3.280 \(\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=547 \[ -\frac{b^2 \left (-84 a^4 A b^3+69 a^2 A b^5+40 a^6 A b+35 a^5 b^2 B-28 a^3 b^4 B-20 a^7 B+8 a b^6 B-20 A b^7\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (-146 a^4 A b^3+167 a^2 A b^5+24 a^6 A b+65 a^5 b^2 B-68 a^3 b^4 B-6 a^7 B+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3}+\frac{\left (a^2 A-8 a b B+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac{\left (-23 a^4 A b^2+27 a^2 A b^4+a^6 A-11 a^3 b^3 B+12 a^5 b B+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{b \left (-53 a^2 A b^3+48 a^4 A b+20 a^3 b^2 B-27 a^5 B-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{b \left (10 a^2 A b-7 a^3 B+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

[Out]

-((b^2*(40*a^6*A*b - 84*a^4*A*b^3 + 69*a^2*A*b^5 - 20*A*b^7 - 20*a^7*B + 35*a^5*b^2*B - 28*a^3*b^4*B + 8*a*b^6
*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d)) + ((a^2*A + 20*A*
b^2 - 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^6*d) - ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*
a^7*B + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b
^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 -
b^2)^3*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(10*a^2
*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2
) + (b*(48*a^4*A*b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*Sec[c + d*x]*Tan[c + d*x])
/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 7.30446, antiderivative size = 547, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3000, 3055, 3001, 3770, 2659, 205} \[ -\frac{b^2 \left (-84 a^4 A b^3+69 a^2 A b^5+40 a^6 A b+35 a^5 b^2 B-28 a^3 b^4 B-20 a^7 B+8 a b^6 B-20 A b^7\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (-146 a^4 A b^3+167 a^2 A b^5+24 a^6 A b+65 a^5 b^2 B-68 a^3 b^4 B-6 a^7 B+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3}+\frac{\left (a^2 A-8 a b B+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac{\left (-23 a^4 A b^2+27 a^2 A b^4+a^6 A-11 a^3 b^3 B+12 a^5 b B+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{b \left (-53 a^2 A b^3+48 a^4 A b+20 a^3 b^2 B-27 a^5 B-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{b \left (10 a^2 A b-7 a^3 B+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]

[Out]

-((b^2*(40*a^6*A*b - 84*a^4*A*b^3 + 69*a^2*A*b^5 - 20*A*b^7 - 20*a^7*B + 35*a^5*b^2*B - 28*a^3*b^4*B + 8*a*b^6
*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d)) + ((a^2*A + 20*A*
b^2 - 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^6*d) - ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*
a^7*B + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b
^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 -
b^2)^3*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(10*a^2
*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2
) + (b*(48*a^4*A*b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*Sec[c + d*x]*Tan[c + d*x])
/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 3000

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((A*b^2 - a*b*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*
Sin[e + f*x])^(1 + n))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m
+ n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^
2, 0] && RationalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n,
-1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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 205

