^2 d (A+2 C+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {(b+a \cos (c+d x))^3 \sec (c) \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (75 a^4 A b^4 \sin (c)-96 a^2 A b^6 \sin (c)+36 A b^8 \sin (c)+27 a^6 b^2 C \sin (c)-18 a^4 b^4 C \sin (c)+6 a^2 b^6 C \sin (c)-60 a^5 A b^3 \sin (d x)+71 a^3 A b^5 \sin (d x)-26 a A b^7 \sin (d x)-18 a^7 b C \sin (d x)+5 a^5 b^3 C \sin (d x)-2 a^3 b^5 C \sin (d x)\right )}{3 a^5 \left (a^2-b^2\right )^3 d (A+2 C+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {2 A (b+a \cos (c+d x))^4 \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \tan (c+d x)}{a^4 d (A+2 C+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]

[Out]

(-8*A*b*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a^5*(A + 2*C + A*Cos[2*c + 2*d*x])*(a
+ b*Sec[c + d*x])^4) + ((-20*a^6*A*b^2 + 35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - 3*a^6*b^2*C)*(b +
a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*
Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2]
+ I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*
(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]
]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]))
)/((-a^2 + b^2)^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[
c + d*x]^2*(A + C*Sec[c + d*x]^2)*(A*b^6*Sin[c] + a^2*b^4*C*Sin[c] - a*A*b^5*Sin[d*x] - a^3*b^3*C*Sin[d*x]))/(
3*a^5*(a^2 - b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Se
c[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(-17*a^2*A*b^5*Sin[c] + 12*A*b^7*Sin[c] - 11*a^4*b^3*C*Sin[c] + 6*a^2*b^5*
C*Sin[c] + 15*a^3*A*b^4*Sin[d*x] - 10*a*A*b^6*Sin[d*x] + 9*a^5*b^2*C*Sin[d*x] - 4*a^3*b^4*C*Sin[d*x]))/(3*a^5*
(a^2 - b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c
+ d*x]^2*(A + C*Sec[c + d*x]^2)*(75*a^4*A*b^4*Sin[c] - 96*a^2*A*b^6*Sin[c] + 36*A*b^8*Sin[c] + 27*a^6*b^2*C*Si
n[c] - 18*a^4*b^4*C*Sin[c] + 6*a^2*b^6*C*Sin[c] - 60*a^5*A*b^3*Sin[d*x] + 71*a^3*A*b^5*Sin[d*x] - 26*a*A*b^7*S
in[d*x] - 18*a^7*b*C*Sin[d*x] + 5*a^5*b^3*C*Sin[d*x] - 2*a^3*b^5*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)^3*d*(A + 2*C
+ A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*A*(b + a*Cos[c + d*x])^4*Sec[c + d*x]*(A + C*Sec[c + d*x]^2
)*Tan[c + d*x])/(a^4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)

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Maple [A]
time = 0.52, size = 503, normalized size = 1.37

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C +3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}+9 a^{6} C +a^{4} b^{2} C \right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C -3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}+2 a^{8} C +3 a^{6} b^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}-\frac {2 A \left (-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+4 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}}{d}\)	\(503\)
default	\(\frac {-\frac {2 \left (\frac {-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C +3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}+9 a^{6} C +a^{4} b^{2} C \right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C -3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}+2 a^{8} C +3 a^{6} b^{2} C \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}-\frac {2 A \left (-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+4 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}}{d}\)	\(503\)
risch	\(\text {Expression too large to display}\)	\(1806\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/a^5*((-1/2*(20*A*a^4*b^2+5*A*a^3*b^3-18*A*a^2*b^4-2*A*a*b^5+6*A*b^6+6*C*a^6+3*C*a^5*b+2*C*a^4*b^2)*a*b
/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5+2/3*(30*A*a^4*b^2-29*A*a^2*b^4+9*A*b^6+9*C*a^6+C*a^4*b^2
)*a*b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(20*A*a^4*b^2-5*A*a^3*b^3-18*A*a^2*b^4+2*A*a*b^
5+6*A*b^6+6*C*a^6-3*C*a^5*b+2*C*a^4*b^2)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(a*tan(1/2*d*
x+1/2*c)^2-b*tan(1/2*d*x+1/2*c)^2-a-b)^3-1/2*(20*A*a^6*b^2-35*A*a^4*b^4+28*A*a^2*b^6-8*A*b^8+2*C*a^8+3*C*a^6*b
^2)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))-2
*A/a^5*(-a*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+4*b*arctan(tan(1/2*d*x+1/2*c))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 934 vs. \(2 (347) = 694\).
time = 2.62, size = 1925, normalized size = 5.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

