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

Optimal. Leaf size=313 \[ \frac{a \left (a^2 b^4 (A-8 C)+7 a^4 b^2 C-2 a^6 C+4 b^6 (A+2 C)\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac{\left (-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+9 a^6 C+4 A b^6\right ) \tan (c+d x)}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{a \left (a^2 b^2 (3 A+8 C)-3 a^4 C+2 A b^4\right ) \tan (c+d x)}{6 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^4 d} \]

[Out]

(C*ArcTanh[Sin[c + d*x]])/(b^4*d) + (a*(a^2*b^4*(A - 8*C) - 2*a^6*C + 7*a^4*b^2*C + 4*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^2*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 - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Ta
n[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((4*A*b^6 + 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 = 1.25142, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4099, 4090, 4080, 3998, 3770, 3831, 2659, 208} \[ \frac{a \left (a^2 b^4 (A-8 C)+7 a^4 b^2 C-2 a^6 C+4 b^6 (A+2 C)\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac{\left (-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+9 a^6 C+4 A b^6\right ) \tan (c+d x)}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{a \left (a^2 b^2 (3 A+8 C)-3 a^4 C+2 A b^4\right ) \tan (c+d x)}{6 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(C*ArcTanh[Sin[c + d*x]])/(b^4*d) + (a*(a^2*b^4*(A - 8*C) - 2*a^6*C + 7*a^4*b^2*C + 4*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^2*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 - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Ta
n[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((4*A*b^6 + 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 4099

Int[((A_.) + 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^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^2*C*(n - 1) + a*b*(A + C)*(m + 1)*Csc[e + f*x] - (A*b^2*(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, 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+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=-\frac{\left (A b^2+a^2 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^2 C\right )-3 a b (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^2 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^4-3 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^4-3 a^4 C+a^2 b^2 (3 A+8 C)\right )+a \left (3 a^4 C+4 b^4 (2 A+3 C)-a^2 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^2 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^4-3 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+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 a b^2 \left (a^2 b^2 (A-2 C)+a^4 C+2 b^4 (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^2 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^4-3 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+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 (a \left (a^2 b^4 (A-8 C)-2 a^6 C+7 a^4 b^2 C+4 b^6 (A+2 C)\right )\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^2 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^4-3 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+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 (a \left (a^2 b^4 (A-8 C)-2 a^6 C+7 a^4 b^2 C+4 b^6 (A+2 C)\right )\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^2 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^4-3 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+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 (a \left (a^2 b^4 (A-8 C)-2 a^6 C+7 a^4 b^2 C+4 b^6 (A+2 C)\right )\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{a \left (a^2 A b^4+4 A b^6-2 a^6 C+7 a^4 b^2 C-8 a^2 b^4 C+8 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^2 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^4-3 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+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 = 7.16969, size = 1092, normalized size = 3.49 \[ -\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)+A\right ) (b+a \cos (c+d x))^4}{b^4 d (\cos (2 c+2 d x) A+A+2 C) (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)+A\right ) (b+a \cos (c+d x))^4}{b^4 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\left (-2 C a^6+7 b^2 C a^4+A b^4 a^2-8 b^4 C a^2+4 A b^6+8 b^6 C\right ) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (\frac{2 i a \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 a \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) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left (C \sec ^2(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+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+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) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (-3 C \sin (d x) a^4+b C \sin (c) a^3+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+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) (a+b \sec (c+d x))^4}-\frac{2 \sec (c) \sec ^2(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (-C \sin (d x) a^3+b C \sin (c) a^2-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) (a+b \sec (c+d x))^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sec[c + d*x]^3*(A + 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 + C*Sec[c + d*x]^2
))/(b^4*d*(A + 2*C + 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 + C*Sec[c + d*x]^2))/(b^4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a
+ b*Sec[c + d*x])^4) + ((a^2*A*b^4 + 4*A*b^6 - 2*a^6*C + 7*a^4*b^2*C - 8*a^2*b^4*C + 8*b^6*C)*(b + a*Cos[c + d
*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(((2*I)*a*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])/(b^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) + (2*a*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 + 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 + C*Sec[c + d*x]^2)*(A*b^3*Sin[c] + a^2*b*C*Sin[c] - a*A*b^2*Sin[d*x] - a^3*C*Sin[d*x]))/(3*a*b*(-a^2 +
b^2)*d*(A + 2*C + 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 + C*Sec[c + d*x]^2)*(-5*a*A*b^3*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] - 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 + 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 + C*Sec[c + d*x]^2)*(-3*a^3*A*b^3*Si
n[c] - 12*a*A*b^5*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] + 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 + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)

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Maple [B]  time = 0.113, size = 2428, normalized size = 7.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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+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+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^2-1/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/(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+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+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^2-28/3/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^2-4/d/b^3/(tan(1/2*d*x+1/2*c)^2*a-ta
n(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-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+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+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-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-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)*t
an(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+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+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)^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)*C-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/(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+4/d*b^2*a/(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/d/b^2*a^5/(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+8/d*b^2*a/(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-2/d/b^4*a^7/(
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+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-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)-8/d*a^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))*C+1/d*a^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))*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+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 [B]  time = 51.2194, size = 4779, normalized size = 15.27 \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+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 - 4*(A + 2*C)*a*b^9 + (2*C*a^10 - 7*C*a^8*b^2 - (A - 8
*C)*a^6*b^4 - 4*(A + 2*C)*a^4*b^6)*cos(d*x + c)^3 + 3*(2*C*a^9*b - 7*C*a^7*b^3 - (A - 8*C)*a^5*b^5 - 4*(A + 2*
C)*a^3*b^7)*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 - 4*(A + 2*C)*a^2*b^8)*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)*
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) - 2*(11*C*a^8*b
^3 - (A + 43*C)*a^6*b^5 + (11*A + 68*C)*a^4*b^7 - 4*(A + 9*C)*a^2*b^9 - 6*A*b^11 + (6*C*a^10*b - 23*C*a^8*b^3
+ (13*A + 43*C)*a^6*b^5 - (11*A + 26*C)*a^4*b^7 - 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 + 5*(2*A + 7*C)*a^5*b^6 - (7*A + 20*C)*a^3*b^8 - 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*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 - 4*(A + 2*C)*a*b^9 + (2*C*a^10 - 7*C*a^8*b^2 - (A - 8*C)*a^6*b^4 - 4*(A + 2*C)*a^4*b^6)*cos(d*x + c)^3 +
3*(2*C*a^9*b - 7*C*a^7*b^3 - (A - 8*C)*a^5*b^5 - 4*(A + 2*C)*a^3*b^7)*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 - 4*(A + 2*C)*a^2*b^8)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*co
s(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 - (A + 43*C)*a^6*b^5 + (11*A + 68*C)*a^
4*b^7 - 4*(A + 9*C)*a^2*b^9 - 6*A*b^11 + (6*C*a^10*b - 23*C*a^8*b^3 + (13*A + 43*C)*a^6*b^5 - (11*A + 26*C)*a^
4*b^7 - 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 + 5*(2*A + 7*C)*a^5*b^6 - (7*A + 20*
C)*a^3*b^8 - 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^1
2)*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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + 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+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**4,x)

[Out]

Integral((A + 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 = 1.38448, size = 1183, normalized size = 3.78 \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+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 - 4*A*a*b^6 - 8*C*a*b^6)*(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 + 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 - 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 - 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 + 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 + 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 - 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 + 1
6*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 + 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) - 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) - 6*C*a^4*b^4*ta
n(1/2*d*x + 1/2*c) + 27*A*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) + 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) + 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