3.1006 \(\int \frac{(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=345 \[ -\frac{\left (-a^4 b^3 (8 A-C)+7 a^2 A b^5+4 a^6 b (2 A+C)-3 a^5 b^2 B-2 a^7 B-2 A b^7\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\sin (c+d x) \left (-13 a^4 b^2 (2 A+C)+17 a^2 A b^4+4 a^3 b^3 B+11 a^5 b B-2 a^6 C-6 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left (-a^2 b^2 (8 A+3 C)+5 a^3 b B-2 a^4 C+3 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d} \]

[Out]

-(((7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B - a^4*b^3*(8*A - C) + 4*a^6*b*(2*A + C))*ArcTan[(Sqrt[a - b]
*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((
A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((3*A*b^4 + 5*a^3*b*B - 2*a^
4*C - a^2*b^2*(8*A + 3*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A
*b^6 + 11*a^5*b*B + 4*a^3*b^3*B - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*
Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.55045, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {3055, 3001, 3770, 2659, 205} \[ -\frac{\left (-a^4 b^3 (8 A-C)+7 a^2 A b^5+4 a^6 b (2 A+C)-3 a^5 b^2 B-2 a^7 B-2 A b^7\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\sin (c+d x) \left (-13 a^4 b^2 (2 A+C)+17 a^2 A b^4+4 a^3 b^3 B+11 a^5 b B-2 a^6 C-6 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left (-a^2 b^2 (8 A+3 C)+5 a^3 b B-2 a^4 C+3 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B - a^4*b^3*(8*A - C) + 4*a^6*b*(2*A + C))*ArcTan[(Sqrt[a - b]
*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((
A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((3*A*b^4 + 5*a^3*b*B - 2*a^
4*C - a^2*b^2*(8*A + 3*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A
*b^6 + 11*a^5*b*B + 4*a^3*b^3*B - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*
Cos[c + d*x]))

