3.328 \(\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=407 \[ \frac{\left (-11 a^2 A b^3+6 a^4 A b+21 a^3 b^2 B-12 a^5 B-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{2 b^4 d \left (a^2-b^2\right )^2}-\frac{\left (-12 a^2 B+6 a A b-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{a^2 \left (-15 a^2 A b^3+6 a^4 A b+29 a^3 b^2 B-12 a^5 B-20 a b^4 B+12 A b^5\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-4 a^3 B+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\left (3 a^3 A b+10 a^2 b^2 B-6 a^4 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2} \]

[Out]

-((6*a*A*b - 12*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) + (a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 -
12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*
(a + b)^(5/2)*d) + ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(
2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Sec[c + d*x]*Tan[c + d*x])/
(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^
2) + (a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b
*Sec[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.95939, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {4029, 4098, 4092, 4082, 3998, 3770, 3831, 2659, 208} \[ \frac{\left (-11 a^2 A b^3+6 a^4 A b+21 a^3 b^2 B-12 a^5 B-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{2 b^4 d \left (a^2-b^2\right )^2}-\frac{\left (-12 a^2 B+6 a A b-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{a^2 \left (-15 a^2 A b^3+6 a^4 A b+29 a^3 b^2 B-12 a^5 B-20 a b^4 B+12 A b^5\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-4 a^3 B+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\left (3 a^3 A b+10 a^2 b^2 B-6 a^4 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]

[Out]

-((6*a*A*b - 12*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) + (a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 -
12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*
(a + b)^(5/2)*d) + ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(
2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Sec[c + d*x]*Tan[c + d*x])/
(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^
2) + (a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b
*Sec[c + d*x]))

Rule 4029

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

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 4092

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[(C*Csc[e + f*x]*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m
+ 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2)
+ A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m
}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4082

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[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m
+ 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) + (b*B*
(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

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 ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx &=\frac{a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\int \frac{\sec ^3(c+d x) \left (3 a (A b-a B)-2 b (A b-a B) \sec (c+d x)-2 \left (a A b-2 a^2 B+b^2 B\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\int \frac{\sec ^2(c+d x) \left (-2 a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right )-b \left (a^2 A b+2 A b^3+a^3 B-4 a b^2 B\right ) \sec (c+d x)+2 \left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\int \frac{\sec (c+d x) \left (2 a \left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right )-2 b \left (a^3 A b-4 a A b^3-2 a^4 B+4 a^2 b^2 B+b^4 B\right ) \sec (c+d x)-2 \left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\int \frac{\sec (c+d x) \left (2 a b \left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right )+2 \left (a^2-b^2\right )^2 \left (6 a A b-12 a^2 B-b^2 B\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (6 a A b-12 a^2 B-b^2 B\right ) \int \sec (c+d x) \, dx}{2 b^5}+\frac{\left (a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (6 a A b-12 a^2 B-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\left (a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right )\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 b^6 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (6 a A b-12 a^2 B-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\left (a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\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^6 \left (a^2-b^2\right )^2 d}\\ &=-\frac{\left (6 a A b-12 a^2 B-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac{\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [A]  time = 2.9472, size = 507, normalized size = 1.25 \[ \frac{\frac{16 a^2 \left (15 a^2 A b^3-6 a^4 A b-29 a^3 b^2 B+12 a^5 B+20 a b^4 B-12 A b^5\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-8 \left (12 a^2 B-6 a A b+b^2 B\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+8 \left (12 a^2 B-6 a A b+b^2 B\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{2 b \tan (c+d x) \sec (c+d x) \left (-2 a b \left (16 a^2 A b^3-9 a^4 A b-32 a^3 b^2 B+18 a^5 B+11 a b^4 B-4 A b^5\right ) \cos (2 (c+d x))+\left (-25 a^4 A b^3-10 a^2 A b^5+18 a^6 A b+47 a^5 b^2 B+14 a^3 b^4 B-36 a^7 B-16 a b^6 B+8 A b^7\right ) \cos (c+d x)-11 a^4 A b^3 \cos (3 (c+d x))+2 a^2 A b^5 \cos (3 (c+d x))+18 a^5 A b^2-32 a^3 A b^4+6 a^6 A b \cos (3 (c+d x))+21 a^5 b^2 B \cos (3 (c+d x))-6 a^3 b^4 B \cos (3 (c+d x))+68 a^4 b^3 B-30 a^2 b^5 B-36 a^6 b B-12 a^7 B \cos (3 (c+d x))+8 a A b^6+4 b^7 B\right )}{\left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}}{16 b^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]

