3.10.24 \(\int \frac {\sec ^3(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [924]

Optimal. Leaf size=358 \[ \frac {C \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {\left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]

[Out]

C*arctanh(sin(d*x+c))/b^4/d-(3*a^2*b^5*B+2*b^7*B-a^3*b^4*(A-8*C)+2*a^7*C-7*a^5*b^2*C-4*a*b^6*(A+2*C))*arctanh(
(a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^4/(a+b)^(7/2)/d-1/3*(A*b^2-a*(B*b-C*a))*sec(d*x+c)^2
*tan(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^3-1/6*a*(2*A*b^4-5*a*b^3*B-3*a^4*C+a^2*b^2*(3*A+8*C))*tan(d*x+c)/b^
3/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2-1/6*(4*A*b^6+a^3*b^3*B-16*a*b^5*B+9*a^6*C+2*a^2*b^4*(7*A+17*C)-a^4*b^2*(3*A
+28*C))*tan(d*x+c)/b^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))

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Rubi [A]
time = 1.74, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4183, 4175, 4165, 4083, 3855, 3916, 2738, 214} \begin {gather*} -\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{6 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}-\frac {\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac {C \tanh ^{-1}(\sin (c+d x))}{b^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]

[Out]

(C*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*
b^6*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*
b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a*(2*A*b^4 - 5
*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((4*A
*b^6 + a^3*b^3*B - 16*a*b^5*B + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Tan[c + d*x])/(6*b^3*
(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 214

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

Rule 2738

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

Rule 3855

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

Rule 3916

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

Rule 4083

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x
_Symbol] :> Dist[B/b, Int[Csc[e + f*x], x], x] + Dist[(A*b - a*B)/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]
, x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]

Rule 4165

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

Rule 4175

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e
+ f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*C
sc[e + f*x])^(m + 1)*Simp[b*(m + 1)*((-a)*(b*B - a*C) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) +
C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^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]

Rule 4183

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

Rubi steps

\begin {align*} \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\sec ^2(c+d x) \left (2 \left (A b^2-a (b B-a C)\right )+3 b (b B-a (A+C)) \sec (c+d x)-3 \left (a^2-b^2\right ) C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\int \frac {\sec (c+d x) \left (-2 b \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right )+\left (a^2 b^3 B-6 b^5 B+3 a^5 C+4 a b^4 (2 A+3 C)-a^3 b^2 (3 A+10 C)\right ) \sec (c+d x)-6 b \left (a^2-b^2\right )^2 C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-3 \left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-2 C)-a^5 b^2 C-2 a b^6 (2 A+3 C)\right )+6 b \left (a^2-b^2\right )^3 C \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {C \int \sec (c+d x) \, dx}{b^4}-\frac {\left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 C)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^3}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^5 \left (a^2-b^2\right )^3}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 C)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right )^3 d}\\ &=\frac {C \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac {\left (a^3 A b^4+4 a A b^6-3 a^2 b^5 B-2 b^7 B-2 a^7 C+7 a^5 b^2 C-8 a^3 b^4 C+8 a b^6 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.43, size = 1302, normalized size = 3.64 \begin {gather*} -\frac {2 C (b+a \cos (c+d x))^4 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{b^4 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {2 C (b+a \cos (c+d x))^4 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{b^4 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {\left (-a^3 A b^4-4 a A b^6+3 a^2 b^5 B+2 b^7 B+2 a^7 C-7 a^5 b^2 C+8 a^3 b^4 C-8 a b^6 C\right ) (b+a \cos (c+d x))^4 \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {2 i \text {ArcTan}\left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \cos (c)}{b^4 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {2 \text {ArcTan}\left (\sec \left (\frac {d x}{2}\right ) \left (\frac {\cos (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}-\frac {i \sin (c)}{\sqrt {a^2-b^2} \sqrt {\cos (2 c)-i \sin (2 c)}}\right ) \left (-i b \sin \left (\frac {d x}{2}\right )+i a \sin \left (c+\frac {d x}{2}\right )\right )\right ) \sin (c)}{b^4 \sqrt {a^2-b^2} d \sqrt {\cos (2 c)-i \sin (2 c)}}\right )}{\left (-a^2+b^2\right )^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}-\frac {2 (b+a \cos (c+d x)) \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (A b^3 \sin (c)-a b^2 B \sin (c)+a^2 b C \sin (c)-a A b^2 \sin (d x)+a^2 b B \sin (d x)-a^3 C \sin (d x)\right )}{3 a b \left (-a^2+b^2\right ) d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}+\frac {(b+a \cos (c+d x))^2 \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-5 a A b^3 \sin (c)+2 a^2 b^2 B \sin (c)+3 b^4 B \sin (c)+a^3 b C \sin (c)-6 a b^3 C \sin (c)+3 a^2 A b^2 \sin (d x)+2 A b^4 \sin (d x)-5 a b^3 B \sin (d x)-3 a^4 C \sin (d x)+8 a^2 b^2 C \sin (d x)\right )}{3 b^2 \left (-a^2+b^2\right )^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}+\frac {(b+a \cos (c+d x))^3 \sec (c) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-3 a^3 A b^3 \sin (c)-12 a A b^5 \sin (c)+9 a^2 b^4 B \sin (c)+6 b^6 B \sin (c)-3 a^5 b C \sin (c)+6 a^3 b^3 C \sin (c)-18 a b^5 C \sin (c)+13 a^2 A b^4 \sin (d x)+2 A b^6 \sin (d x)-4 a^3 b^3 B \sin (d x)-11 a b^5 B \sin (d x)+6 a^6 C \sin (d x)-17 a^4 b^2 C \sin (d x)+26 a^2 b^4 C \sin (d x)\right )}{3 b^3 \left (-a^2+b^2\right )^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} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]

