3.279 \(\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=420 \[ \frac{b \left (-35 a^4 A b^3+28 a^2 A b^5+20 a^6 A b+8 a^5 b^2 B-7 a^3 b^4 B-8 a^7 B+2 a b^6 B-8 A b^7\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left (-65 a^4 A b^2+68 a^2 A b^4+6 a^6 A-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}+\frac{b \left (-11 a^2 A b^3+12 a^4 A b+2 a^3 b^2 B-6 a^5 B-a b^4 B+4 A b^5\right ) \tan (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{b \left (9 a^2 A b-6 a^3 B+a b^2 B-4 A b^3\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d} \]

[Out]

(b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*a^3*b^4*B + 2*a*b^6*B)*ArcT
an[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) - ((4*A*b - a*B)*ArcTanh[S
in[c + d*x]])/(a^5*d) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b
^5*B)*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d
*x])^3) + (b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x
])^2) + (b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Tan[c + d*x])/(2*a^3*(a^2 -
b^2)^3*d*(a + b*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 6.22096, antiderivative size = 420, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{b \left (-35 a^4 A b^3+28 a^2 A b^5+20 a^6 A b+8 a^5 b^2 B-7 a^3 b^4 B-8 a^7 B+2 a b^6 B-8 A b^7\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left (-65 a^4 A b^2+68 a^2 A b^4+6 a^6 A-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}+\frac{b \left (-11 a^2 A b^3+12 a^4 A b+2 a^3 b^2 B-6 a^5 B-a b^4 B+4 A b^5\right ) \tan (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{b \left (9 a^2 A b-6 a^3 B+a b^2 B-4 A b^3\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*a^3*b^4*B + 2*a*b^6*B)*ArcT
an[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) - ((4*A*b - a*B)*ArcTanh[S
in[c + d*x]])/(a^5*d) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b
^5*B)*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d
*x])^3) + (b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x
])^2) + (b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Tan[c + d*x])/(2*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 ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac{b (A b-a B) \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-4 A b^2+a b B-3 a (A b-a B) \cos (c+d x)+3 b (A b-a B) \cos ^2(c+d x)\right ) \sec ^2(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) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \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 (6 a^4 A-23 a^2 A b^2+12 A b^4+8 a^3 b B-3 a b^3 B-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)+2 b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \cos ^2(c+d x)\right ) \sec ^2(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) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \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 (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B-a \left (18 a^4 A b-7 a^2 A b^3+4 A b^5-6 a^5 B-8 a^3 b^2 B-a b^4 B\right ) \cos (c+d x)+3 b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac{\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \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 (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 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 (4 A b-a B)+3 a b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \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 (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{(4 A b-a B) \int \sec (c+d x) \, dx}{a^5}+\frac{\left (b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \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 (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 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^5 \left (a^2-b^2\right )^3 d}\\ &=\frac{b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 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^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac{b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \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 (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 3.02178, size = 549, normalized size = 1.31 \[ \frac{\frac{2 a \tan (c+d x) \left (6 a b^2 \left (-53 a^4 A b^2+57 a^2 A b^4+6 a^6 A-15 a^3 b^3 B+20 a^5 b B+5 a b^5 B-20 A b^6\right ) \cos (2 (c+d x))+b \left (-438 a^6 A b^2+305 a^4 A b^4+28 a^2 A b^6+72 a^8 A-50 a^5 b^3 B-7 a^3 b^5 B+144 a^7 b B+18 a b^7 B-72 A b^8\right ) \cos (c+d x)+6 a^6 A b^3 \cos (3 (c+d x))-65 a^4 A b^5 \cos (3 (c+d x))+68 a^2 A b^7 \cos (3 (c+d x))-36 a^7 A b^2-246 a^5 A b^4+318 a^3 A b^6+24 a^9 A+26 a^5 b^4 B \cos (3 (c+d x))-17 a^3 b^6 B \cos (3 (c+d x))+120 a^6 b^3 B-90 a^4 b^5 B+30 a^2 b^7 B-120 a A b^8+6 a b^8 B \cos (3 (c+d x))-24 A b^9 \cos (3 (c+d x))\right )}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}-\frac{48 b \left (35 a^4 A b^3-28 a^2 A b^5-20 a^6 A b-8 a^5 b^2 B+7 a^3 b^4 B+8 a^7 B-2 a b^6 B+8 A b^7\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}+48 (4 A b-a B) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+48 (a B-4 A b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{48 a^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-48*b*(-20*a^6*A*b + 35*a^4*A*b^3 - 28*a^2*A*b^5 + 8*A*b^7 + 8*a^7*B - 8*a^5*b^2*B + 7*a^3*b^4*B - 2*a*b^6*B
)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + 48*(4*A*b - a*B)*Log[Cos[(c + d*x
)/2] - Sin[(c + d*x)/2]] + 48*(-4*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(24*a^9*A - 36*a^
7*A*b^2 - 246*a^5*A*b^4 + 318*a^3*A*b^6 - 120*a*A*b^8 + 120*a^6*b^3*B - 90*a^4*b^5*B + 30*a^2*b^7*B + b*(72*a^
8*A - 438*a^6*A*b^2 + 305*a^4*A*b^4 + 28*a^2*A*b^6 - 72*A*b^8 + 144*a^7*b*B - 50*a^5*b^3*B - 7*a^3*b^5*B + 18*
a*b^7*B)*Cos[c + d*x] + 6*a*b^2*(6*a^6*A - 53*a^4*A*b^2 + 57*a^2*A*b^4 - 20*A*b^6 + 20*a^5*b*B - 15*a^3*b^3*B
+ 5*a*b^5*B)*Cos[2*(c + d*x)] + 6*a^6*A*b^3*Cos[3*(c + d*x)] - 65*a^4*A*b^5*Cos[3*(c + d*x)] + 68*a^2*A*b^7*Co
s[3*(c + d*x)] - 24*A*b^9*Cos[3*(c + d*x)] + 26*a^5*b^4*B*Cos[3*(c + d*x)] - 17*a^3*b^6*B*Cos[3*(c + d*x)] + 6
*a*b^8*B*Cos[3*(c + d*x)])*Tan[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(48*a^5*d)

