3.10.34 \(\int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx\) [934]

Optimal. Leaf size=336 \[ \frac {(b B-a C) x}{a^4}-\frac {b \left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]

[Out]

(B*b-C*a)*x/a^4-b*(8*B*a^6*b-8*B*a^4*b^3+7*B*a^2*b^5-2*B*b^7-10*C*a^7+5*C*a^5*b^2-7*C*a^3*b^4+2*C*a*b^6)*arcta
nh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^4/(a-b)^(7/2)/(a+b)^(7/2)/d+1/3*b^2*(B*b-2*C*a)*tan(d*x+c)/a/
(a^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*b^2*(8*B*a^2*b-3*B*b^3-13*C*a^3+3*C*a*b^2)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+
b*sec(d*x+c))^2+1/6*b^2*(26*B*a^4*b-17*B*a^2*b^3+6*B*b^5-37*C*a^5+13*C*a^3*b^2-6*C*a*b^4)*tan(d*x+c)/a^3/(a^2-
b^2)^3/d/(a+b*sec(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 3.18, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {24, 4008, 4145, 4004, 3916, 2738, 214} \begin {gather*} \frac {x (b B-a C)}{a^4}+\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {b^2 \left (-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {b^2 \left (-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {b \left (-10 a^7 C+8 a^6 b B+5 a^5 b^2 C-8 a^4 b^3 B-7 a^3 b^4 C+7 a^2 b^5 B+2 a b^6 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 )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^5,x]

[Out]

((b*B - a*C)*x)/a^4 - (b*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C - 7*a^3*b^4
*C + 2*a*b^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b^
2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(8*a^2*b*B - 3*b^3*B - 13*a^3*
C + 3*a*b^2*C)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(26*a^4*b*B - 17*a^2*b^3*B
+ 6*b^5*B - 37*a^5*C + 13*a^3*b^2*C - 6*a*b^4*C)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

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 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 4004

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

Rule 4008

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

Rule 4145

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

Rubi steps

\begin {align*} \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx &=\frac {\int \frac {b^2 (b B-a C)+b^3 C \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx}{b^2}\\ &=\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {-3 b^2 \left (a^2-b^2\right ) (b B-a C)+3 a b^3 (b B-2 a C) \sec (c+d x)-2 b^4 (b B-2 a C) \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx}{3 a b^2 \left (a^2-b^2\right )}\\ &=\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {6 b^2 \left (a^2-b^2\right )^2 (b B-a C)-2 a b^3 \left (6 a^2 b B-b^3 B-9 a^3 C-a b^2 C\right ) \sec (c+d x)+b^4 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {-6 b^2 \left (a^2-b^2\right )^3 (b B-a C)+3 a b^3 \left (6 a^4 b B-2 a^2 b^3 B+b^5 B-8 a^5 C-a^3 b^2 C-a b^4 C\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^3 b^2 \left (a^2-b^2\right )^3}\\ &=\frac {(b B-a C) x}{a^4}+\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (b \left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {(b B-a C) x}{a^4}+\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac {(b B-a C) x}{a^4}+\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 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 )}{a^4 \left (a^2-b^2\right )^3 d}\\ &=\frac {(b B-a C) x}{a^4}-\frac {b \left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 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^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1097\) vs. \(2(336)=672\).
time = 5.45, size = 1097, normalized size = 3.26 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec ^3(c+d x) (b B-a C+b C \sec (c+d x)) \left (\frac {24 b \left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac {36 a^8 b^2 B c-84 a^6 b^4 B c+36 a^4 b^6 B c+36 a^2 b^8 B c-24 b^{10} B c-36 a^9 b c C+84 a^7 b^3 c C-36 a^5 b^5 c C-36 a^3 b^7 c C+24 a b^9 c C+36 a^8 b^2 B d x-84 a^6 b^4 B d x+36 a^4 b^6 B d x+36 a^2 b^8 B d x-24 b^{10} B d x-36 a^9 b C d x+84 a^7 b^3 C d x-36 a^5 b^5 C d x-36 a^3 b^7 C d x+24 a b^9 C d x-18 a \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) (-b B+a C) (c+d x) \cos (c+d x)-36 a^2 b \left (a^2-b^2\right )^3 (-b B+a C) (c+d x) \cos (2 (c+d x))+6 a^9 b B c \cos (3 (c+d x))-18 a^7 b^3 B c \cos (3 (c+d x))+18 a^5 b^5 B c \cos (3 (c+d x))-6 a^3 b^7 B c \cos (3 (c+d x))-6 a^{10} c C \cos (3 (c+d x))+18 a^8 b^2 c C \cos (3 (c+d x))-18 a^6 b^4 c C \cos (3 (c+d x))+6 a^4 b^6 c C \cos (3 (c+d x))+6 a^9 b B d x \cos (3 (c+d x))-18 a^7 b^3 B d x \cos (3 (c+d x))+18 a^5 b^5 B d x \cos (3 (c+d x))-6 a^3 b^7 B d x \cos (3 (c+d x))-6 a^{10} C d x \cos (3 (c+d x))+18 a^8 b^2 C d x \cos (3 (c+d x))-18 a^6 b^4 C d x \cos (3 (c+d x))+6 a^4 b^6 C d x \cos (3 (c+d x))+36 a^7 b^3 B \sin (c+d x)+72 a^5 b^5 B \sin (c+d x)-57 a^3 b^7 B \sin (c+d x)+24 a b^9 B \sin (c+d x)-54 a^8 b^2 C \sin (c+d x)-111 a^6 b^4 C \sin (c+d x)+39 a^4 b^6 C \sin (c+d x)-24 a^2 b^8 C \sin (c+d x)+120 a^6 b^4 B \sin (2 (c+d x))-90 a^4 b^6 B \sin (2 (c+d x))+30 a^2 b^8 B \sin (2 (c+d x))-174 a^7 b^3 C \sin (2 (c+d x))+84 a^5 b^5 C \sin (2 (c+d x))-30 a^3 b^7 C \sin (2 (c+d x))+36 a^7 b^3 B \sin (3 (c+d x))-32 a^5 b^5 B \sin (3 (c+d x))+11 a^3 b^7 B \sin (3 (c+d x))-54 a^8 b^2 C \sin (3 (c+d x))+37 a^6 b^4 C \sin (3 (c+d x))-13 a^4 b^6 C \sin (3 (c+d x))}{\left (a^2-b^2\right )^3}\right )}{24 a^4 d (b C+(b B-a C) \cos (c+d x)) (a+b \sec (c+d x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^5,x]

