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

Optimal. Leaf size=336 \[ \frac {A x}{a^4}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}-\frac {\left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))} \]

[Out]

A*x/a^4-(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))*arctanh((a-b)^(1/2)*tan(1/2*
d*x+1/2*c)/(a+b)^(1/2))/a^4/(a-b)^(7/2)/(a+b)^(7/2)/d+1/3*(A*b^2-a*(B*b-C*a))*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*se
c(d*x+c))^3-1/6*(3*A*b^4+5*a^3*b*B-2*a^4*C-a^2*b^2*(8*A+3*C))*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2-
1/6*(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))*tan(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b
*sec(d*x+c))

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Rubi [A]  time = 2.14, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4060, 3919, 3831, 2659, 208} \[ -\frac {\left (-a^4 b^3 (8 A-C)+7 a^2 A b^5+4 a^6 b (2 A+C)-3 a^5 b^2 B-2 a^7 B-2 A b^7\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac {\tan (c+d x) \left (-13 a^4 b^2 (2 A+C)+17 a^2 A b^4+4 a^3 b^3 B+11 a^5 b B-2 a^6 C-6 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {\tan (c+d x) \left (-a^2 b^2 (8 A+3 C)+5 a^3 b B-2 a^4 C+3 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {A x}{a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*x)/a^4 - ((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))*ArcTanh[(
Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan
[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[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))
*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[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))*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 208

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3831

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

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4060

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(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))/(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)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {-3 A \left (a^2-b^2\right )+3 a (A b-a B+b C) \sec (c+d x)-2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (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 ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {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 ) \sec (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 ) \sec ^2(c+d x)}{(a+b \sec (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 ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (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 ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (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 ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {-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 ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {A x}{a^4}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (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 ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (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 ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (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 ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {A x}{a^4}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (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 ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (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 ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (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 ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^4 b \left (a^2-b^2\right )^3}\\ &=\frac {A x}{a^4}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (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 ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (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 ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (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}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^4 b \left (a^2-b^2\right )^3 d}\\ &=\frac {A x}{a^4}-\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 ) \tanh ^{-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 {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (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 ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (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 ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 7.87, size = 1230, normalized size = 3.66 \[ \frac {2 A x \sec ^2(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^4}{a^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac {\left (2 B a^7-8 A b a^6-4 b C a^6+3 b^2 B a^5+8 A b^3 a^4-b^3 C a^4-7 A b^5 a^2+2 A b^7\right ) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac {2 i \tan ^{-1}\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 a \sin \left (c+\frac {d x}{2}\right )-i b \sin \left (\frac {d x}{2}\right )\right )\right ) \cos (c)}{a^4 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}+\frac {2 \tan ^{-1}\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 a \sin \left (c+\frac {d x}{2}\right )-i b \sin \left (\frac {d x}{2}\right )\right )\right ) \sin (c)}{a^4 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) (b+a \cos (c+d x))^4}{\left (b^2-a^2\right )^3 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac {\sec (c) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (6 C \sin (d x) a^7-12 b C \sin (c) a^6-18 b B \sin (d x) a^6+27 b^2 B \sin (c) a^5+36 A b^2 \sin (d x) a^5+10 b^2 C \sin (d x) a^5-48 A b^3 \sin (c) a^4-3 b^3 C \sin (c) a^4+5 b^3 B \sin (d x) a^4-18 b^4 B \sin (c) a^3-32 A b^4 \sin (d x) a^3-b^4 C \sin (d x) a^3+51 A b^5 \sin (c) a^2-2 b^5 B \sin (d x) a^2+6 b^6 B \sin (c) a+11 A b^6 \sin (d x) a-18 A b^7 \sin (c)\right ) (b+a \cos (c+d x))^3}{3 a^4 \left (a^2-b^2\right )^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac {\sec (c) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-9 A \sin (c) b^6+6 a B \sin (c) b^5+7 a A \sin (d x) b^5+14 a^2 A \sin (c) b^4-3 a^2 C \sin (c) b^4-4 a^2 B \sin (d x) b^4-11 a^3 B \sin (c) b^3-12 a^3 A \sin (d x) b^3+a^3 C \sin (d x) b^3+8 a^4 C \sin (c) b^2+9 a^4 B \sin (d x) b^2-6 a^5 C \sin (d x) b\right ) (b+a \cos (c+d x))^2}{3 a^4 \left (a^2-b^2\right )^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}-\frac {2 \sec (c) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (A \sin (c) b^5-a B \sin (c) b^4-a A \sin (d x) b^4+a^2 C \sin (c) b^3+a^2 B \sin (d x) b^3-a^3 C \sin (d x) b^2\right ) (b+a \cos (c+d x))}{3 a^4 \left (a^2-b^2\right ) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*A*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a^4*(A + 2*C + 2*B*Cos[
c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-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)*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*S
ec[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^4*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^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + 2*B*Cos[c + d
*x] + 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 + B*Sec[c
+ d*x] + C*Sec[c + d*x]^2)*(A*b^5*Sin[c] - a*b^4*B*Sin[c] + a^2*b^3*C*Sin[c] - a*A*b^4*Sin[d*x] + a^2*b^3*B*S
in[d*x] - a^3*b^2*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*S
ec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(14*a^
2*A*b^4*Sin[c] - 9*A*b^6*Sin[c] - 11*a^3*b^3*B*Sin[c] + 6*a*b^5*B*Sin[c] + 8*a^4*b^2*C*Sin[c] - 3*a^2*b^4*C*Si
n[c] - 12*a^3*A*b^3*Sin[d*x] + 7*a*A*b^5*Sin[d*x] + 9*a^4*b^2*B*Sin[d*x] - 4*a^2*b^4*B*Sin[d*x] - 6*a^5*b*C*Si
n[d*x] + a^3*b^3*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + 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 + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-48*
a^4*A*b^3*Sin[c] + 51*a^2*A*b^5*Sin[c] - 18*A*b^7*Sin[c] + 27*a^5*b^2*B*Sin[c] - 18*a^3*b^4*B*Sin[c] + 6*a*b^6
*B*Sin[c] - 12*a^6*b*C*Sin[c] - 3*a^4*b^3*C*Sin[c] + 36*a^5*A*b^2*Sin[d*x] - 32*a^3*A*b^4*Sin[d*x] + 11*a*A*b^
6*Sin[d*x] - 18*a^6*b*B*Sin[d*x] + 5*a^4*b^3*B*Sin[d*x] - 2*a^2*b^5*B*Sin[d*x] + 6*a^7*C*Sin[d*x] + 10*a^5*b^2
*C*Sin[d*x] - a^3*b^4*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a
+ b*Sec[c + d*x])^4)

