\(\int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\) [290]

Optimal result

Integrand size = 31, antiderivative size = 351 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a \left (4 a^2 A b^5-4 A b^7+a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+10 a b^6 B\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^4 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

(A*a^4-6*A*a^2*b^2+A*b^4+4*B*a^3*b-4*B*a*b^3)*x/(a^2+b^2)^4+(4*A*a^3*b-4*A*a*b^3-B*a^4+6*B*a^2*b^2-B*b^4)*ln(c
os(d*x+c))/(a^2+b^2)^4/d+a*(4*A*a^2*b^5-4*A*b^7+B*a^7+4*B*a^5*b^2+5*B*a^3*b^4+10*B*a*b^6)*ln(a+b*tan(d*x+c))/b
^4/(a^2+b^2)^4/d+1/3*a*(A*b-B*a)*tan(d*x+c)^3/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^3+1/2*a*(2*A*b^3-a*(a^2+3*b^2)*B)
*tan(d*x+c)^2/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2+a^2*(A*a^2*b^3-3*A*b^5+B*a^5+3*B*a^3*b^2+6*B*a*b^4)/b^4/(a^
2+b^2)^3/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3686, 3726, 3716, 3707, 3698, 31, 3556} \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {a (A b-a B) \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {a^2 \left (a^5 B+3 a^3 b^2 B+a^2 A b^3+6 a b^4 B-3 A b^5\right )}{b^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {\left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4}+\frac {a \left (a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+4 a^2 A b^5+10 a b^6 B-4 A b^7\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^4} \]

[In]

Int[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]

[Out]

((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*x)/(a^2 + b^2)^4 + ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*
a^2*b^2*B - b^4*B)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) + (a*(4*a^2*A*b^5 - 4*A*b^7 + a^7*B + 4*a^5*b^2*B + 5*
a^3*b^4*B + 10*a*b^6*B)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^4*d) + (a*(A*b - a*B)*Tan[c + d*x]^3)/(3*b*(
a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a*(2*A*b^3 - a*(a^2 + 3*b^2)*B)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d
*(a + b*Tan[c + d*x])^2) + (a^2*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B))/(b^4*(a^2 + b^2)^3*d*
(a + b*Tan[c + d*x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3556

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

Rule 3686

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

Rule 3698

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[
A/(b*f), Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]

Rule 3707

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

Rule 3716

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

Rule 3726

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

Rubi steps \begin{align*} \text {integral}& = \frac {a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {\tan ^2(c+d x) \left (-3 a (A b-a B)+3 b (A b-a B) \tan (c+d x)+3 \left (a^2+b^2\right ) B \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )} \\ & = \frac {a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan (c+d x) \left (-6 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )-6 b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+6 \left (a^2+b^2\right )^2 B \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 b^2 \left (a^2+b^2\right )^2} \\ & = \frac {a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {6 a \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )-6 b^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan (c+d x)+6 \left (a^2+b^2\right )^3 B \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{6 b^3 \left (a^2+b^2\right )^3} \\ & = \frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}+\frac {a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^4}+\frac {\left (a \left (4 a^2 A b^5-4 A b^7+a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+10 a b^6 B\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^3 \left (a^2+b^2\right )^4} \\ & = \frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (a \left (4 a^2 A b^5-4 A b^7+a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+10 a b^6 B\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^4 \left (a^2+b^2\right )^4 d} \\ & = \frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a \left (4 a^2 A b^5-4 A b^7+a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+10 a b^6 B\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^4 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a^2 \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.56 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.87 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 (-i A+B) \log (i-\tan (c+d x))}{(a+i b)^4}+\frac {3 (i A+B) \log (i+\tan (c+d x))}{(a-i b)^4}+\frac {6 a \left (4 a^2 A b^5-4 A b^7+a^7 B+4 a^5 b^2 B+5 a^3 b^4 B+10 a b^6 B\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^4}+\frac {2 a^4 (-A b+a B)}{b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {3 a^3 \left (2 a^2 A b+4 A b^3-3 a^3 B-5 a b^2 B\right )}{b^4 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {6 a^2 \left (-a^4 A b-3 a^2 A b^3-6 A b^5+3 a^5 B+9 a^3 b^2 B+10 a b^4 B\right )}{b^4 \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}}{6 d} \]