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(b*B - a*C + b*C*Sec[c + d*x])*((24*b*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^
5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C - 7*a^3*b^4*C + 2*a*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2
- b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (36*a^8*b^2*B*c - 84*a^6*b^4*B*c + 36*a^4*b^6*B*c + 36*a^
2*b^8*B*c - 24*b^10*B*c - 36*a^9*b*c*C + 84*a^7*b^3*c*C - 36*a^5*b^5*c*C - 36*a^3*b^7*c*C + 24*a*b^9*c*C + 36*
a^8*b^2*B*d*x - 84*a^6*b^4*B*d*x + 36*a^4*b^6*B*d*x + 36*a^2*b^8*B*d*x - 24*b^10*B*d*x - 36*a^9*b*C*d*x + 84*a
^7*b^3*C*d*x - 36*a^5*b^5*C*d*x - 36*a^3*b^7*C*d*x + 24*a*b^9*C*d*x - 18*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(-(b*B)
+ a*C)*(c + d*x)*Cos[c + d*x] - 36*a^2*b*(a^2 - b^2)^3*(-(b*B) + a*C)*(c + d*x)*Cos[2*(c + d*x)] + 6*a^9*b*B*
c*Cos[3*(c + d*x)] - 18*a^7*b^3*B*c*Cos[3*(c + d*x)] + 18*a^5*b^5*B*c*Cos[3*(c + d*x)] - 6*a^3*b^7*B*c*Cos[3*(
c + d*x)] - 6*a^10*c*C*Cos[3*(c + d*x)] + 18*a^8*b^2*c*C*Cos[3*(c + d*x)] - 18*a^6*b^4*c*C*Cos[3*(c + d*x)] +
6*a^4*b^6*c*C*Cos[3*(c + d*x)] + 6*a^9*b*B*d*x*Cos[3*(c + d*x)] - 18*a^7*b^3*B*d*x*Cos[3*(c + d*x)] + 18*a^5*b
^5*B*d*x*Cos[3*(c + d*x)] - 6*a^3*b^7*B*d*x*Cos[3*(c + d*x)] - 6*a^10*C*d*x*Cos[3*(c + d*x)] + 18*a^8*b^2*C*d*
x*Cos[3*(c + d*x)] - 18*a^6*b^4*C*d*x*Cos[3*(c + d*x)] + 6*a^4*b^6*C*d*x*Cos[3*(c + d*x)] + 36*a^7*b^3*B*Sin[c
+ d*x] + 72*a^5*b^5*B*Sin[c + d*x] - 57*a^3*b^7*B*Sin[c + d*x] + 24*a*b^9*B*Sin[c + d*x] - 54*a^8*b^2*C*Sin[c
+ d*x] - 111*a^6*b^4*C*Sin[c + d*x] + 39*a^4*b^6*C*Sin[c + d*x] - 24*a^2*b^8*C*Sin[c + d*x] + 120*a^6*b^4*B*S
in[2*(c + d*x)] - 90*a^4*b^6*B*Sin[2*(c + d*x)] + 30*a^2*b^8*B*Sin[2*(c + d*x)] - 174*a^7*b^3*C*Sin[2*(c + d*x
)] + 84*a^5*b^5*C*Sin[2*(c + d*x)] - 30*a^3*b^7*C*Sin[2*(c + d*x)] + 36*a^7*b^3*B*Sin[3*(c + d*x)] - 32*a^5*b^
5*B*Sin[3*(c + d*x)] + 11*a^3*b^7*B*Sin[3*(c + d*x)] - 54*a^8*b^2*C*Sin[3*(c + d*x)] + 37*a^6*b^4*C*Sin[3*(c +
d*x)] - 13*a^4*b^6*C*Sin[3*(c + d*x)])/(a^2 - b^2)^3))/(24*a^4*d*(b*C + (b*B - a*C)*Cos[c + d*x])*(a + b*Sec[
c + d*x])^4)