[Out]

((16*a^2*(-6*a^4*A*b + 15*a^2*A*b^3 - 12*A*b^5 + 12*a^5*B - 29*a^3*b^2*B + 20*a*b^4*B)*ArcTanh[((-a + b)*Tan[(
c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 8*(-6*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] - Sin[
(c + d*x)/2]] + 8*(-6*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*b*(18*a^5*A*b^2
- 32*a^3*A*b^4 + 8*a*A*b^6 - 36*a^6*b*B + 68*a^4*b^3*B - 30*a^2*b^5*B + 4*b^7*B + (18*a^6*A*b - 25*a^4*A*b^3 -
10*a^2*A*b^5 + 8*A*b^7 - 36*a^7*B + 47*a^5*b^2*B + 14*a^3*b^4*B - 16*a*b^6*B)*Cos[c + d*x] - 2*a*b*(-9*a^4*A*
b + 16*a^2*A*b^3 - 4*A*b^5 + 18*a^5*B - 32*a^3*b^2*B + 11*a*b^4*B)*Cos[2*(c + d*x)] + 6*a^6*A*b*Cos[3*(c + d*x
)] - 11*a^4*A*b^3*Cos[3*(c + d*x)] + 2*a^2*A*b^5*Cos[3*(c + d*x)] - 12*a^7*B*Cos[3*(c + d*x)] + 21*a^5*b^2*B*C
os[3*(c + d*x)] - 6*a^3*b^4*B*Cos[3*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])
^2))/(16*b^5*d)

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Maple [B]  time = 0.106, size = 1599, normalized size = 3.9 \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)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x)

[Out]