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

Rubi steps

\begin{align*} \int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac{b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\int \frac{\left (3 a^2 A-5 A b^2+2 a b B-3 a (A b-a B) \cos (c+d x)+4 b (A b-a B) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\int \frac{\left (2 \left (3 a^4 A-18 a^2 A b^2+10 A b^4+9 a^3 b B-4 a b^3 B\right )-2 a \left (6 a^2 A b-A b^3-3 a^3 B-2 a b^2 B\right ) \cos (c+d x)+3 b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\left (6 \left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right )-a \left (18 a^4 A b-8 a^2 A b^3+5 A b^5-6 a^5 B-7 a^3 b^2 B-2 a b^4 B\right ) \cos (c+d x)+2 b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac{\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\left (-2 \left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right )+2 a \left (3 a^6 A+27 a^4 A b^2-25 a^2 A b^4+10 A b^6-18 a^5 b B+7 a^3 b^3 B-4 a b^5 B\right ) \cos (c+d x)+6 b \left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^4 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac{\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\left (6 \left (a^2-b^2\right )^3 \left (a^2 A+20 A b^2-8 a b B\right )+6 a b \left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac{\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (a^2 A+20 A b^2-8 a b B\right ) \int \sec (c+d x) \, dx}{2 a^6}-\frac{\left (b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac{\left (a^2 A+20 A b^2-8 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac{\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\left (b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^6 \left (a^2-b^2\right )^3 d}\\ &=-\frac{b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 (a-b)^{7/2} (a+b)^{7/2} d}+\frac{\left (a^2 A+20 A b^2-8 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac{\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 4.93706, size = 781, normalized size = 1.43 \[ \frac{\frac{96 b^2 \left (84 a^4 A b^3-69 a^2 A b^5-40 a^6 A b-35 a^5 b^2 B+28 a^3 b^4 B+20 a^7 B-8 a b^6 B+20 A b^7\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}-48 \left (a^2 A-8 a b B+20 A b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+48 \left (a^2 A-8 a b B+20 A b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{2 a \tan (c+d x) \sec (c+d x) \left (6 a \left (-9 a^6 A b^3+309 a^4 A b^5-400 a^2 A b^7-20 a^8 A b-6 a^7 b^2 B-135 a^5 b^4 B+163 a^3 b^6 B+8 a^9 B-60 a b^8 B+150 A b^9\right ) \cos (c+d x)+12 b \left (85 a^6 A b^3-55 a^4 A b^5-19 a^2 A b^7-21 a^8 A b-36 a^7 b^2 B+20 a^5 b^4 B+8 a^3 b^6 B+6 a^9 B-8 a b^8 B+20 A b^9\right ) \cos (2 (c+d x))-138 a^7 A b^3 \cos (3 (c+d x))-24 a^6 A b^4 \cos (4 (c+d x))+738 a^5 A b^5 \cos (3 (c+d x))+146 a^4 A b^6 \cos (4 (c+d x))-840 a^3 A b^7 \cos (3 (c+d x))-167 a^2 A b^8 \cos (4 (c+d x))-324 a^8 A b^2+1116 a^6 A b^4-830 a^4 A b^6-61 a^2 A b^8+24 a^{10} A+36 a^8 b^2 B \cos (3 (c+d x))+6 a^7 b^3 B \cos (4 (c+d x))-318 a^6 b^4 B \cos (3 (c+d x))-65 a^5 b^5 B \cos (4 (c+d x))+342 a^4 b^6 B \cos (3 (c+d x))+68 a^3 b^7 B \cos (4 (c+d x))-120 a^2 b^8 B \cos (3 (c+d x))-438 a^7 b^3 B+305 a^5 b^5 B+28 a^3 b^7 B+72 a^9 b B+300 a A b^9 \cos (3 (c+d x))-24 a b^9 B \cos (4 (c+d x))-72 a b^9 B+60 A b^{10} \cos (4 (c+d x))+180 A b^{10}\right )}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}}{96 a^6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]

[Out]