[Out]

(-2*C*(b + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] +
C*Sec[c + d*x]^2))/(b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*C*(b
+ a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c
+ d*x]^2))/(b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-(a^3*A*b^4)
- 4*a*A*b^6 + 3*a^2*b^5*B + 2*b^7*B + 2*a^7*C - 7*a^5*b^2*C + 8*a^3*b^4*C - 8*a*b^6*C)*(b + a*Cos[c + d*x])^4*
Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*S
qrt[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])/(b^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])/(b^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^3*Sin[c] - a*b^2*B*Sin[c] + a^2*b*C*
Sin[c] - a*A*b^2*Sin[d*x] + a^2*b*B*Sin[d*x] - a^3*C*Sin[d*x]))/(3*a*b*(-a^2 + b^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])^2*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c
+ d*x] + C*Sec[c + d*x]^2)*(-5*a*A*b^3*Sin[c] + 2*a^2*b^2*B*Sin[c] + 3*b^4*B*Sin[c] + a^3*b*C*Sin[c] - 6*a*b^
3*C*Sin[c] + 3*a^2*A*b^2*Sin[d*x] + 2*A*b^4*Sin[d*x] - 5*a*b^3*B*Sin[d*x] - 3*a^4*C*Sin[d*x] + 8*a^2*b^2*C*Sin
[d*x]))/(3*b^2*(-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)*(-3*a^3*A*b^3*Sin[c] - 12*
a*A*b^5*Sin[c] + 9*a^2*b^4*B*Sin[c] + 6*b^6*B*Sin[c] - 3*a^5*b*C*Sin[c] + 6*a^3*b^3*C*Sin[c] - 18*a*b^5*C*Sin[
c] + 13*a^2*A*b^4*Sin[d*x] + 2*A*b^6*Sin[d*x] - 4*a^3*b^3*B*Sin[d*x] - 11*a*b^5*B*Sin[d*x] + 6*a^6*C*Sin[d*x]
- 17*a^4*b^2*C*Sin[d*x] + 26*a^2*b^4*C*Sin[d*x]))/(3*b^3*(-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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Maple [A]
time = 1.31, size = 584, normalized size = 1.63