[In]

Integrate[(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]

[Out]

((3*((-I)*A + B)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + (3*(I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (6*a*(
4*a^2*A*b^5 - 4*A*b^7 + a^7*B + 4*a^5*b^2*B + 5*a^3*b^4*B + 10*a*b^6*B)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b
^2)^4) + (2*a^4*(-(A*b) + a*B))/(b^4*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (3*a^3*(2*a^2*A*b + 4*A*b^3 - 3*a^3
*B - 5*a*b^2*B))/(b^4*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) + (6*a^2*(-(a^4*A*b) - 3*a^2*A*b^3 - 6*A*b^5 + 3*a
^5*B + 9*a^3*b^2*B + 10*a*b^4*B))/(b^4*(a^2 + b^2)^3*(a + b*Tan[c + d*x])))/(6*d)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a \left (4 A \,a^{2} b^{5}-4 A \,b^{7}+B \,a^{7}+4 B \,a^{5} b^{2}+5 B \,a^{3} b^{4}+10 B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} b^{4}}-\frac {a^{2} \left (A \,a^{4} b +3 A \,a^{2} b^{3}+6 A \,b^{5}-3 B \,a^{5}-9 B \,a^{3} b^{2}-10 B a \,b^{4}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{4} \left (A b -B a \right )}{3 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {a^{3} \left (2 A \,a^{2} b +4 A \,b^{3}-3 B \,a^{3}-5 B a \,b^{2}\right )}{2 b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(343\)
default	\(\frac {\frac {\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a \left (4 A \,a^{2} b^{5}-4 A \,b^{7}+B \,a^{7}+4 B \,a^{5} b^{2}+5 B \,a^{3} b^{4}+10 B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} b^{4}}-\frac {a^{2} \left (A \,a^{4} b +3 A \,a^{2} b^{3}+6 A \,b^{5}-3 B \,a^{5}-9 B \,a^{3} b^{2}-10 B a \,b^{4}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{4} \left (A b -B a \right )}{3 b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {a^{3} \left (2 A \,a^{2} b +4 A \,b^{3}-3 B \,a^{3}-5 B a \,b^{2}\right )}{2 b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(343\)
norman	\(\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a^{3} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{3} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {a^{3} \left (2 A \,a^{5} b +4 A \,a^{3} b^{3}+26 A a \,b^{5}-11 B \,a^{6}-34 B \,a^{4} b^{2}-47 B \,a^{2} b^{4}\right )}{6 d \,b^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {a^{2} \left (2 A \,a^{5} b +6 A \,a^{3} b^{3}+20 A a \,b^{5}-9 B \,a^{6}-28 B \,a^{4} b^{2}-35 B \,a^{2} b^{4}\right ) \tan \left (d x +c \right )}{2 b^{3} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {a \left (A \,a^{5} b +3 A \,a^{3} b^{3}+6 A a \,b^{5}-3 B \,a^{6}-9 B \,a^{4} b^{2}-10 B \,a^{2} b^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \,b^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 b \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {3 b^{2} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {a \left (4 A \,a^{2} b^{5}-4 A \,b^{7}+B \,a^{7}+4 B \,a^{5} b^{2}+5 B \,a^{3} b^{4}+10 B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right ) d \,b^{4}}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(780\)
risch	\(\text {Expression too large to display}\)	\(1614\)
parallelrisch	\(\text {Expression too large to display}\)	\(1808\)