________________________________________________________________________________________

Maple [A]
time = 0.55, size = 526, normalized size = 1.57

method	result	size
derivativedivides	\(\frac {\frac {2 \left (b B -a C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b \left (\frac {-\frac {\left (12 a^{4} b B +4 a^{3} b^{2} B -6 a^{2} b^{3} B -a \,b^{4} B +2 b^{5} B -18 a^{5} C -7 a^{4} b C +4 a^{3} b^{2} C +C \,a^{2} b^{3}-2 C a \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (18 a^{4} b B -11 a^{2} b^{3} B +3 b^{5} B -27 a^{5} C +10 a^{3} b^{2} C -3 C a \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (12 a^{4} b B -4 a^{3} b^{2} B -6 a^{2} b^{3} B +a \,b^{4} B +2 b^{5} B -18 a^{5} C +7 a^{4} b C +4 a^{3} b^{2} C -C \,a^{2} b^{3}-2 C a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (8 B \,a^{6} b -8 B \,a^{4} b^{3}+7 a^{2} b^{5} B -2 b^{7} B -10 a^{7} C +5 a^{5} b^{2} C -7 C \,a^{3} b^{4}+2 C a \,b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4}}}{d}\)	\(526\)
default	\(\frac {\frac {2 \left (b B -a C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b \left (\frac {-\frac {\left (12 a^{4} b B +4 a^{3} b^{2} B -6 a^{2} b^{3} B -a \,b^{4} B +2 b^{5} B -18 a^{5} C -7 a^{4} b C +4 a^{3} b^{2} C +C \,a^{2} b^{3}-2 C a \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (18 a^{4} b B -11 a^{2} b^{3} B +3 b^{5} B -27 a^{5} C +10 a^{3} b^{2} C -3 C a \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (12 a^{4} b B -4 a^{3} b^{2} B -6 a^{2} b^{3} B +a \,b^{4} B +2 b^{5} B -18 a^{5} C +7 a^{4} b C +4 a^{3} b^{2} C -C \,a^{2} b^{3}-2 C a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (8 B \,a^{6} b -8 B \,a^{4} b^{3}+7 a^{2} b^{5} B -2 b^{7} B -10 a^{7} C +5 a^{5} b^{2} C -7 C \,a^{3} b^{4}+2 C a \,b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4}}}{d}\)	\(526\)
risch	\(\text {Expression too large to display}\)	\(2131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*(B*b-C*a)/a^4*arctan(tan(1/2*d*x+1/2*c))+2*b/a^4*((-1/2*(12*B*a^4*b+4*B*a^3*b^2-6*B*a^2*b^3-B*a*b^4+2*B
*b^5-18*C*a^5-7*C*a^4*b+4*C*a^3*b^2+C*a^2*b^3-2*C*a*b^4)*a*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*(18*B*a^4*b-11*B*a^2*b^3+3*B*b^5-27*C*a^5+10*C*a^3*b^2-3*C*a*b^4)*a*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*(12*B*a^4*b-4*B*a^3*b^2-6*B*a^2*b^3+B*a*b^4+2*B*b^5-18*C*a^5+7*C*a^4*b+4*C*a^3*b^2-C*
a^2*b^3-2*C*a*b^4)*a*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*(8*B*a^6*b-8*B*a^4*b^3+7*B*a^2*b^5-2*B*b^7-10*C*a^7+5*C*a^5*b^2-7*C*a^3*b^4+2*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))))