6/d*a^6/b^4/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-15
/d*a^4/b^2/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-12/
d*a^7/b^5/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B+29/d
*a^5/b^3/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B-20/d*
a^3/b/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B+10/d*a^4
/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-1/d*a^5/b^3/(tan
(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+1/d*a^4/b^2/(tan(1/2*d*x+
1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-8/d*a^3/b/(tan(1/2*d*x+1/2*c)^2*a-
tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-4/d*a^5/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d
*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+4/d*a^5/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*
d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-1/d*a^5/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c
)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-10/d*a^4/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2
*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+1/d*a^4/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/
2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+8/d*a^3/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2
*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+6/d*a^6/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/
2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-6/d*a^6/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1
/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+1/2/d/b^3/(tan(1/2*d*x+1/2*c)-1)*B-1/2/d*B/b^3/(tan(1/2*d*
x+1/2*c)+1)^2-1/d/b^3/(tan(1/2*d*x+1/2*c)+1)*A+1/2/d/b^3/(tan(1/2*d*x+1/2*c)+1)*B-1/2/d/b^3*ln(tan(1/2*d*x+1/2
*c)-1)*B+1/2/d/b^3*ln(tan(1/2*d*x+1/2*c)+1)*B+1/2/d*B/b^3/(tan(1/2*d*x+1/2*c)-1)^2-1/d/b^3/(tan(1/2*d*x+1/2*c)
-1)*A+3/d/b^4*ln(tan(1/2*d*x+1/2*c)-1)*A*a-6/d/b^5*ln(tan(1/2*d*x+1/2*c)-1)*B*a^2+3/d/b^4/(tan(1/2*d*x+1/2*c)+
1)*B*a-3/d/b^4*ln(tan(1/2*d*x+1/2*c)+1)*A*a+6/d/b^5*ln(tan(1/2*d*x+1/2*c)+1)*B*a^2+3/d/b^4/(tan(1/2*d*x+1/2*c)
-1)*B*a+12/d*a^2/(a^4-2*a^2*b^2+b^4)/((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)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 171.058, size = 5414, normalized size = 13.3 \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)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/4*(((12*B*a^9 - 6*A*a^8*b - 29*B*a^7*b^2 + 15*A*a^6*b^3 + 20*B*a^5*b^4 - 12*A*a^4*b^5)*cos(d*x + c)^4 + 2*
(12*B*a^8*b - 6*A*a^7*b^2 - 29*B*a^6*b^3 + 15*A*a^5*b^4 + 20*B*a^4*b^5 - 12*A*a^3*b^6)*cos(d*x + c)^3 + (12*B*
a^7*b^2 - 6*A*a^6*b^3 - 29*B*a^5*b^4 + 15*A*a^4*b^5 + 20*B*a^3*b^6 - 12*A*a^2*b^7)*cos(d*x + c)^2)*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)) - ((12*B*a^10 - 6*A*a^9*b - 35*B*a^8*b^2 +
18*A*a^7*b^3 + 33*B*a^6*b^4 - 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4 + 2*(12*B*
a^9*b - 6*A*a^8*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*a^2*b^8 -
B*a*b^9)*cos(d*x + c)^3 + (12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 - 18*A*a^3*
b^7 - 9*B*a^2*b^8 + 6*A*a*b^9 - B*b^10)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + ((12*B*a^10 - 6*A*a^9*b - 35*B
*a^8*b^2 + 18*A*a^7*b^3 + 33*B*a^6*b^4 - 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4
+ 2*(12*B*a^9*b - 6*A*a^8*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*
a^2*b^8 - B*a*b^9)*cos(d*x + c)^3 + (12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 -
18*A*a^3*b^7 - 9*B*a^2*b^8 + 6*A*a*b^9 - B*b^10)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(B*a^6*b^4 - 3*B*
a^4*b^6 + 3*B*a^2*b^8 - B*b^10 - (12*B*a^9*b - 6*A*a^8*b^2 - 33*B*a^7*b^3 + 17*A*a^6*b^4 + 27*B*a^5*b^5 - 13*A
*a^4*b^6 - 6*B*a^3*b^7 + 2*A*a^2*b^8)*cos(d*x + c)^3 - (18*B*a^8*b^2 - 9*A*a^7*b^3 - 50*B*a^6*b^4 + 25*A*a^5*b
^5 + 43*B*a^4*b^6 - 20*A*a^3*b^7 - 11*B*a^2*b^8 + 4*A*a*b^9)*cos(d*x + c)^2 - 2*(2*B*a^7*b^3 - A*a^6*b^4 - 6*B
*a^5*b^5 + 3*A*a^4*b^6 + 6*B*a^3*b^7 - 3*A*a^2*b^8 - 2*B*a*b^9 + A*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^5
- 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*d*cos(d*x + c)^4 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*
x + c)^3 + (a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2), -1/4*(2*((12*B*a^9 - 6*A*a^8*b - 29*B*
a^7*b^2 + 15*A*a^6*b^3 + 20*B*a^5*b^4 - 12*A*a^4*b^5)*cos(d*x + c)^4 + 2*(12*B*a^8*b - 6*A*a^7*b^2 - 29*B*a^6*
b^3 + 15*A*a^5*b^4 + 20*B*a^4*b^5 - 12*A*a^3*b^6)*cos(d*x + c)^3 + (12*B*a^7*b^2 - 6*A*a^6*b^3 - 29*B*a^5*b^4
+ 15*A*a^4*b^5 + 20*B*a^3*b^6 - 12*A*a^2*b^7)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos
(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - ((12*B*a^10 - 6*A*a^9*b - 35*B*a^8*b^2 + 18*A*a^7*b^3 + 33*B*a^6*
b^4 - 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4 + 2*(12*B*a^9*b - 6*A*a^8*b^2 - 35*
B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*a^2*b^8 - B*a*b^9)*cos(d*x + c)^3 +
(12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 - 18*A*a^3*b^7 - 9*B*a^2*b^8 + 6*A*a
*b^9 - B*b^10)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + ((12*B*a^10 - 6*A*a^9*b - 35*B*a^8*b^2 + 18*A*a^7*b^3 +
33*B*a^6*b^4 - 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4 + 2*(12*B*a^9*b - 6*A*a^8
*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*a^2*b^8 - B*a*b^9)*cos(d*
x + c)^3 + (12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 - 18*A*a^3*b^7 - 9*B*a^2*b
^8 + 6*A*a*b^9 - B*b^10)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(B*a^6*b^4 - 3*B*a^4*b^6 + 3*B*a^2*b^8 - B
*b^10 - (12*B*a^9*b - 6*A*a^8*b^2 - 33*B*a^7*b^3 + 17*A*a^6*b^4 + 27*B*a^5*b^5 - 13*A*a^4*b^6 - 6*B*a^3*b^7 +
2*A*a^2*b^8)*cos(d*x + c)^3 - (18*B*a^8*b^2 - 9*A*a^7*b^3 - 50*B*a^6*b^4 + 25*A*a^5*b^5 + 43*B*a^4*b^6 - 20*A*
a^3*b^7 - 11*B*a^2*b^8 + 4*A*a*b^9)*cos(d*x + c)^2 - 2*(2*B*a^7*b^3 - A*a^6*b^4 - 6*B*a^5*b^5 + 3*A*a^4*b^6 +
6*B*a^3*b^7 - 3*A*a^2*b^8 - 2*B*a*b^9 + A*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9
- a^2*b^11)*d*cos(d*x + c)^4 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*x + c)^3 + (a^6*b^7 - 3*a
^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2)]