method	result	size
derivativedivides	\(\frac {-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {\left (A \,a^{3} b^{3}+6 a^{2} A \,b^{4}+2 A a \,b^{5}+2 A \,b^{6}-2 a^{3} b^{3} B -3 B \,a^{2} b^{4}-6 a \,b^{5} B +2 a^{6} C -C \,a^{5} b -6 a^{4} b^{2} C +4 C \,a^{3} b^{3}+12 C \,a^{2} b^{4}\right ) b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (7 a^{2} A \,b^{4}+3 A \,b^{6}-a^{3} b^{3} B -9 a \,b^{5} B +3 a^{6} C -11 a^{4} b^{2} C +18 C \,a^{2} b^{4}\right ) b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+2 A a \,b^{5}-2 A \,b^{6}+2 a^{3} b^{3} B -3 B \,a^{2} b^{4}+6 a \,b^{5} B -2 a^{6} C -C \,a^{5} b +6 a^{4} b^{2} C +4 C \,a^{3} b^{3}-12 C \,a^{2} b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (A \,a^{3} b^{4}+4 A a \,b^{6}-3 a^{2} b^{5} B -2 b^{7} B -2 a^{7} C +7 a^{5} b^{2} C -8 C \,a^{3} b^{4}+8 C a \,b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}}{d}\)	\(584\)
default	\(\frac {-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {\left (A \,a^{3} b^{3}+6 a^{2} A \,b^{4}+2 A a \,b^{5}+2 A \,b^{6}-2 a^{3} b^{3} B -3 B \,a^{2} b^{4}-6 a \,b^{5} B +2 a^{6} C -C \,a^{5} b -6 a^{4} b^{2} C +4 C \,a^{3} b^{3}+12 C \,a^{2} b^{4}\right ) b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (7 a^{2} A \,b^{4}+3 A \,b^{6}-a^{3} b^{3} B -9 a \,b^{5} B +3 a^{6} C -11 a^{4} b^{2} C +18 C \,a^{2} b^{4}\right ) b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+2 A a \,b^{5}-2 A \,b^{6}+2 a^{3} b^{3} B -3 B \,a^{2} b^{4}+6 a \,b^{5} B -2 a^{6} C -C \,a^{5} b +6 a^{4} b^{2} C +4 C \,a^{3} b^{3}-12 C \,a^{2} b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (A \,a^{3} b^{4}+4 A a \,b^{6}-3 a^{2} b^{5} B -2 b^{7} B -2 a^{7} C +7 a^{5} b^{2} C -8 C \,a^{3} b^{4}+8 C a \,b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}}{d}\)	\(584\)
risch	\(\text {Expression too large to display}\)	\(2271\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-C/b^4*ln(tan(1/2*d*x+1/2*c)-1)+C/b^4*ln(tan(1/2*d*x+1/2*c)+1)-2/b^4*((-1/2*(A*a^3*b^3+6*A*a^2*b^4+2*A*a*
b^5+2*A*b^6-2*B*a^3*b^3-3*B*a^2*b^4-6*B*a*b^5+2*C*a^6-C*a^5*b-6*C*a^4*b^2+4*C*a^3*b^3+12*C*a^2*b^4)*b/(a-b)/(a
^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5+2/3*(7*A*a^2*b^4+3*A*b^6-B*a^3*b^3-9*B*a*b^5+3*C*a^6-11*C*a^4*b^2
+18*C*a^2*b^4)*b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+1/2*(A*a^3*b^3-6*A*a^2*b^4+2*A*a*b^5-2*A
*b^6+2*B*a^3*b^3-3*B*a^2*b^4+6*B*a*b^5-2*C*a^6-C*a^5*b+6*C*a^4*b^2+4*C*a^3*b^3-12*C*a^2*b^4)*b/(a+b)/(a^3-3*a^
2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(a*tan(1/2*d*x+1/2*c)^2-b*tan(1/2*d*x+1/2*c)^2-a-b)^3-1/2*(A*a^3*b^4+4*A*
a*b^6-3*B*a^2*b^5-2*B*b^7-2*C*a^7+7*C*a^5*b^2-8*C*a^3*b^4+8*C*a*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b
))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (343) = 686\).