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fricas [B]  time = 0.72, size = 2056, normalized size = 6.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(12*(A*a^11 - 4*A*a^9*b^2 + 6*A*a^7*b^4 - 4*A*a^5*b^6 + A*a^3*b^8)*d*x*cos(d*x + c)^3 + 36*(A*a^10*b - 4
*A*a^8*b^3 + 6*A*a^6*b^5 - 4*A*a^4*b^7 + A*a^2*b^9)*d*x*cos(d*x + c)^2 + 36*(A*a^9*b^2 - 4*A*a^7*b^4 + 6*A*a^5
*b^6 - 4*A*a^3*b^8 + A*a*b^10)*d*x*cos(d*x + c) + 12*(A*a^8*b^3 - 4*A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*
b^11)*d*x - 3*(2*B*a^7*b^3 - 4*(2*A + C)*a^6*b^4 + 3*B*a^5*b^5 + (8*A - C)*a^4*b^6 - 7*A*a^2*b^8 + 2*A*b^10 +
(2*B*a^10 - 4*(2*A + C)*a^9*b + 3*B*a^8*b^2 + (8*A - C)*a^7*b^3 - 7*A*a^5*b^5 + 2*A*a^3*b^7)*cos(d*x + c)^3 +
3*(2*B*a^9*b - 4*(2*A + C)*a^8*b^2 + 3*B*a^7*b^3 + (8*A - C)*a^6*b^4 - 7*A*a^4*b^6 + 2*A*a^2*b^8)*cos(d*x + c)
^2 + 3*(2*B*a^8*b^2 - 4*(2*A + C)*a^7*b^3 + 3*B*a^6*b^4 + (8*A - C)*a^5*b^5 - 7*A*a^3*b^7 + 2*A*a*b^9)*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*(2*C*a^9*b^2 - 11*B*
a^8*b^3 + (26*A + 11*C)*a^7*b^4 + 7*B*a^6*b^5 - (43*A + 13*C)*a^5*b^6 + 4*B*a^4*b^7 + 23*A*a^3*b^8 - 6*A*a*b^1
0 + (6*C*a^11 - 18*B*a^10*b + 4*(9*A + C)*a^9*b^2 + 23*B*a^8*b^3 - (68*A + 11*C)*a^7*b^4 - 7*B*a^6*b^5 + (43*A
+ C)*a^5*b^6 + 2*B*a^4*b^7 - 11*A*a^3*b^8)*cos(d*x + c)^2 + 3*(2*C*a^10*b - 9*B*a^9*b^2 + (20*A + 7*C)*a^8*b^
3 + 8*B*a^7*b^4 - 5*(7*A + 2*C)*a^6*b^5 + B*a^5*b^6 + (20*A + C)*a^4*b^7 - 5*A*a^2*b^9)*cos(d*x + c))*sin(d*x
+ c))/((a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^
10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 + 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*
d*cos(d*x + c) + (a^12*b^3 - 4*a^10*b^5 + 6*a^8*b^7 - 4*a^6*b^9 + a^4*b^11)*d), 1/6*(6*(A*a^11 - 4*A*a^9*b^2 +
6*A*a^7*b^4 - 4*A*a^5*b^6 + A*a^3*b^8)*d*x*cos(d*x + c)^3 + 18*(A*a^10*b - 4*A*a^8*b^3 + 6*A*a^6*b^5 - 4*A*a^
4*b^7 + A*a^2*b^9)*d*x*cos(d*x + c)^2 + 18*(A*a^9*b^2 - 4*A*a^7*b^4 + 6*A*a^5*b^6 - 4*A*a^3*b^8 + A*a*b^10)*d*
x*cos(d*x + c) + 6*(A*a^8*b^3 - 4*A*a^6*b^5 + 6*A*a^4*b^7 - 4*A*a^2*b^9 + A*b^11)*d*x + 3*(2*B*a^7*b^3 - 4*(2*
A + C)*a^6*b^4 + 3*B*a^5*b^5 + (8*A - C)*a^4*b^6 - 7*A*a^2*b^8 + 2*A*b^10 + (2*B*a^10 - 4*(2*A + C)*a^9*b + 3*
B*a^8*b^2 + (8*A - C)*a^7*b^3 - 7*A*a^5*b^5 + 2*A*a^3*b^7)*cos(d*x + c)^3 + 3*(2*B*a^9*b - 4*(2*A + C)*a^8*b^2
+ 3*B*a^7*b^3 + (8*A - C)*a^6*b^4 - 7*A*a^4*b^6 + 2*A*a^2*b^8)*cos(d*x + c)^2 + 3*(2*B*a^8*b^2 - 4*(2*A + C)*
a^7*b^3 + 3*B*a^6*b^4 + (8*A - C)*a^5*b^5 - 7*A*a^3*b^7 + 2*A*a*b^9)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sq
rt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (2*C*a^9*b^2 - 11*B*a^8*b^3 + (26*A + 11*C)*
a^7*b^4 + 7*B*a^6*b^5 - (43*A + 13*C)*a^5*b^6 + 4*B*a^4*b^7 + 23*A*a^3*b^8 - 6*A*a*b^10 + (6*C*a^11 - 18*B*a^1
0*b + 4*(9*A + C)*a^9*b^2 + 23*B*a^8*b^3 - (68*A + 11*C)*a^7*b^4 - 7*B*a^6*b^5 + (43*A + C)*a^5*b^6 + 2*B*a^4*
b^7 - 11*A*a^3*b^8)*cos(d*x + c)^2 + 3*(2*C*a^10*b - 9*B*a^9*b^2 + (20*A + 7*C)*a^8*b^3 + 8*B*a^7*b^4 - 5*(7*A
+ 2*C)*a^6*b^5 + B*a^5*b^6 + (20*A + C)*a^4*b^7 - 5*A*a^2*b^9)*cos(d*x + c))*sin(d*x + c))/((a^15 - 4*a^13*b^
2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6
*b^9)*d*cos(d*x + c)^2 + 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) + (a^12*b
^3 - 4*a^10*b^5 + 6*a^8*b^7 - 4*a^6*b^9 + a^4*b^11)*d)]