________________________________________________________________________________________

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((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (322) = 644\).
time = 5.03, size = 2408, normalized size = 7.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,x, algorithm="fricas")

[Out]

[-1/12*(12*(C*a^12 - B*a^11*b - 4*C*a^10*b^2 + 4*B*a^9*b^3 + 6*C*a^8*b^4 - 6*B*a^7*b^5 - 4*C*a^6*b^6 + 4*B*a^5
*b^7 + C*a^4*b^8 - B*a^3*b^9)*d*x*cos(d*x + c)^3 + 36*(C*a^11*b - B*a^10*b^2 - 4*C*a^9*b^3 + 4*B*a^8*b^4 + 6*C
*a^7*b^5 - 6*B*a^6*b^6 - 4*C*a^5*b^7 + 4*B*a^4*b^8 + C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x + c)^2 + 36*(C*a^10*b
^2 - B*a^9*b^3 - 4*C*a^8*b^4 + 4*B*a^7*b^5 + 6*C*a^6*b^6 - 6*B*a^5*b^7 - 4*C*a^4*b^8 + 4*B*a^3*b^9 + C*a^2*b^1
0 - B*a*b^11)*d*x*cos(d*x + c) + 12*(C*a^9*b^3 - B*a^8*b^4 - 4*C*a^7*b^5 + 4*B*a^6*b^6 + 6*C*a^5*b^7 - 6*B*a^4
*b^8 - 4*C*a^3*b^9 + 4*B*a^2*b^10 + C*a*b^11 - B*b^12)*d*x + 3*(10*C*a^7*b^4 - 8*B*a^6*b^5 - 5*C*a^5*b^6 + 8*B
*a^4*b^7 + 7*C*a^3*b^8 - 7*B*a^2*b^9 - 2*C*a*b^10 + 2*B*b^11 + (10*C*a^10*b - 8*B*a^9*b^2 - 5*C*a^8*b^3 + 8*B*
a^7*b^4 + 7*C*a^6*b^5 - 7*B*a^5*b^6 - 2*C*a^4*b^7 + 2*B*a^3*b^8)*cos(d*x + c)^3 + 3*(10*C*a^9*b^2 - 8*B*a^8*b^
3 - 5*C*a^7*b^4 + 8*B*a^6*b^5 + 7*C*a^5*b^6 - 7*B*a^4*b^7 - 2*C*a^3*b^8 + 2*B*a^2*b^9)*cos(d*x + c)^2 + 3*(10*
C*a^8*b^3 - 8*B*a^7*b^4 - 5*C*a^6*b^5 + 8*B*a^5*b^6 + 7*C*a^4*b^7 - 7*B*a^3*b^8 - 2*C*a^2*b^9 + 2*B*a*b^10)*co
s(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(
d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + 2*(37*C*a^8*b^4 -
26*B*a^7*b^5 - 50*C*a^6*b^6 + 43*B*a^5*b^7 + 19*C*a^4*b^8 - 23*B*a^3*b^9 - 6*C*a^2*b^10 + 6*B*a*b^11 + (54*C*
a^10*b^2 - 36*B*a^9*b^3 - 91*C*a^8*b^4 + 68*B*a^7*b^5 + 50*C*a^6*b^6 - 43*B*a^5*b^7 - 13*C*a^4*b^8 + 11*B*a^3*
b^9)*cos(d*x + c)^2 + 3*(29*C*a^9*b^3 - 20*B*a^8*b^4 - 43*C*a^7*b^5 + 35*B*a^6*b^6 + 19*C*a^5*b^7 - 20*B*a^4*b
^8 - 5*C*a^3*b^9 + 5*B*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^
7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 + 3*(a^1
3*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) + (a^12*b^3 - 4*a^10*b^5 + 6*a^8*b^7 - 4
*a^6*b^9 + a^4*b^11)*d), -1/6*(6*(C*a^12 - B*a^11*b - 4*C*a^10*b^2 + 4*B*a^9*b^3 + 6*C*a^8*b^4 - 6*B*a^7*b^5 -