((96*b^2*(-40*a^6*A*b + 84*a^4*A*b^3 - 69*a^2*A*b^5 + 20*A*b^7 + 20*a^7*B - 35*a^5*b^2*B + 28*a^3*b^4*B - 8*a*
b^6*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - 48*(a^2*A + 20*A*b^2 - 8*a*b
*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(a^2*A + 20*A*b^2 - 8*a*b*B)*Log[Cos[(c + d*x)/2] + Sin[(c +
d*x)/2]] + (2*a*(24*a^10*A - 324*a^8*A*b^2 + 1116*a^6*A*b^4 - 830*a^4*A*b^6 - 61*a^2*A*b^8 + 180*A*b^10 + 72*
a^9*b*B - 438*a^7*b^3*B + 305*a^5*b^5*B + 28*a^3*b^7*B - 72*a*b^9*B + 6*a*(-20*a^8*A*b - 9*a^6*A*b^3 + 309*a^4
*A*b^5 - 400*a^2*A*b^7 + 150*A*b^9 + 8*a^9*B - 6*a^7*b^2*B - 135*a^5*b^4*B + 163*a^3*b^6*B - 60*a*b^8*B)*Cos[c
+ d*x] + 12*b*(-21*a^8*A*b + 85*a^6*A*b^3 - 55*a^4*A*b^5 - 19*a^2*A*b^7 + 20*A*b^9 + 6*a^9*B - 36*a^7*b^2*B +
20*a^5*b^4*B + 8*a^3*b^6*B - 8*a*b^8*B)*Cos[2*(c + d*x)] - 138*a^7*A*b^3*Cos[3*(c + d*x)] + 738*a^5*A*b^5*Cos
[3*(c + d*x)] - 840*a^3*A*b^7*Cos[3*(c + d*x)] + 300*a*A*b^9*Cos[3*(c + d*x)] + 36*a^8*b^2*B*Cos[3*(c + d*x)]
- 318*a^6*b^4*B*Cos[3*(c + d*x)] + 342*a^4*b^6*B*Cos[3*(c + d*x)] - 120*a^2*b^8*B*Cos[3*(c + d*x)] - 24*a^6*A*
b^4*Cos[4*(c + d*x)] + 146*a^4*A*b^6*Cos[4*(c + d*x)] - 167*a^2*A*b^8*Cos[4*(c + d*x)] + 60*A*b^10*Cos[4*(c +
d*x)] + 6*a^7*b^3*B*Cos[4*(c + d*x)] - 65*a^5*b^5*B*Cos[4*(c + d*x)] + 68*a^3*b^7*B*Cos[4*(c + d*x)] - 24*a*b^
9*B*Cos[4*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(96*a^6*d)

________________________________________________________________________________________

Maple [B]  time = 0.219, size = 3042, normalized size = 5.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x)

[Out]

12/d*b^8/a^5/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x
+1/2*c)*A-6/d*b^7/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*ta
n(1/2*d*x+1/2*c)*B-6/d*b^7/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^
2+b^3)*tan(1/2*d*x+1/2*c)^5*B+24/d*b^8/a^5/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^
2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+1/2/d*A/a^4/(tan(1/2*d*x+1/2*c)-1)+1/2/d*A/a^4/(tan(1/2*d*x+1/2*c)+1
)-4/d/a^5*ln(tan(1/2*d*x+1/2*c)+1)*B*b+116/3/d*b^5/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(
a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-12/d*b^7/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^
2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+12/d*b^8/a^5/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2
*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+28/d*b^6/a^3/(a^6-3*a^4*b^2+3*a^
2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+20/d*b^9/a^6/(a^6-3*a^4*
b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-8/d*b^8/a^5/(a^6
-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-212/3/d/a
^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^6/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c
)^3*A+30/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2
*d*x+1/2*c)^5*A+6/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^5/(a-b)/(a^3+3*a^2*b+3*a*b^2+b
^3)*tan(1/2*d*x+1/2*c)^5*A-34/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^6/(a+b)/(a^3-3*a^2
*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-34/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*b^6/(a-b)/
(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A-6/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3
*b^5/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+60/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*
b+a+b)^3*b^4/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+30/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x
+1/2*c)^2*b+a+b)^3*b^4/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+84/d/a^2/(a^6-3*a^4*b^2+3*a^2*b^4-
b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^5-69/d/a^4/(a^6-3*a^4*b^2+3*
a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A*b^7-1/d/a^4/(tan(1/2*d
*x+1/2*c)+1)*B+1/2/d*A/a^4/(tan(1/2*d*x+1/2*c)-1)^2-1/d/a^4/(tan(1/2*d*x+1/2*c)-1)*B-1/2/d*A/a^4/(tan(1/2*d*x+
1/2*c)+1)^2-5/d*b^4/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*ta
n(1/2*d*x+1/2*c)^5*B+18/d*b^5/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a
*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-2/d*b^6/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-
3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-3/d*b^7/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a
-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+18/d*b^5/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2
*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+3/d*b^7/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x
+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+5/d*b^4/a/(tan(1/2*d*x+1/2*c)^2*a-tan(
1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+2/d*b^6/a^3/(tan(1/2*d*x+1/2*c)
^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+1/2/d/a^4*A*ln(tan(1
/2*d*x+1/2*c)+1)-1/2/d/a^4*A*ln(tan(1/2*d*x+1/2*c)-1)-40/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)
^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B*b^3-20/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^
2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B*b^3+20/d*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a
-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B*a-20/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2
*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B*b^3+4/d/a^5/(tan(1/2*d*x+1/2*c)-
1)*A*b-10/d/a^6*ln(tan(1/2*d*x+1/2*c)-1)*A*b^2+4/d/a^5*ln(tan(1/2*d*x+1/2*c)-1)*B*b+4/d/a^5/(tan(1/2*d*x+1/2*c
)+1)*A*b+10/d/a^6*ln(tan(1/2*d*x+1/2*c)+1)*A*b^2-35/d*b^4/a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*
arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-40/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1
/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)**3/(a+b*cos(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.78131, size = 1472, normalized size = 2.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