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giac [B]  time = 0.42, size = 1106, normalized size = 3.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(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*(d*x + c)*A/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*ta
n(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*t
an(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

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maple [B]  time = 0.91, size = 3223, normalized size = 9.59 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+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 details)Is 4*a^2-4*b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 18.65, size = 11934, normalized size = 35.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*A*atan(((A*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*
A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*
A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*
a^12*b^2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2*a^12*b^2 - 32*A*B*a^13*b - 16*B*C*a^13*b + 12*A*B*a^5*b^9 - 3
4*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3 - 4*A*C*a^4*b^10 - 2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 - 48*A*C*a^
10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*C*a^11*b^3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9
+ 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) + (A*((8*(4*A*a
^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*A*a^10*b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^13*b^8
+ 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^5 + 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*
b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^6 + 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2
- 2*C*a^11*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 + 2*C*a^14*b^7 + 18*C*a^15*b^6 - 18*C*a^16*b^5 - 22*C*a^17*b^4
+ 22*C*a^18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b - 8*C*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10
+ 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2)
- (A*tan(c/2 + (d*x)/2)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10 + 12
0*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^
2)*8i)/(a^4*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12
*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*1i)/a^4))/a^4 + (A*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*
b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A
^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*
a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2*a^12*b^2
- 32*A*B*a^13*b - 16*B*C*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3 - 4*A*C*
a^4*b^10 - 2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 - 48*A*C*a^10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*C*a^11*b^
3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 1
0*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) - (A*((8*(4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*A*a^10*
b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^5 + 20*A
*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^6 + 6*B*
a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 2*C*a^11*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 + 2*C*a^14
*b^7 + 18*C*a^15*b^6 - 18*C*a^16*b^5 - 22*C*a^17*b^4 + 22*C*a^18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b
- 8*C*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 +
10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) + (A*tan(c/2 + (d*x)/2)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^
13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 +
120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^2)*8i)/(a^4*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*
b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*1i)/a^4))
/a^4)/((16*(4*A^3*b^13 + 4*A*B^2*a^13 - 4*A^2*B*a^13 - 2*A^3*a*b^12 + 16*A^3*a^12*b - 26*A^3*a^2*b^11 + 11*A^3
*a^3*b^10 + 70*A^3*a^4*b^9 - 34*A^3*a^5*b^8 - 110*A^3*a^6*b^7 + 66*A^3*a^7*b^6 + 110*A^3*a^8*b^5 - 64*A^3*a^9*
b^4 - 64*A^3*a^10*b^3 + 48*A^3*a^11*b^2 - 28*A^2*B*a^12*b + 8*A^2*C*a^12*b + 9*A*B^2*a^9*b^4 + 12*A*B^2*a^11*b
^2 + 6*A^2*B*a^4*b^9 + 6*A^2*B*a^5*b^8 - 20*A^2*B*a^6*b^7 - 14*A^2*B*a^7*b^6 + 14*A^2*B*a^8*b^5 + 6*A^2*B*a^9*
b^4 - 22*A^2*B*a^10*b^3 + 6*A^2*B*a^11*b^2 + A*C^2*a^7*b^6 + 8*A*C^2*a^9*b^4 + 16*A*C^2*a^11*b^2 - 2*A^2*C*a^3
*b^10 - 2*A^2*C*a^4*b^9 - 2*A^2*C*a^6*b^7 + 22*A^2*C*a^7*b^6 + 18*A^2*C*a^8*b^5 - 26*A^2*C*a^9*b^4 - 22*A^2*C*
a^10*b^3 + 56*A^2*C*a^11*b^2 - 16*A*B*C*a^12*b - 6*A*B*C*a^8*b^5 - 28*A*B*C*a^10*b^3))/(a^19*b + a^20 - a^9*b^
11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3
- 5*a^18*b^2) - (A*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b
- 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 +
156*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12
*B^2*a^12*b^2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2*a^12*b^2 - 32*A*B*a^13*b - 16*B*C*a^13*b + 12*A*B*a^5*b^
9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3 - 4*A*C*a^4*b^10 - 2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 - 48*A
*C*a^10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*C*a^11*b^3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8
*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) + (A*((8*(
4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*A*a^10*b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^1
3*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^5 + 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*
a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^6 + 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^1
9*b^2 - 2*C*a^11*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 + 2*C*a^14*b^7 + 18*C*a^15*b^6 - 18*C*a^16*b^5 - 22*C*a^17
*b^4 + 22*C*a^18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b - 8*C*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^10