4*C*a^6*b^6 + 4*B*a^5*b^7 + C*a^4*b^8 - B*a^3*b^9)*d*x*cos(d*x + c)^3 + 18*(C*a^11*b - B*a^10*b^2 - 4*C*a^9*b
^3 + 4*B*a^8*b^4 + 6*C*a^7*b^5 - 6*B*a^6*b^6 - 4*C*a^5*b^7 + 4*B*a^4*b^8 + C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x
+ c)^2 + 18*(C*a^10*b^2 - B*a^9*b^3 - 4*C*a^8*b^4 + 4*B*a^7*b^5 + 6*C*a^6*b^6 - 6*B*a^5*b^7 - 4*C*a^4*b^8 + 4
*B*a^3*b^9 + C*a^2*b^10 - B*a*b^11)*d*x*cos(d*x + c) + 6*(C*a^9*b^3 - B*a^8*b^4 - 4*C*a^7*b^5 + 4*B*a^6*b^6 +
6*C*a^5*b^7 - 6*B*a^4*b^8 - 4*C*a^3*b^9 + 4*B*a^2*b^10 + C*a*b^11 - B*b^12)*d*x - 3*(10*C*a^7*b^4 - 8*B*a^6*b^
5 - 5*C*a^5*b^6 + 8*B*a^4*b^7 + 7*C*a^3*b^8 - 7*B*a^2*b^9 - 2*C*a*b^10 + 2*B*b^11 + (10*C*a^10*b - 8*B*a^9*b^2
- 5*C*a^8*b^3 + 8*B*a^7*b^4 + 7*C*a^6*b^5 - 7*B*a^5*b^6 - 2*C*a^4*b^7 + 2*B*a^3*b^8)*cos(d*x + c)^3 + 3*(10*C
*a^9*b^2 - 8*B*a^8*b^3 - 5*C*a^7*b^4 + 8*B*a^6*b^5 + 7*C*a^5*b^6 - 7*B*a^4*b^7 - 2*C*a^3*b^8 + 2*B*a^2*b^9)*co
s(d*x + c)^2 + 3*(10*C*a^8*b^3 - 8*B*a^7*b^4 - 5*C*a^6*b^5 + 8*B*a^5*b^6 + 7*C*a^4*b^7 - 7*B*a^3*b^8 - 2*C*a^2
*b^9 + 2*B*a*b^10)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*s
in(d*x + c))) + (37*C*a^8*b^4 - 26*B*a^7*b^5 - 50*C*a^6*b^6 + 43*B*a^5*b^7 + 19*C*a^4*b^8 - 23*B*a^3*b^9 - 6*C
*a^2*b^10 + 6*B*a*b^11 + (54*C*a^10*b^2 - 36*B*a^9*b^3 - 91*C*a^8*b^4 + 68*B*a^7*b^5 + 50*C*a^6*b^6 - 43*B*a^5
*b^7 - 13*C*a^4*b^8 + 11*B*a^3*b^9)*cos(d*x + c)^2 + 3*(29*C*a^9*b^3 - 20*B*a^8*b^4 - 43*C*a^7*b^5 + 35*B*a^6*
b^6 + 19*C*a^5*b^7 - 20*B*a^4*b^8 - 5*C*a^3*b^9 + 5*B*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^15 - 4*a^13*b^
2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6
*b^9)*d*cos(d*x + c)^2 + 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) + (a^12*b
^3 - 4*a^10*b^5 + 6*a^8*b^7 - 4*a^6*b^9 + a^4*b^11)*d)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {B b}{a^{4} + 4 a^{3} b \sec {\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\right )\, dx - \int \frac {C a}{a^{4} + 4 a^{3} b \sec {\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\, dx - \int \left (- \frac {C b \sec {\left (c + d x \right )}}{a^{4} + 4 a^{3} b \sec {\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a**2*C+b**2*B*sec(d*x+c)+b**2*C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**5,x)