time = 135.84, size = 2466, normalized size = 6.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(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*(3*(2*C*a^7*b^3 - 7*C*a^5*b^5 - (A - 8*C)*a^3*b^7 + 3*B*a^2*b^8 - 4*(A + 2*C)*a*b^9 + 2*B*b^10 + (2*C*a
^10 - 7*C*a^8*b^2 - (A - 8*C)*a^6*b^4 + 3*B*a^5*b^5 - 4*(A + 2*C)*a^4*b^6 + 2*B*a^3*b^7)*cos(d*x + c)^3 + 3*(2
*C*a^9*b - 7*C*a^7*b^3 - (A - 8*C)*a^5*b^5 + 3*B*a^4*b^6 - 4*(A + 2*C)*a^3*b^7 + 2*B*a^2*b^8)*cos(d*x + c)^2 +
3*(2*C*a^8*b^2 - 7*C*a^6*b^4 - (A - 8*C)*a^4*b^6 + 3*B*a^3*b^7 - 4*(A + 2*C)*a^2*b^8 + 2*B*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)) - 6*(C*a^8*b^3 - 4*C*a^6*b^5
+ 6*C*a^4*b^7 - 4*C*a^2*b^9 + C*b^11 + (C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4*C*a^5*b^6 + C*a^3*b^8)*cos(d*x
+ c)^3 + 3*(C*a^10*b - 4*C*a^8*b^3 + 6*C*a^6*b^5 - 4*C*a^4*b^7 + C*a^2*b^9)*cos(d*x + c)^2 + 3*(C*a^9*b^2 - 4
*C*a^7*b^4 + 6*C*a^5*b^6 - 4*C*a^3*b^8 + C*a*b^10)*cos(d*x + c))*log(sin(d*x + c) + 1) + 6*(C*a^8*b^3 - 4*C*a^
6*b^5 + 6*C*a^4*b^7 - 4*C*a^2*b^9 + C*b^11 + (C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4*C*a^5*b^6 + C*a^3*b^8)*co
s(d*x + c)^3 + 3*(C*a^10*b - 4*C*a^8*b^3 + 6*C*a^6*b^5 - 4*C*a^4*b^7 + C*a^2*b^9)*cos(d*x + c)^2 + 3*(C*a^9*b^
2 - 4*C*a^7*b^4 + 6*C*a^5*b^6 - 4*C*a^3*b^8 + C*a*b^10)*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(11*C*a^8*b^3
- 2*B*a^7*b^4 - (A + 43*C)*a^6*b^5 + 7*B*a^5*b^6 + (11*A + 68*C)*a^4*b^7 - 23*B*a^3*b^8 - 4*(A + 9*C)*a^2*b^9
+ 18*B*a*b^10 - 6*A*b^11 + (6*C*a^10*b - 23*C*a^8*b^3 - 4*B*a^7*b^4 + (13*A + 43*C)*a^6*b^5 - 7*B*a^5*b^6 - (
11*A + 26*C)*a^4*b^7 + 11*B*a^3*b^8 - 2*A*a^2*b^9)*cos(d*x + c)^2 + 3*(5*C*a^9*b^2 - (A + 20*C)*a^7*b^4 - B*a^
6*b^5 + 5*(2*A + 7*C)*a^5*b^6 - 8*B*a^4*b^7 - (7*A + 20*C)*a^3*b^8 + 9*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x + c))*s
in(d*x + c))/((a^11*b^4 - 4*a^9*b^6 + 6*a^7*b^8 - 4*a^5*b^10 + a^3*b^12)*d*cos(d*x + c)^3 + 3*(a^10*b^5 - 4*a^
8*b^7 + 6*a^6*b^9 - 4*a^4*b^11 + a^2*b^13)*d*cos(d*x + c)^2 + 3*(a^9*b^6 - 4*a^7*b^8 + 6*a^5*b^10 - 4*a^3*b^12
+ a*b^14)*d*cos(d*x + c) + (a^8*b^7 - 4*a^6*b^9 + 6*a^4*b^11 - 4*a^2*b^13 + b^15)*d), -1/6*(3*(2*C*a^7*b^3 -
7*C*a^5*b^5 - (A - 8*C)*a^3*b^7 + 3*B*a^2*b^8 - 4*(A + 2*C)*a*b^9 + 2*B*b^10 + (2*C*a^10 - 7*C*a^8*b^2 - (A -
8*C)*a^6*b^4 + 3*B*a^5*b^5 - 4*(A + 2*C)*a^4*b^6 + 2*B*a^3*b^7)*cos(d*x + c)^3 + 3*(2*C*a^9*b - 7*C*a^7*b^3 -
(A - 8*C)*a^5*b^5 + 3*B*a^4*b^6 - 4*(A + 2*C)*a^3*b^7 + 2*B*a^2*b^8)*cos(d*x + c)^2 + 3*(2*C*a^8*b^2 - 7*C*a^6
*b^4 - (A - 8*C)*a^4*b^6 + 3*B*a^3*b^7 - 4*(A + 2*C)*a^2*b^8 + 2*B*a*b^9)*cos(d*x + c))*sqrt(-a^2 + b^2)*arcta
n(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - 3*(C*a^8*b^3 - 4*C*a^6*b^5 + 6*C*a^4*b^
7 - 4*C*a^2*b^9 + C*b^11 + (C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4*C*a^5*b^6 + C*a^3*b^8)*cos(d*x + c)^3 + 3*(
C*a^10*b - 4*C*a^8*b^3 + 6*C*a^6*b^5 - 4*C*a^4*b^7 + C*a^2*b^9)*cos(d*x + c)^2 + 3*(C*a^9*b^2 - 4*C*a^7*b^4 +