[-1/12*(48*(A*a^11*b - 4*A*a^9*b^3 + 6*A*a^7*b^5 - 4*A*a^5*b^7 + A*a^3*b^9)*d*x*cos(d*x + c)^3 + 144*(A*a^10*b
^2 - 4*A*a^8*b^4 + 6*A*a^6*b^6 - 4*A*a^4*b^8 + A*a^2*b^10)*d*x*cos(d*x + c)^2 + 144*(A*a^9*b^3 - 4*A*a^7*b^5 +
6*A*a^5*b^7 - 4*A*a^3*b^9 + A*a*b^11)*d*x*cos(d*x + c) + 48*(A*a^8*b^4 - 4*A*a^6*b^6 + 6*A*a^4*b^8 - 4*A*a^2*
b^10 + A*b^12)*d*x - 3*(2*C*a^8*b^3 + (20*A + 3*C)*a^6*b^5 - 35*A*a^4*b^7 + 28*A*a^2*b^9 - 8*A*b^11 + (2*C*a^1
1 + (20*A + 3*C)*a^9*b^2 - 35*A*a^7*b^4 + 28*A*a^5*b^6 - 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(2*C*a^10*b + (20*A +
3*C)*a^8*b^3 - 35*A*a^6*b^5 + 28*A*a^4*b^7 - 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*(2*C*a^9*b^2 + (20*A + 3*C)*a^7*
b^4 - 35*A*a^5*b^6 + 28*A*a^3*b^8 - 8*A*a*b^10)*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*((6*A - 11*C)*a^9*b^3 - (71*A - 7*C)*a^7*b^5 + (133*A + 4*C)*a^5*b^7 - 92*A
*a^3*b^9 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10*b^2 + 6*A*a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)*cos(d*x + c)^3 + (1
8*(A - C)*a^11*b - (132*A - 23*C)*a^9*b^3 + (239*A - 7*C)*a^7*b^5 - (169*A - 2*C)*a^5*b^7 + 44*A*a^3*b^9)*cos(
d*x + c)^2 + 3*(3*(2*A - 3*C)*a^10*b^2 - (59*A - 8*C)*a^8*b^4 + (110*A + C)*a^6*b^6 - 77*A*a^4*b^8 + 20*A*a^2*
b^10)*cos(d*x + c))*sin(d*x + c))/((a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 + a^8*b^8)*d*cos(d*x + c)^3 +
3*(a^15*b - 4*a^13*b^3 + 6*a^11*b^5 - 4*a^9*b^7 + a^7*b^9)*d*cos(d*x + c)^2 + 3*(a^14*b^2 - 4*a^12*b^4 + 6*a^1
0*b^6 - 4*a^8*b^8 + a^6*b^10)*d*cos(d*x + c) + (a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7 - 4*a^7*b^9 + a^5*b^11)*d),
-1/6*(24*(A*a^11*b - 4*A*a^9*b^3 + 6*A*a^7*b^5 - 4*A*a^5*b^7 + A*a^3*b^9)*d*x*cos(d*x + c)^3 + 72*(A*a^10*b^2
- 4*A*a^8*b^4 + 6*A*a^6*b^6 - 4*A*a^4*b^8 + A*a^2*b^10)*d*x*cos(d*x + c)^2 + 72*(A*a^9*b^3 - 4*A*a^7*b^5 + 6*A
*a^5*b^7 - 4*A*a^3*b^9 + A*a*b^11)*d*x*cos(d*x + c) + 24*(A*a^8*b^4 - 4*A*a^6*b^6 + 6*A*a^4*b^8 - 4*A*a^2*b^10
+ A*b^12)*d*x - 3*(2*C*a^8*b^3 + (20*A + 3*C)*a^6*b^5 - 35*A*a^4*b^7 + 28*A*a^2*b^9 - 8*A*b^11 + (2*C*a^11 +
(20*A + 3*C)*a^9*b^2 - 35*A*a^7*b^4 + 28*A*a^5*b^6 - 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(2*C*a^10*b + (20*A + 3*C
)*a^8*b^3 - 35*A*a^6*b^5 + 28*A*a^4*b^7 - 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*(2*C*a^9*b^2 + (20*A + 3*C)*a^7*b^4
- 35*A*a^5*b^6 + 28*A*a^3*b^8 - 8*A*a*b^10)*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))) - ((6*A - 11*C)*a^9*b^3 - (71*A - 7*C)*a^7*b^5 + (133*A + 4*C)*a^5*b^7
- 92*A*a^3*b^9 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10*b^2 + 6*A*a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)*cos(d*x + c)^
3 + (18*(A - C)*a^11*b - (132*A - 23*C)*a^9*b^3 + (239*A - 7*C)*a^7*b^5 - (169*A - 2*C)*a^5*b^7 + 44*A*a^3*b^9
)*cos(d*x + c)^2 + 3*(3*(2*A - 3*C)*a^10*b^2 - (59*A - 8*C)*a^8*b^4 + (110*A + C)*a^6*b^6 - 77*A*a^4*b^8 + 20*
A*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 + a^8*b^8)*d*cos(d*x + c
)^3 + 3*(a^15*b - 4*a^13*b^3 + 6*a^11*b^5 - 4*a^9*b^7 + a^7*b^9)*d*cos(d*x + c)^2 + 3*(a^14*b^2 - 4*a^12*b^4 +
6*a^10*b^6 - 4*a^8*b^8 + a^6*b^10)*d*cos(d*x + c) + (a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7 - 4*a^7*b^9 + a^5*b^11
)*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(A+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**4,x)