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{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \sin (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 \left (a^2-b^2\right )-3 a (A b-a B+b C) \cos (c+d x)+2 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (3 A b^4+5 a^3 b B-2 a^4 C-a^2 b^2 (8 A+3 C)\right ) \sin (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 \left (a^2-b^2\right )^2+2 a \left (A b^3+3 a^3 B+2 a b^2 B-a^2 b (6 A+5 C)\right ) \cos (c+d x)-\left (3 A b^4+5 a^3 b B-2 a^4 C-a^2 b^2 (8 A+3 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (3 A b^4+5 a^3 b B-2 a^4 C-a^2 b^2 (8 A+3 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (17 a^2 A b^4-6 A b^6+11 a^5 b B+4 a^3 b^3 B-2 a^6 C-13 a^4 b^2 (2 A+C)\right ) \sin (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 A \left (a^2-b^2\right )^3-3 a \left (A b^5-2 a^5 B-3 a^3 b^2 B-a^2 b^3 (2 A-C)+2 a^4 b (3 A+2 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (3 A b^4+5 a^3 b B-2 a^4 C-a^2 b^2 (8 A+3 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (17 a^2 A b^4-6 A b^6+11 a^5 b B+4 a^3 b^3 B-2 a^6 C-13 a^4 b^2 (2 A+C)\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{A \int \sec (c+d x) \, dx}{a^4}-\frac{\left (7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B-a^4 b^3 (8 A-C)+4 a^6 b (2 A+C)\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (3 A b^4+5 a^3 b B-2 a^4 C-a^2 b^2 (8 A+3 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (17 a^2 A b^4-6 A b^6+11 a^5 b B+4 a^3 b^3 B-2 a^6 C-13 a^4 b^2 (2 A+C)\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\left (7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B-a^4 b^3 (8 A-C)+4 a^6 b (2 A+C)\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^4 \left (a^2-b^2\right )^3 d}\\ &=-\frac{\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B+4 a^6 b C+a^4 b^3 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (3 A b^4+5 a^3 b B-2 a^4 C-a^2 b^2 (8 A+3 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac{\left (17 a^2 A b^4-6 A b^6+11 a^5 b B+4 a^3 b^3 B-2 a^6 C-13 a^4 b^2 (2 A+C)\right ) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [C]  time = 6.7203, size = 587, normalized size = 1.7 \[ \frac{\cos (c+d x) (A \sec (c+d x)+B+C \cos (c+d x)) \left (-\frac{6 i (\cos (c)-i \sin (c)) \left (a^4 b^3 (8 A-C)-7 a^2 A b^5-4 a^6 b (2 A+C)+3 a^5 b^2 B+2 a^7 B+2 A b^7\right ) \tan ^{-1}\left (\frac{(\sin (c)+i \cos (c)) \left (\tan \left (\frac{d x}{2}\right ) (b \cos (c)-a)+b \sin (c)\right )}{\sqrt{-\left (a^2-b^2\right ) (\cos (c)-i \sin (c))^2}}\right )}{a^4 \left (a^2-b^2\right )^3 \sqrt{-\left (a^2-b^2\right ) (\cos (c)-i \sin (c))^2}}+\frac{2 \sec (c) \left (a (a C-b B)+A b^2\right ) (b \sin (d x)-a \sin (c))}{a b \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{\sec (c) \sin (d x) \left (13 a^4 b^2 (2 A+C)-17 a^2 A b^4-4 a^3 b^3 B-11 a^5 b B+2 a^6 C+6 A b^6\right )+3 a \tan (c) \left (a^2 b^3 (2 A-C)-2 a^4 b (3 A+2 C)+3 a^3 b^2 B+2 a^5 B-A b^5\right )}{\left (a^3-a b^2\right )^3 (a+b \cos (c+d x))}+\frac{\sec (c) \sin (d x) \left (a^2 b^2 (8 A+3 C)-5 a^3 b B+2 a^4 C-3 A b^4\right )+a \tan (c) \left (-a^2 b (6 A+5 C)+3 a^3 B+2 a b^2 B+A b^3\right )}{a^2 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{6 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{a^4}+\frac{6 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a^4}\right )}{3 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[c + d*x]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((-6*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/a^4 + (6*
A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/a^4 - ((6*I)*(-7*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B + 3*a^5*b^2*B + a^4
*b^3*(8*A - C) - 4*a^6*b*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqr
t[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(Cos[c] - I*Sin[c]))/(a^4*(a^2 - b^2)^3*Sqrt[-((a^2 - b^2)*(Cos[c] -
I*Sin[c])^2)]) + (2*(A*b^2 + a*(-(b*B) + a*C))*Sec[c]*(-(a*Sin[c]) + b*Sin[d*x]))/(a*b*(a^2 - b^2)*(a + b*Cos[
c + d*x])^3) + ((-17*a^2*A*b^4 + 6*A*b^6 - 11*a^5*b*B - 4*a^3*b^3*B + 2*a^6*C + 13*a^4*b^2*(2*A + C))*Sec[c]*S
in[d*x] + 3*a*(-(A*b^5) + 2*a^5*B + 3*a^3*b^2*B + a^2*b^3*(2*A - C) - 2*a^4*b*(3*A + 2*C))*Tan[c])/((a^3 - a*b
^2)^3*(a + b*Cos[c + d*x])) + ((-3*A*b^4 - 5*a^3*b*B + 2*a^4*C + a^2*b^2*(8*A + 3*C))*Sec[c]*Sin[d*x] + a*(A*b
^3 + 3*a^3*B + 2*a*b^2*B - a^2*b*(6*A + 5*C))*Tan[c])/(a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2)))/(3*d*(2*A +
C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))

________________________________________________________________________________________

Maple [B]  time = 0.098, size = 3121, normalized size = 9.1 \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)+C*cos(d*x+c)^2)*sec(d*x+c)/(a+b*cos(d*x+c))^4,x)

[Out]

1/d/(a*tan(1/2*d*x+1/2*c)^2-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
*C*b^3-1/d/(a*tan(1/2*d*x+1/2*c)^2-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)*C*b^3-1/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b)
)^(1/2))*C*b^3+12/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3
)*tan(1/2*d*x+1/2*c)*A+12/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a/(a-b)/(a^3+3*a^2*b+3*a
*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+1/d/a^4*A*ln(tan(1/2*d*x+1/2*c)+1)-1/d/a^4*A*ln(tan(1/2*d*x+1/2*c)-1)+6/d/(a*
tan(1/2*d*x+1/2*c)^2-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*C*a*b^
2+28/3/d/(a*tan(1/2*d*x+1/2*c)^2-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*C*a*b^2+6/d/(a*tan(1/2*d*x+1/2*c)^2-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)*C*a*b^2-7/d/a^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)
/((a+b)*(a-b))^(1/2))*A*b^5+3/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-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-6/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^2/(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/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^3/(a+b
)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+2/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^3
/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C-4/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-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+4/d*b^3/(a*tan(1/2*d*x+1/2*c)^2-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-3/d*b^2/(a*tan(1/2*d*x+1/2*c)^2-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-2/d*b/(a*tan(1/2*d*x+1/2*c)^2-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)*C*a^2+2/d*b/(a*tan(1/2*d*x+1/2*c
)^2-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*C*a^2+2/d/a^3/(a*tan(1/
2*d*x+1/2*c)^2-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^6-6/d/a/
(a*tan(1/2*d*x+1/2*c)^2-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^4
+1/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^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^5+2/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-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^6-6/d/a/(a*tan(1/2*d*x+1/2*c)^2-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^4-1/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^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^5+2/d/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(
1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a^3*B+8/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-
b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-44/3/d/a/(a*tan(1/2*d*x+1/2*c)^2-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^4+4/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-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^6+2/d/a^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+
b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A*b^7-4/3/d/(a*tan(1/2*d*x+1/2*c)^2-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^3*B-2/d/(a*tan(1/2*d*x+1/2*c)^
2-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^3*B-2/d/(a*tan(1/2*d*x+
1/2*c)^2-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*B-12/d*b/(a*tan(
1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^2/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+24/d
*b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c
)^3*A-6/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^2/(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/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan
(1/2*d*x+1/2*c)^3*C-8/d*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((
a+b)*(a-b))^(1/2))*a^2*A-4/d*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*
c)/((a+b)*(a-b))^(1/2))*C*a^2+3/d*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d
*x+1/2*c)/((a+b)*(a-b))^(1/2))*a*B