6*C*a^5*b^6 - 4*C*a^3*b^8 + C*a*b^10)*cos(d*x + c))*log(sin(d*x + c) + 1) + 3*(C*a^8*b^3 - 4*C*a^6*b^5 + 6*C*a
^4*b^7 - 4*C*a^2*b^9 + C*b^11 + (C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4*C*a^5*b^6 + C*a^3*b^8)*cos(d*x + c)^3
+ 3*(C*a^10*b - 4*C*a^8*b^3 + 6*C*a^6*b^5 - 4*C*a^4*b^7 + C*a^2*b^9)*cos(d*x + c)^2 + 3*(C*a^9*b^2 - 4*C*a^7*b
^4 + 6*C*a^5*b^6 - 4*C*a^3*b^8 + C*a*b^10)*cos(d*x + c))*log(-sin(d*x + c) + 1) + (11*C*a^8*b^3 - 2*B*a^7*b^4
- (A + 43*C)*a^6*b^5 + 7*B*a^5*b^6 + (11*A + 68*C)*a^4*b^7 - 23*B*a^3*b^8 - 4*(A + 9*C)*a^2*b^9 + 18*B*a*b^10
- 6*A*b^11 + (6*C*a^10*b - 23*C*a^8*b^3 - 4*B*a^7*b^4 + (13*A + 43*C)*a^6*b^5 - 7*B*a^5*b^6 - (11*A + 26*C)*a^
4*b^7 + 11*B*a^3*b^8 - 2*A*a^2*b^9)*cos(d*x + c)^2 + 3*(5*C*a^9*b^2 - (A + 20*C)*a^7*b^4 - B*a^6*b^5 + 5*(2*A
+ 7*C)*a^5*b^6 - 8*B*a^4*b^7 - (7*A + 20*C)*a^3*b^8 + 9*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x + c))*sin(d*x + c))/((
a^11*b^4 - 4*a^9*b^6 + 6*a^7*b^8 - 4*a^5*b^10 + a^3*b^12)*d*cos(d*x + c)^3 + 3*(a^10*b^5 - 4*a^8*b^7 + 6*a^6*b
^9 - 4*a^4*b^11 + a^2*b^13)*d*cos(d*x + c)^2 + 3*(a^9*b^6 - 4*a^7*b^8 + 6*a^5*b^10 - 4*a^3*b^12 + a*b^14)*d*co
s(d*x + c) + (a^8*b^7 - 4*a^6*b^9 + 6*a^4*b^11 - 4*a^2*b^13 + b^15)*d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (343) = 686\).
time = 0.57, size = 1135, normalized size = 3.17 \begin {gather*} -\frac {\frac {3 \, {\left (2 \, C a^{7} - 7 \, C a^{5} b^{2} - A a^{3} b^{4} + 8 \, C a^{3} b^{4} + 3 \, B a^{2} b^{5} - 4 \, A a b^{6} - 8 \, C a b^{6} + 2 \, B b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {3 \, C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} + \frac {3 \, C \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac {6 \, C a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, C a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, A a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, C a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, B a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, A a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, C a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27 \, B a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, C a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 18 \, B a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 56 \, C a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, B a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 28 \, A a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 116 \, C a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, B a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, A a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, C a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, C a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, C a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, C