[Out]

Integral((A + C*sec(c + d*x)**2)*cos(c + d*x)/(a + b*sec(c + d*x))**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (347) = 694\).
time = 0.60, size = 847, normalized size = 2.31 \begin {gather*} \frac {\frac {3 \, {\left (2 \, C a^{8} + 20 \, A a^{6} b^{2} + 3 \, C a^{6} b^{2} - 35 \, A a^{4} b^{4} + 28 \, A a^{2} b^{6} - 8 \, A b^{8}\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^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {12 \, {\left (d x + c\right )} A b}{a^{5}} + \frac {18 \, C a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 117 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 42 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, C a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 236 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 152 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, C a^{8} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, A a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, A a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 117 \, A a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 42 \, A a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\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 )}^{3}} + \frac {6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x, algorithm="giac")

[Out]

1/3*(3*(2*C*a^8 + 20*A*a^6*b^2 + 3*C*a^6*b^2 - 35*A*a^4*b^4 + 28*A*a^2*b^6 - 8*A*b^8)*(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^1
1 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(-a^2 + b^2)) - 12*(d*x + c)*A*b/a^5 + (18*C*a^8*b*tan(1/2*d*x + 1/2*
c)^5 - 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 + 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^6*b^3*tan(1/2*d*x + 1
/2*c)^5 - 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 24*A*a^4*b^5*tan(1/2*d*x
+ 1/2*c)^5 + 6*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 24*A*a^2*b^7*tan(1/2
*d*x + 1/2*c)^5 - 42*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 18*A*b^9*tan(1/2*d*x + 1/2*c)^5 - 36*C*a^8*b*tan(1/2*d*x
+ 1/2*c)^3 - 120*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 32*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 236*A*a^4*b^5*tan(1
/2*d*x + 1/2*c)^3 + 4*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 152*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 + 36*A*b^9*tan(1
/2*d*x + 1/2*c)^3 + 18*C*a^8*b*tan(1/2*d*x + 1/2*c) + 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c) + 60*A*a^6*b^3*tan(1/2
*d*x + 1/2*c) + 6*C*a^6*b^3*tan(1/2*d*x + 1/2*c) + 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c) + 3*C*a^5*b^4*tan(1/2*d*
x + 1/2*c) - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*C*a^4*b^5*tan(1/2*d*x + 1/2*c) - 117*A*a^3*b^6*tan(1/2*d*x
+ 1/2*c) - 24*A*a^2*b^7*tan(1/2*d*x + 1/2*c) + 42*A*a*b^8*tan(1/2*d*x + 1/2*c) + 18*A*b^9*tan(1/2*d*x + 1/2*c)
)/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) +
6*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^4))/d

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Mupad [B]
time = 17.27, size = 2500, normalized size = 6.81 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)*(A + C/cos(c + d*x)^2))/(a + b/cos(c + d*x))^4,x)

[Out]