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.3989, size = 1521, normalized size = 4.41 \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)+C*cos(d*x+c)^2)*sec(d*x+c)/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/3*(3*(2*B*a^7 - 8*A*a^6*b - 4*C*a^6*b + 3*B*a^5*b^2 + 8*A*a^4*b^3 - C*a^4*b^3 - 7*A*a^2*b^5 + 2*A*b^7)*(pi*f
loor(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^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(a^2 - b^2)) + 3*A*log(abs(tan(1/2*d*x + 1/2*c) + 1))
/a^4 - 3*A*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^4 + (6*C*a^8*tan(1/2*d*x + 1/2*c)^5 - 18*B*a^7*b*tan(1/2*d*x +
1/2*c)^5 - 6*C*a^7*b*tan(1/2*d*x + 1/2*c)^5 + 36*A*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 27*B*a^6*b^2*tan(1/2*d*x
+ 1/2*c)^5 + 12*C*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 - 60*A*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^5*b^3*tan(1/2*d
*x + 1/2*c)^5 - 27*C*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 + 3*B*a^4*b^4*tan(1/2
*d*x + 1/2*c)^5 + 12*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 + 45*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^3*b^5*tan(
1/2*d*x + 1/2*c)^5 + 3*C*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 15*A*a*b^7*tan(
1/2*d*x + 1/2*c)^5 + 6*A*b^8*tan(1/2*d*x + 1/2*c)^5 + 12*C*a^8*tan(1/2*d*x + 1/2*c)^3 - 36*B*a^7*b*tan(1/2*d*x
+ 1/2*c)^3 + 72*A*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 16*C*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 + 32*B*a^5*b^3*tan(1/2
*d*x + 1/2*c)^3 - 116*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 - 28*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 4*B*a^3*b^5*tan
(1/2*d*x + 1/2*c)^3 + 56*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^8*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^8*tan(1/2*
d*x + 1/2*c) - 18*B*a^7*b*tan(1/2*d*x + 1/2*c) + 6*C*a^7*b*tan(1/2*d*x + 1/2*c) + 36*A*a^6*b^2*tan(1/2*d*x + 1
/2*c) - 27*B*a^6*b^2*tan(1/2*d*x + 1/2*c) + 12*C*a^6*b^2*tan(1/2*d*x + 1/2*c) + 60*A*a^5*b^3*tan(1/2*d*x + 1/2
*c) - 6*B*a^5*b^3*tan(1/2*d*x + 1/2*c) + 27*C*a^5*b^3*tan(1/2*d*x + 1/2*c) - 6*A*a^4*b^4*tan(1/2*d*x + 1/2*c)
- 3*B*a^4*b^4*tan(1/2*d*x + 1/2*c) + 12*C*a^4*b^4*tan(1/2*d*x + 1/2*c) - 45*A*a^3*b^5*tan(1/2*d*x + 1/2*c) - 6
*B*a^3*b^5*tan(1/2*d*x + 1/2*c) - 3*C*a^3*b^5*tan(1/2*d*x + 1/2*c) - 6*A*a^2*b^6*tan(1/2*d*x + 1/2*c) + 15*A*a
*b^7*tan(1/2*d*x + 1/2*c) + 6*A*b^8*tan(1/2*d*x + 1/2*c))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*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))/d