[Out]

-Integral(-B*b/(a**4 + 4*a**3*b*sec(c + d*x) + 6*a**2*b**2*sec(c + d*x)**2 + 4*a*b**3*sec(c + d*x)**3 + b**4*s
ec(c + d*x)**4), x) - Integral(C*a/(a**4 + 4*a**3*b*sec(c + d*x) + 6*a**2*b**2*sec(c + d*x)**2 + 4*a*b**3*sec(
c + d*x)**3 + b**4*sec(c + d*x)**4), x) - Integral(-C*b*sec(c + d*x)/(a**4 + 4*a**3*b*sec(c + d*x) + 6*a**2*b*
*2*sec(c + d*x)**2 + 4*a*b**3*sec(c + d*x)**3 + b**4*sec(c + d*x)**4), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (322) = 644\).
time = 0.67, size = 860, normalized size = 2.56 \begin {gather*} \frac {\frac {3 \, {\left (10 \, C a^{7} b - 8 \, B a^{6} b^{2} - 5 \, C a^{5} b^{3} + 8 \, B a^{4} b^{4} + 7 \, C a^{3} b^{5} - 7 \, B a^{2} b^{6} - 2 \, C a b^{7} + 2 \, B b^{8}\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^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {3 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{4}} + \frac {54 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, B a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 87 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, B a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 42 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, B a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, C a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, B a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 108 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, B a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 148 \, C a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 116 \, B a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 52 \, C a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 56 \, B a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, B b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54 \, C a^{7} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 87 \, C a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 60 \, B a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 42 \, C a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, B a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\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((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,x, algorithm="giac")

[Out]

1/3*(3*(10*C*a^7*b - 8*B*a^6*b^2 - 5*C*a^5*b^3 + 8*B*a^4*b^4 + 7*C*a^3*b^5 - 7*B*a^2*b^6 - 2*C*a*b^7 + 2*B*b^8
)*(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^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(-a^2 + b^2)) - 3*(C*a - B*b)*(d*x + c)/a^4
+ (54*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 - 87*C*a^6*b^3*tan(1/2*d*x + 1/2
*c)^5 + 60*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 42*C*a^4*b^5*tan(1/2*d*x +
1/2*c)^5 - 45*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^2*b^7*tan(1/2*d*x
+ 1/2*c)^5 + 15*B*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a*b^8*tan(1/2*d*x + 1/2*c)^5 - 6*B*b^9*tan(1/2*d*x + 1/2
*c)^5 - 108*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 + 72*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 148*C*a^5*b^4*tan(1/2*d*x
+ 1/2*c)^3 - 116*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 52*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 + 56*B*a^2*b^7*tan(1/
2*d*x + 1/2*c)^3 + 12*C*a*b^8*tan(1/2*d*x + 1/2*c)^3 - 12*B*b^9*tan(1/2*d*x + 1/2*c)^3 + 54*C*a^7*b^2*tan(1/2*
d*x + 1/2*c) - 36*B*a^6*b^3*tan(1/2*d*x + 1/2*c) + 87*C*a^6*b^3*tan(1/2*d*x + 1/2*c) - 60*B*a^5*b^4*tan(1/2*d*
x + 1/2*c) + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c) - 42*C*a^4*b^5*tan(1/2*d*x + 1/2*c) + 45*B*a^3*b^6*tan(1/2*d*x +
1/2*c) + 6*B*a^2*b^7*tan(1/2*d*x + 1/2*c) + 15*C*a^2*b^7*tan(1/2*d*x + 1/2*c) - 15*B*a*b^8*tan(1/2*d*x + 1/2*
c) + 6*C*a*b^8*tan(1/2*d*x + 1/2*c) - 6*B*b^9*tan(1/2*d*x + 1/2*c))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(
a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3))/d

________________________________________________________________________________________

Mupad [B]
time = 16.75, size = 2500, normalized size = 7.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((B*b^2)/cos(c + d*x) - C*a^2 + (C*b^2)/cos(c + d*x)^2 + B*a*b)/(a + b/cos(c + d*x))^5,x)

