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

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

[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]))

________________________________________________________________________________________

Rubi [A]  time = 2.50854, antiderivative size = 358, normalized size of antiderivative = 1., 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 = {4098, 4090, 4080, 3998, 3770, 3831, 2659, 208} \[ -\frac{\left (-a^3 b^4 (A-8 C)+3 a^2 b^5 B-7 a^5 b^2 C+2 a^7 C-4 a b^6 (A+2 C)+2 b^7 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac{\tan (c+d x) \left (-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+a^3 b^3 B+9 a^6 C-16 a b^5 B+4 A b^6\right )}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{a \tan (c+d x) \left (a^2 b^2 (3 A+8 C)-3 a^4 C-5 a b^3 B+2 A b^4\right )}{6 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^4 d} \]

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 4098

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]

Rule 4090

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 4080

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 3998

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 3770

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

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 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 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]

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 ) \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 )}{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*}

Mathematica [C]  time = 7.3465, size = 1302, normalized size = 3.64 \[ -\frac{2 C \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 (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^4}{b^4 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 C \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 (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^4}{b^4 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{\left (2 C a^7-7 b^2 C a^5-A b^4 a^3+8 b^4 C a^3+3 b^5 B a^2-4 A b^6 a-8 b^6 C a+2 b^7 B\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)}{b^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)}{b^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^6-3 b C \sin (c) a^5-17 b^2 C \sin (d x) a^4-3 A b^3 \sin (c) a^3+6 b^3 C \sin (c) a^3-4 b^3 B \sin (d x) a^3+9 b^4 B \sin (c) a^2+13 A b^4 \sin (d x) a^2+26 b^4 C \sin (d x) a^2-12 A b^5 \sin (c) a-18 b^5 C \sin (c) a-11 b^5 B \sin (d x) a+6 b^6 B \sin (c)+2 A b^6 \sin (d x)\right ) (b+a \cos (c+d x))^3}{3 b^3 \left (b^2-a^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 (-3 C \sin (d x) a^4+b C \sin (c) a^3+2 b^2 B \sin (c) a^2+3 A b^2 \sin (d x) a^2+8 b^2 C \sin (d x) a^2-5 A b^3 \sin (c) a-6 b^3 C \sin (c) a-5 b^3 B \sin (d x) a+3 b^4 B \sin (c)+2 A b^4 \sin (d x)\right ) (b+a \cos (c+d x))^2}{3 b^2 \left (b^2-a^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 (-C \sin (d x) a^3+b C \sin (c) a^2+b B \sin (d x) a^2-b^2 B \sin (c) a-A b^2 \sin (d x) a+A b^3 \sin (c)\right ) (b+a \cos (c+d x))}{3 a b \left (b^2-a^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[(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)

________________________________________________________________________________________

Maple [B]  time = 0.125, size = 3244, 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(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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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]  time = 1.57917, size = 1532, normalized size = 4.28 \begin{align*} \text{result too large to display} \end{align*}

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