*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*
b^2) - (A*tan(c/2 + (d*x)/2)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10
+ 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^
20*b^2)*8i)/(a^4*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10
*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*1i)/a^4)*1i)/a^4 + (A*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14
+ 8*A^2*b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10
- 120*A^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 +
48*A^2*a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2*
a^12*b^2 - 32*A*B*a^13*b - 16*B*C*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3
- 4*A*C*a^4*b^10 - 2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 - 48*A*C*a^10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*C
*a^11*b^3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12
*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) - (A*((8*(4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26
*A*a^10*b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^
5 + 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^
6 + 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 2*C*a^11*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 +
2*C*a^14*b^7 + 18*C*a^15*b^6 - 18*C*a^16*b^5 - 22*C*a^17*b^4 + 22*C*a^18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4*
B*a^20*b - 8*C*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^1
4*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) + (A*tan(c/2 + (d*x)/2)*(8*a^21*b - 8*a^8*b^14 +
8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^
16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^2)*8i)/(a^4*(a^16*b + a^17 - a^6*b^11 - a^7*b^10
+ 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*1
i)/a^4)*1i)/a^4)))/(a^4*d) - ((tan(c/2 + (d*x)/2)*(2*A*b^6 + 2*C*a^6 - 6*A*a^2*b^4 - 4*A*a^3*b^3 + 12*A*a^4*b^
2 - 2*B*a^3*b^3 + 3*B*a^4*b^2 - C*a^3*b^3 + 6*C*a^4*b^2 + A*a*b^5 - 6*B*a^5*b - 2*C*a^5*b))/((a + b)*(3*a^5*b
- a^6 + a^3*b^3 - 3*a^4*b^2)) + (4*tan(c/2 + (d*x)/2)^3*(3*A*b^6 + 3*C*a^6 - 11*A*a^2*b^4 + 18*A*a^4*b^2 - B*a
^3*b^3 + 7*C*a^4*b^2 - 9*B*a^5*b))/(3*(a + b)^2*(a^5 - 2*a^4*b + a^3*b^2)) + (tan(c/2 + (d*x)/2)^5*(2*A*b^6 +
2*C*a^6 - 6*A*a^2*b^4 + 4*A*a^3*b^3 + 12*A*a^4*b^2 - 2*B*a^3*b^3 - 3*B*a^4*b^2 + C*a^3*b^3 + 6*C*a^4*b^2 - A*a
*b^5 - 6*B*a^5*b + 2*C*a^5*b))/((a^3*b - a^4)*(a + b)^3))/(d*(tan(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3
+ 3*b^3) - tan(c/2 + (d*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 - tan(c/2
+ (d*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) + (atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B
^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^5*b^9
- 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 +
44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2*a^12*b^2 - 32*A*B*
a^13*b - 16*B*C*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3 - 4*A*C*a^4*b^10 -
2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 - 48*A*C*a^10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*C*a^11*b^3))/(a^16*
b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4
- 5*a^14*b^3 - 5*a^15*b^2) + (((8*(4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*A*a^10*b^11 - 14*A*
a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^5 + 20*A*a^17*b^4 +
64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^6 + 6*B*a^16*b^5 - 6
*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 2*C*a^11*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 + 2*C*a^14*b^7 + 18*C*
a^15*b^6 - 18*C*a^16*b^5 - 22*C*a^17*b^4 + 22*C*a^18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b - 8*C*a^20*
b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5
+ 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) - (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^
7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b)*(8*a^21*b - 8*a^8*b^14 + 8*a^
9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b
^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^2))/((a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35
*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b
^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*((a + b)^7*(a - b)^7)^
(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b))/(2*(a
^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)))*((a + b)^7
*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a
^6*b)*1i)/(2*(a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^
2)) + (((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A^2*a^2
*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A^2*a^8
*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^
2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2*a^12*b^2 - 32*A*B*a^13*b - 16*B*C*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a
^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3 - 4*A*C*a^4*b^10 - 2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 - 48*A*C*a^10*b^4
+ 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*C*a^11*b^3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9
*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) - (((8*(4*A*a^21 + 4*B
*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*A*a^10*b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*
a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^5 + 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b^9 - 6*B
*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^6 + 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 2*C*a^
11*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 + 2*C*a^14*b^7 + 18*C*a^15*b^6 - 18*C*a^16*b^5 - 22*C*a^17*b^4 + 22*C*a^
18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b - 8*C*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11
*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) + (4*tan(
c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^