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Maple [B]  time = 0.194, size = 2844, normalized size = 6.8 \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)^2/(a+b*cos(d*x+c))^4,x)

[Out]

-1/d*A/a^4/(tan(1/2*d*x+1/2*c)-1)-1/d*A/a^4/(tan(1/2*d*x+1/2*c)+1)+1/d/a^4*ln(tan(1/2*d*x+1/2*c)+1)*B+20/d*a*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))*A+12
/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*
a*b^2+24/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*a*b^2+12/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*a*b^2-5/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+18/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-2/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+2/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+18/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+5/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-8/d*a^2*b/(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-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))*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))*A-7/d*b^5
/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))*B+
2/d*b^7/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))*B-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))*A-6/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)^5*B-1/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-44/3/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^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*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*A+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*A-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*A+1/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+4/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^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3
*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)*A-6/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)*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-4/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^3-1/d/a^4*ln(tan(1/2*d*x+1/2*c)-1)*B-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*A*b^3-40/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*A*b^3+4/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^3-20/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)*A*b^3-4/d/a^5*ln(
tan(1/2*d*x+1/2*c)+1)*A*b+4/d/a^5*ln(tan(1/2*d*x+1/2*c)-1)*A*b+8/d/(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*b^3

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

[Out]

Timed out

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Giac [B]  time = 1.71497, size = 1345, normalized size = 3.2 \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)^2/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/3*(3*(8*B*a^7*b - 20*A*a^6*b^2 - 8*B*a^5*b^3 + 35*A*a^4*b^4 + 7*B*a^3*b^5 - 28*A*a^2*b^6 - 2*B*a*b^7 + 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^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(a^2 - b^2)) + (36*B*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 - 60*B*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 - 6*B*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 + 45*B*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 - 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 24*A*a^2*b^7*t
an(1/2*d*x + 1/2*c)^5 - 15*B*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 + 6*B*a*b^8*ta
n(1/2*d*x + 1/2*c)^5 - 18*A*b^9*tan(1/2*d*x + 1/2*c)^5 + 72*B*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 - 120*A*a^6*b^3*t
an(1/2*d*x + 1/2*c)^3 - 116*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 + 236*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 56*B*a^3
*b^6*tan(1/2*d*x + 1/2*c)^3 - 152*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 - 12*B*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 36*A*
b^9*tan(1/2*d*x + 1/2*c)^3 + 36*B*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) + 60*B*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) - 6*B*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) - 45*B*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) - 6*B*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) + 15*B*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) + 6*B*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) + 3*(B*a - 4*A*b)*log(abs(tan(
1/2*d*x + 1/2*c) + 1))/a^5 - 3*(B*a - 4*A*b)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^5 - 6*A*tan(1/2*d*x + 1/2*c)
/((tan(1/2*d*x + 1/2*c)^2 - 1)*a^4))/d