((tan(c/2 + (d*x)/2)*(8*A*b^7 - 2*A*a^7 - 24*A*a^2*b^5 - 11*A*a^3*b^4 + 26*A*a^4*b^3 + 6*A*a^5*b^2 + 2*C*a^4*b
^3 - 3*C*a^5*b^2 + 4*A*a*b^6 - 2*A*a^6*b + 6*C*a^6*b))/((a + b)*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2)) + (tan(
c/2 + (d*x)/2)^7*(2*A*a^7 + 8*A*b^7 - 24*A*a^2*b^5 + 11*A*a^3*b^4 + 26*A*a^4*b^3 - 6*A*a^5*b^2 + 2*C*a^4*b^3 +
3*C*a^5*b^2 - 4*A*a*b^6 - 2*A*a^6*b + 6*C*a^6*b))/((a^4*b - a^5)*(a + b)^3) + (tan(c/2 + (d*x)/2)^3*(18*A*a^8
+ 72*A*b^8 - 236*A*a^2*b^6 + 47*A*a^3*b^5 + 273*A*a^4*b^4 - 60*A*a^5*b^3 - 72*A*a^6*b^2 + 10*C*a^4*b^4 - 7*C*
a^5*b^3 + 45*C*a^6*b^2 - 12*A*a*b^7 - 18*C*a^7*b))/(3*(a + b)^2*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2)) - (tan(
c/2 + (d*x)/2)^5*(18*A*a^8 + 72*A*b^8 - 236*A*a^2*b^6 - 47*A*a^3*b^5 + 273*A*a^4*b^4 + 60*A*a^5*b^3 - 72*A*a^6
*b^2 + 10*C*a^4*b^4 + 7*C*a^5*b^3 + 45*C*a^6*b^2 + 12*A*a*b^7 + 18*C*a^7*b))/(3*(a^4*b - a^5)*(a + b)^3*(a - b
)))/(d*(3*a*b^2 + 3*a^2*b - tan(c/2 + (d*x)/2)^4*(6*a^2*b - 6*b^3) + tan(c/2 + (d*x)/2)^2*(6*a*b^2 - 2*a^3 + 4
*b^3) + tan(c/2 + (d*x)/2)^6*(2*a^3 - 6*a*b^2 + 4*b^3) + a^3 + b^3 - tan(c/2 + (d*x)/2)^8*(3*a*b^2 - 3*a^2*b +
a^3 - b^3))) - (8*A*b*atan(((4*A*b*((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 768*A
^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9
+ 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b
^3 + 64*A^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6
- 20*A*C*a^12*b^4 + 80*A*C*a^14*b^2))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12
*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) + (A*b*((16*(2*C*a^24 + 8*A*a^10*b^1
4 - 4*A*a^11*b^13 - 52*A*a^12*b^12 + 25*A*a^13*b^11 + 143*A*a^14*b^10 - 63*A*a^15*b^9 - 217*A*a^16*b^8 + 87*A*
a^17*b^7 + 193*A*a^18*b^6 - 73*A*a^19*b^5 - 95*A*a^20*b^4 + 36*A*a^21*b^3 + 20*A*a^22*b^2 + 3*C*a^15*b^9 - 3*C
*a^16*b^8 - 7*C*a^17*b^7 + 7*C*a^18*b^6 + 3*C*a^19*b^5 - 3*C*a^20*b^4 + 3*C*a^21*b^3 - 3*C*a^22*b^2 - 8*A*a^23
*b - 2*C*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6
+ 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) - (A*b*tan(c/2 + (d*x)/2)*(8*a^23*b - 8*a^10*b^14 + 8*
a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^1
8*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2)*32i)/(a^5*(a^18*b + a^19 - a^8*b^11 - a^9*b^10