4*b^3 - 8*A*a^6*b - 4*C*a^6*b)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^
10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*
a^20*b^2))/((a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2
)*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*
a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b
^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b))/(2*(a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^1
0*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5
+ 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b)*1i)/(2*(a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^
8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)))/((16*(4*A^3*b^13 + 4*A*B^2*a^13 - 4*A^2*B*a^1
3 - 2*A^3*a*b^12 + 16*A^3*a^12*b - 26*A^3*a^2*b^11 + 11*A^3*a^3*b^10 + 70*A^3*a^4*b^9 - 34*A^3*a^5*b^8 - 110*A
^3*a^6*b^7 + 66*A^3*a^7*b^6 + 110*A^3*a^8*b^5 - 64*A^3*a^9*b^4 - 64*A^3*a^10*b^3 + 48*A^3*a^11*b^2 - 28*A^2*B*
a^12*b + 8*A^2*C*a^12*b + 9*A*B^2*a^9*b^4 + 12*A*B^2*a^11*b^2 + 6*A^2*B*a^4*b^9 + 6*A^2*B*a^5*b^8 - 20*A^2*B*a
^6*b^7 - 14*A^2*B*a^7*b^6 + 14*A^2*B*a^8*b^5 + 6*A^2*B*a^9*b^4 - 22*A^2*B*a^10*b^3 + 6*A^2*B*a^11*b^2 + A*C^2*
a^7*b^6 + 8*A*C^2*a^9*b^4 + 16*A*C^2*a^11*b^2 - 2*A^2*C*a^3*b^10 - 2*A^2*C*a^4*b^9 - 2*A^2*C*a^6*b^7 + 22*A^2*
C*a^7*b^6 + 18*A^2*C*a^8*b^5 - 26*A^2*C*a^9*b^4 - 22*A^2*C*a^10*b^3 + 56*A^2*C*a^11*b^2 - 16*A*B*C*a^12*b - 6*
A*B*C*a^8*b^5 - 28*A*B*C*a^10*b^3))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*
b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) - (((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14
+ 8*A^2*b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^1
0 - 120*A^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 156*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4
+ 48*A^2*a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2
*a^12*b^2 - 32*A*B*a^13*b - 16*B*C*a^13*b + 12*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3
- 4*A*C*a^4*b^10 - 2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 - 48*A*C*a^10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*
C*a^11*b^3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^1
2*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) + (((8*(4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*
A*a^10*b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^5
+ 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^6
+ 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 2*C*a^11*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 + 2
*C*a^14*b^7 + 18*C*a^15*b^6 - 18*C*a^16*b^5 - 22*C*a^17*b^4 + 22*C*a^18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4*B
*a^20*b - 8*C*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14
*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) - (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2
)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b)*(8*a^21*b
- 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^
15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^2))/((a^18 - a^4*b^14 + 7*a^6*b^12
- 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 +
5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*((a
+ b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b
- 4*C*a^6*b))/(2*(a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^1
6*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3
- 8*A*a^6*b - 4*C*a^6*b))/(2*(a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14
*b^4 - 7*a^16*b^2)) + (((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B^2*a^14 - 8*A^2*a*b^13 - 8*A^2*a^1
3*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 117*A^2*a^4*b^10 - 120*A^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b
^7 + 156*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 + 44*A^2*a^12*b^2 + 9*B^2*a^10*b^4
+ 12*B^2*a^12*b^2 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 + 16*C^2*a^12*b^2 - 32*A*B*a^13*b - 16*B*C*a^13*b + 12*A*B*a^
5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3 - 4*A*C*a^4*b^10 - 2*A*C*a^6*b^8 + 40*A*C*a^8*b^6 -
48*A*C*a^10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 28*B*C*a^11*b^3))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5
*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) - (((8
*(4*A*a^21 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*A*a^10*b^11 - 14*A*a^11*b^10 - 70*A*a^12*b^9 + 30*A*a
^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110*A*a^16*b^5 + 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*
B*a^12*b^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15*b^6 + 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a
^19*b^2 - 2*C*a^11*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 + 2*C*a^14*b^7 + 18*C*a^15*b^6 - 18*C*a^16*b^5 - 22*C*a^
17*b^4 + 22*C*a^18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4*B*a^20*b - 8*C*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^
10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^1
8*b^2) + (4*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*
B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^
11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 -
48*a^19*b^3 - 8*a^20*b^2))/((a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*
b^4 - 7*a^16*b^2)*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 1
0*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a^7 - 7*A*a^2
*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b))/(2*(a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a
^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(2*A*b^7 + 2*B*a
^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b))/(2*(a^18 - a^4*b^14 + 7*a^6
*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2))))*((a + b)^7*(a - b)^7)^(1/2)*(2*
A*b^7 + 2*B*a^7 - 7*A*a^2*b^5 + 8*A*a^4*b^3 + 3*B*a^5*b^2 - C*a^4*b^3 - 8*A*a^6*b - 4*C*a^6*b)*1i)/(d*(a^18 -
a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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