[Out]

(log(tan(c/2 + (d*x)/2) + 1i)*(B*b - C*a)*1i)/(a^4*d) - ((4*tan(c/2 + (d*x)/2)^3*(3*B*b^7 - 11*B*a^2*b^5 + 18*
B*a^4*b^3 + 10*C*a^3*b^4 - 27*C*a^5*b^2 - 3*C*a*b^6))/(3*(a + b)^2*(a^5 - 2*a^4*b + a^3*b^2)) + (tan(c/2 + (d*
x)/2)*(2*B*b^7 - 6*B*a^2*b^5 - 4*B*a^3*b^4 + 12*B*a^4*b^3 - C*a^2*b^5 + 4*C*a^3*b^4 + 7*C*a^4*b^3 - 18*C*a^5*b
^2 + B*a*b^6 - 2*C*a*b^6))/((a + b)*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)) + (tan(c/2 + (d*x)/2)^5*(2*B*b^7 -
6*B*a^2*b^5 + 4*B*a^3*b^4 + 12*B*a^4*b^3 + C*a^2*b^5 + 4*C*a^3*b^4 - 7*C*a^4*b^3 - 18*C*a^5*b^2 - B*a*b^6 - 2*
C*a*b^6))/((a^3*b - a^4)*(a + b)^3))/(d*(tan(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3) - tan(c/2 +
(d*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 - tan(c/2 + (d*x)/2)^6*(3*a*b^2
- 3*a^2*b + a^3 - b^3))) - (log(tan(c/2 + (d*x)/2) - 1i)*(B*b*1i - C*a*1i))/(a^4*d) + (b*atan(((b*((a + b)^7*
(a - b)^7)^(1/2)*((8*tan(c/2 + (d*x)/2)*(8*B^2*b^16 + 4*C^2*a^16 - 8*B^2*a*b^15 - 8*C^2*a^15*b - 48*B^2*a^2*b^
14 + 48*B^2*a^3*b^13 + 117*B^2*a^4*b^12 - 120*B^2*a^5*b^11 - 164*B^2*a^6*b^10 + 160*B^2*a^7*b^9 + 156*B^2*a^8*
b^8 - 120*B^2*a^9*b^7 - 92*B^2*a^10*b^6 + 48*B^2*a^11*b^5 + 44*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 4*B^2*a^14*b^2
+ 8*C^2*a^2*b^14 - 8*C^2*a^3*b^13 - 48*C^2*a^4*b^12 + 48*C^2*a^5*b^11 + 105*C^2*a^6*b^10 - 120*C^2*a^7*b^9 - 1
30*C^2*a^8*b^8 + 160*C^2*a^9*b^7 + 145*C^2*a^10*b^6 - 120*C^2*a^11*b^5 - 64*C^2*a^12*b^4 + 48*C^2*a^13*b^3 + 8
0*C^2*a^14*b^2 - 16*B*C*a*b^15 - 8*B*C*a^15*b + 16*B*C*a^2*b^14 + 96*B*C*a^3*b^13 - 96*B*C*a^4*b^12 - 222*B*C*
a^5*b^11 + 240*B*C*a^6*b^10 + 294*B*C*a^7*b^9 - 320*B*C*a^8*b^8 - 292*B*C*a^9*b^7 + 240*B*C*a^10*b^6 + 168*B*C
*a^11*b^5 - 96*B*C*a^12*b^4 - 120*B*C*a^13*b^3 + 16*B*C*a^14*b^2))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^
8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) + (b*((8*
(4*C*a^22 + 4*B*a^8*b^14 - 2*B*a^9*b^13 - 26*B*a^10*b^12 + 14*B*a^11*b^11 + 70*B*a^12*b^10 - 30*B*a^13*b^9 - 1
10*B*a^14*b^8 + 30*B*a^15*b^7 + 110*B*a^16*b^6 - 20*B*a^17*b^5 - 64*B*a^18*b^4 + 12*B*a^19*b^3 + 16*B*a^20*b^2
- 4*C*a^9*b^13 + 2*C*a^10*b^12 + 26*C*a^11*b^11 - 20*C*a^12*b^10 - 64*C*a^13*b^9 + 44*C*a^14*b^8 + 96*C*a^15*
b^7 - 36*C*a^16*b^6 - 104*C*a^17*b^5 + 14*C*a^18*b^4 + 70*C*a^19*b^3 - 8*C*a^20*b^2 - 4*B*a^21*b - 20*C*a^21*b
))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 +
10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) - (4*b*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(2*B*b^7 + 10*C*