-1/6*(6*(20*B*a^7*b^2 - 40*A*a^6*b^3 - 35*B*a^5*b^4 + 84*A*a^4*b^5 + 28*B*a^3*b^6 - 69*A*a^2*b^7 - 8*B*a*b^8 +
20*A*b^9)*(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^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*sqrt(a^2 - b^2)) + 2*(60*B*a^7*b^3*tan
(1/2*d*x + 1/2*c)^5 - 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 + 162*A*a^5*b
^5*tan(1/2*d*x + 1/2*c)^5 - 24*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 + 117*B*
a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 24*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 +
48*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 42*B*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 81*A*a*b^9*tan(1/2*d*x + 1/2*c)^5
+ 18*B*a*b^9*tan(1/2*d*x + 1/2*c)^5 - 36*A*b^10*tan(1/2*d*x + 1/2*c)^5 + 120*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^3
- 180*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 - 236*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^3 + 392*A*a^4*b^6*tan(1/2*d*x + 1/
2*c)^3 + 152*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^3 - 284*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 - 36*B*a*b^9*tan(1/2*d*x
+ 1/2*c)^3 + 72*A*b^10*tan(1/2*d*x + 1/2*c)^3 + 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c) - 90*A*a^6*b^4*tan(1/2*d*x +
1/2*c) + 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c) - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c) - 24*B*a^5*b^5*tan(1/2*d*x +
1/2*c) + 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c) - 117*B*a^4*b^6*tan(1/2*d*x + 1/2*c) + 213*A*a^3*b^7*tan(1/2*d*x +
1/2*c) - 24*B*a^3*b^7*tan(1/2*d*x + 1/2*c) + 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c) + 42*B*a^2*b^8*tan(1/2*d*x + 1
/2*c) - 81*A*a*b^9*tan(1/2*d*x + 1/2*c) + 18*B*a*b^9*tan(1/2*d*x + 1/2*c) - 36*A*b^10*tan(1/2*d*x + 1/2*c))/((
a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*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) - 3*(
A*a^2 - 8*B*a*b + 20*A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^6 + 3*(A*a^2 - 8*B*a*b + 20*A*b^2)*log(abs(ta
n(1/2*d*x + 1/2*c) - 1))/a^6 - 6*(A*a*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*tan(1/2*d*x + 1/2*c)^3 + 8*A*b*tan(1/2*d*
x + 1/2*c)^3 + A*a*tan(1/2*d*x + 1/2*c) + 2*B*a*tan(1/2*d*x + 1/2*c) - 8*A*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d
*x + 1/2*c)^2 - 1)^2*a^5))/d