+ 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))
*4i)/a^5))/a^5 + (4*A*b*((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 768*A^2*a^2*b^14
+ 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a
^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*
a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6 - 20*A*C*a^1
2*b^4 + 80*A*C*a^14*b^2))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^
13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) - (A*b*((16*(2*C*a^24 + 8*A*a^10*b^14 - 4*A*a^11
*b^13 - 52*A*a^12*b^12 + 25*A*a^13*b^11 + 143*A*a^14*b^10 - 63*A*a^15*b^9 - 217*A*a^16*b^8 + 87*A*a^17*b^7 + 1
93*A*a^18*b^6 - 73*A*a^19*b^5 - 95*A*a^20*b^4 + 36*A*a^21*b^3 + 20*A*a^22*b^2 + 3*C*a^15*b^9 - 3*C*a^16*b^8 -
7*C*a^17*b^7 + 7*C*a^18*b^6 + 3*C*a^19*b^5 - 3*C*a^20*b^4 + 3*C*a^21*b^3 - 3*C*a^22*b^2 - 8*A*a^23*b - 2*C*a^2
3*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b
^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (A*b*tan(c/2 + (d*x)/2)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 +
48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*
a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2)*32i)/(a^5*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9
+ 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*4i)/a^5))/a
^5)/((32*(128*A^3*b^16 - 64*A^3*a*b^15 - 832*A^3*a^2*b^14 + 400*A^3*a^3*b^13 + 2288*A^3*a^4*b^12 - 1088*A^3*a^
5*b^11 - 3472*A^3*a^6*b^10 + 1602*A^3*a^7*b^9 + 3088*A^3*a^8*b^8 - 1280*A^3*a^9*b^7 - 1520*A^3*a^10*b^6 + 480*
A^3*a^11*b^5 + 320*A^3*a^12*b^4 + 8*A*C^2*a^15*b + 18*A*C^2*a^11*b^5 + 24*A*C^2*a^13*b^3 - 48*A^2*C*a^5*b^11 -
48*A^2*C*a^6*b^10 + 160*A^2*C*a^7*b^9 + 112*A^2*C*a^8*b^8 - 148*A^2*C*a^9*b^7 - 48*A^2*C*a^10*b^6 + 8*A^2*C*a
^11*b^5 - 48*A^2*C*a^12*b^4 + 128*A^2*C*a^13*b^3 + 32*A^2*C*a^14*b^2))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10
+ 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) +
(A*b*((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*C^2*a^16 - 128*A^2*a*b^15 - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13
+ 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600...

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