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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 )}\right ) \sec ^{5}{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**3,x)

[Out]

Integral((A + B*sec(c + d*x))*sec(c + d*x)**5/(a + b*sec(c + d*x))**3, x)

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Giac [B]  time = 1.70943, size = 1878, normalized size = 4.61 \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)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(2*(12*B*a^7 - 6*A*a^6*b - 29*B*a^5*b^2 + 15*A*a^4*b^3 + 20*B*a^3*b^4 - 12*A*a^2*b^5)*(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^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(-a^2 + b^2)) - 2*(12*B*a^7*tan(1/2*d*x + 1/2*c)^7 - 6*A*a^6*b*tan(1/2*d*x +
1/2*c)^7 - 18*B*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 17*B*a^5*b^2*tan(1/2*d*x +
1/2*c)^7 + 9*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 16*A*a^3*b^4*tan(1/2*d*
x + 1/2*c)^7 - 2*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 2*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 13*B*a^2*b^5*tan(1/2*
d*x + 1/2*c)^7 + 4*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 2*A*b^7*tan(1/2*d*x + 1
/2*c)^7 + B*b^7*tan(1/2*d*x + 1/2*c)^7 - 36*B*a^7*tan(1/2*d*x + 1/2*c)^5 + 18*A*a^6*b*tan(1/2*d*x + 1/2*c)^5 +
18*B*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 67*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^5
- 35*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 29*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 16*A*a^3*b^4*tan(1/2*d*x + 1/2*c
)^5 - 26*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 10*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 5*B*a^2*b^5*tan(1/2*d*x + 1/
2*c)^5 - 4*A*a*b^6*tan(1/2*d*x + 1/2*c)^5 + 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 2*A*b^7*tan(1/2*d*x + 1/2*c)^5
+ 3*B*b^7*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^7*tan(1/2*d*x + 1/2*c)^3 - 18*A*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 18*B*
a^6*b*tan(1/2*d*x + 1/2*c)^3 - 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 67*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 35*A
*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 16*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 +
26*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 - 10*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^3
- 4*A*a*b^6*tan(1/2*d*x + 1/2*c)^3 - 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^7*tan(1/2*d*x + 1/2*c)^3 + 3*B*
b^7*tan(1/2*d*x + 1/2*c)^3 - 12*B*a^7*tan(1/2*d*x + 1/2*c) + 6*A*a^6*b*tan(1/2*d*x + 1/2*c) - 18*B*a^6*b*tan(1
/2*d*x + 1/2*c) + 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c) + 17*B*a^5*b^2*tan(1/2*d*x + 1/2*c) - 9*A*a^4*b^3*tan(1/2*d
*x + 1/2*c) + 33*B*a^4*b^3*tan(1/2*d*x + 1/2*c) - 16*A*a^3*b^4*tan(1/2*d*x + 1/2*c) + 2*B*a^3*b^4*tan(1/2*d*x
+ 1/2*c) - 2*A*a^2*b^5*tan(1/2*d*x + 1/2*c) - 13*B*a^2*b^5*tan(1/2*d*x + 1/2*c) + 4*A*a*b^6*tan(1/2*d*x + 1/2*
c) - 4*B*a*b^6*tan(1/2*d*x + 1/2*c) + 2*A*b^7*tan(1/2*d*x + 1/2*c) + B*b^7*tan(1/2*d*x + 1/2*c))/((a^4*b^4 - 2
*a^2*b^6 + b^8)*(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b)^2)
- (12*B*a^2 - 6*A*a*b + B*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^5 + (12*B*a^2 - 6*A*a*b + B*b^2)*log(abs(t
an(1/2*d*x + 1/2*c) - 1))/b^5)/d