a^7 - 7*B*a^2*b^5 + 8*B*a^4*b^3 + 7*C*a^3*b^4 - 5*C*a^5*b^2 - 8*B*a^6*b - 2*C*a*b^6)*(8*a^21*b - 8*a^8*b^14 +
8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^
16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^2))/((a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10
+ 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a
^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*((a + b)^7*(a - b)
^7)^(1/2)*(2*B*b^7 + 10*C*a^7 - 7*B*a^2*b^5 + 8*B*a^4*b^3 + 7*C*a^3*b^4 - 5*C*a^5*b^2 - 8*B*a^6*b - 2*C*a*b^6)
)/(2*(a^18 - a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)))*(2*
B*b^7 + 10*C*a^7 - 7*B*a^2*b^5 + 8*B*a^4*b^3 + 7*C*a^3*b^4 - 5*C*a^5*b^2 - 8*B*a^6*b - 2*C*a*b^6)*1i)/(2*(a^18
- a^4*b^14 + 7*a^6*b^12 - 21*a^8*b^10 + 35*a^10*b^8 - 35*a^12*b^6 + 21*a^14*b^4 - 7*a^16*b^2)) + (b*((a + b)^
7*(a - b)^7)^(1/2)*((8*tan(c/2 + (d*x)/2)*(8*B^2*b^16 + 4*C^2*a^16 - 8*B^2*a*b^15 - 8*C^2*a^15*b - 48*B^2*a^2*
b^14 + 48*B^2*a^3*b^13 + 117*B^2*a^4*b^12 - 120*B^2*a^5*b^11 - 164*B^2*a^6*b^10 + 160*B^2*a^7*b^9 + 156*B^2*a^
8*b^8 - 120*B^2*a^9*b^7 - 92*B^2*a^10*b^6 + 48*B^2*a^11*b^5 + 44*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 4*B^2*a^14*b^
2 + 8*C^2*a^2*b^14 - 8*C^2*a^3*b^13 - 48*C^2*a^4*b^12 + 48*C^2*a^5*b^11 + 105*C^2*a^6*b^10 - 120*C^2*a^7*b^9 -
130*C^2*a^8*b^8 + 160*C^2*a^9*b^7 + 145*C^2*a^10*b^6 - 120*C^2*a^11*b^5 - 64*C^2*a^12*b^4 + 48*C^2*a^13*b^3 +
80*C^2*a^14*b^2 - 16*B*C*a*b^15 - 8*B*C*a^15*b + 16*B*C*a^2*b^14 + 96*B*C*a^3*b^13 - 96*B*C*a^4*b^12 - 222*B*
C*a^5*b^11 + 240*B*C*a^6*b^10 + 294*B*C*a^7*b^9 - 320*B*C*a^8*b^8 - 292*B*C*a^9*b^7 + 240*B*C*a^10*b^6 + 168*B
*C*a^11*b^5 - 96*B*C*a^12*b^4 - 120*B*C*a^13*b^3 + 16*B*C*a^14*b^2))/(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*
a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) - (b*((
8*(4*C*a^22 + 4*B*a^8*b^14 - 2*B*a^9*b^13 - 26*B*a^10*b^12 + 14*B*a^11*b^11 + 70*B*a^12*b^10 - 30*B*a^13*b^9 -
110*B*a^14*b^8 + 30*B*a^15*b^7 + 110*B*a^16*b^6 - 20*B*a^17*b^5 - 64*B*a^18*b^4 + 12*B*a^19*b^3 + 16*B*a^20*b
^2 - 4*C*a^9*b^13 + 2*C*a^10*b^12 + 26*C*a^11*b^11 - 20*C*a^12*b^10 - 64*C*a^13*b^9 + 44*C*a^14*b^8 + 96*C*a^1
5*b^7 - 36*C*a^16*b^6 - 104*C*a^17*b^5 + 14*C*a^18*b^4 + 70*C*a^19*b^3 - 8*C*a^20*b^2 - 4*B*a^21*b - 20*C*a^21
*b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*...

________________________________________________________________________________________