a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, A a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 27 \, B a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, C a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 18 \, B a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{3}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(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*C*a^7 - 7*C*a^5*b^2 - A*a^3*b^4 + 8*C*a^3*b^4 + 3*B*a^2*b^5 - 4*A*a*b^6 - 8*C*a*b^6 + 2*B*b^7)*(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^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*sqrt(-a^2 + b^2)) - 3*C*log(abs(tan(1/2*d*x + 1/2*c)
+ 1))/b^4 + 3*C*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^4 - (6*C*a^8*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^7*b*tan(1/2*
d*x + 1/2*c)^5 - 6*C*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 3*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 + 45*C*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 12*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 - 6*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 27*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^3*b^5*ta
n(1/2*d*x + 1/2*c)^5 - 60*C*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 12*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 27*B*a^2*b^
6*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*A*a*b^7*tan(1/2*d*x + 1/2*c)^5 - 18*B*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 + 56*C*a^6*b^2*tan(
1/2*d*x + 1/2*c)^3 + 4*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 - 28*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 - 116*C*a^4*b^4*
tan(1/2*d*x + 1/2*c)^3 + 32*B*a^3*b^5*tan(1/2*d*x + 1/2*c)^3 + 16*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 + 72*C*a^2*
b^6*tan(1/2*d*x + 1/2*c)^3 - 36*B*a*b^7*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) + 15*C*a^7*b*tan(1/2*d*x + 1/2*c) - 6*C*a^6*b^2*tan(1/2*d*x + 1/2*c) - 3*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) - 45*C*a^5*b^3*tan(1/2*d*x + 1/2*c) + 12*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) - 6*C*a^4*b^4*tan(1/2*d*x + 1/2*c) + 27*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) + 60*C*a^3*b^5*tan(1/2*d*x + 1/2*c) + 12*A*a^2*b^6*tan(1/2*d*x + 1/2*c
) - 27*B*a^2*b^6*tan(1/2*d*x + 1/2*c) + 36*C*a^2*b^6*tan(1/2*d*x + 1/2*c) + 6*A*a*b^7*tan(1/2*d*x + 1/2*c) - 1
8*B*a*b^7*tan(1/2*d*x + 1/2*c) + 6*A*b^8*tan(1/2*d*x + 1/2*c))/((a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*(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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Mupad [B]
time = 19.09, size = 2500, normalized size = 6.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/(cos(c + d*x)^3*(a + b/cos(c + d*x))^4),x)

[Out]

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

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