[In]

int(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a^2+b^2)^4*(1/2*(-4*A*a^3*b+4*A*a*b^3+B*a^4-6*B*a^2*b^2+B*b^4)*ln(1+tan(d*x+c)^2)+(A*a^4-6*A*a^2*b^2+A
*b^4+4*B*a^3*b-4*B*a*b^3)*arctan(tan(d*x+c)))+a*(4*A*a^2*b^5-4*A*b^7+B*a^7+4*B*a^5*b^2+5*B*a^3*b^4+10*B*a*b^6)
/(a^2+b^2)^4/b^4*ln(a+b*tan(d*x+c))-a^2*(A*a^4*b+3*A*a^2*b^3+6*A*b^5-3*B*a^5-9*B*a^3*b^2-10*B*a*b^4)/b^4/(a^2+
b^2)^3/(a+b*tan(d*x+c))-1/3*a^4*(A*b-B*a)/b^4/(a^2+b^2)/(a+b*tan(d*x+c))^3+1/2*a^3*(2*A*a^2*b+4*A*b^3-3*B*a^3-
5*B*a*b^2)/b^4/(a^2+b^2)^2/(a+b*tan(d*x+c))^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1113 vs. \(2 (346) = 692\).

Time = 0.36 (sec) , antiderivative size = 1113, normalized size of antiderivative = 3.17 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(3*B*a^9*b^2 + 6*B*a^7*b^4 + 18*A*a^6*b^5 + 47*B*a^5*b^6 - 26*A*a^4*b^7 - (11*B*a^8*b^3 - 2*A*a^7*b^4 + 42
*B*a^6*b^5 - 6*A*a^5*b^6 + 75*B*a^4*b^7 - 48*A*a^3*b^8 - 6*(A*a^4*b^7 + 4*B*a^3*b^8 - 6*A*a^2*b^9 - 4*B*a*b^10
+ A*b^11)*d*x)*tan(d*x + c)^3 + 6*(A*a^7*b^4 + 4*B*a^6*b^5 - 6*A*a^5*b^6 - 4*B*a^4*b^7 + A*a^3*b^8)*d*x - 3*(
5*B*a^9*b^2 + 18*B*a^7*b^4 + 2*A*a^6*b^5 + 37*B*a^5*b^6 - 30*A*a^4*b^7 - 20*B*a^3*b^8 + 12*A*a^2*b^9 - 6*(A*a^
5*b^6 + 4*B*a^4*b^7 - 6*A*a^3*b^8 - 4*B*a^2*b^9 + A*a*b^10)*d*x)*tan(d*x + c)^2 + 3*(B*a^11 + 4*B*a^9*b^2 + 5*
B*a^7*b^4 + 4*A*a^6*b^5 + 10*B*a^5*b^6 - 4*A*a^4*b^7 + (B*a^8*b^3 + 4*B*a^6*b^5 + 5*B*a^4*b^7 + 4*A*a^3*b^8 +
10*B*a^2*b^9 - 4*A*a*b^10)*tan(d*x + c)^3 + 3*(B*a^9*b^2 + 4*B*a^7*b^4 + 5*B*a^5*b^6 + 4*A*a^4*b^7 + 10*B*a^3*
b^8 - 4*A*a^2*b^9)*tan(d*x + c)^2 + 3*(B*a^10*b + 4*B*a^8*b^3 + 5*B*a^6*b^5 + 4*A*a^5*b^6 + 10*B*a^4*b^7 - 4*A
*a^3*b^8)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - 3*(B*a^11
+ 4*B*a^9*b^2 + 6*B*a^7*b^4 + 4*B*a^5*b^6 + B*a^3*b^8 + (B*a^8*b^3 + 4*B*a^6*b^5 + 6*B*a^4*b^7 + 4*B*a^2*b^9 +
B*b^11)*tan(d*x + c)^3 + 3*(B*a^9*b^2 + 4*B*a^7*b^4 + 6*B*a^5*b^6 + 4*B*a^3*b^8 + B*a*b^10)*tan(d*x + c)^2 +
3*(B*a^10*b + 4*B*a^8*b^3 + 6*B*a^6*b^5 + 4*B*a^4*b^7 + B*a^2*b^9)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) -
3*(2*B*a^10*b + 5*B*a^8*b^3 + 2*A*a^7*b^4 + 12*B*a^6*b^5 - 22*A*a^5*b^6 - 35*B*a^4*b^7 + 20*A*a^3*b^8 - 6*(A*
a^6*b^5 + 4*B*a^5*b^6 - 6*A*a^4*b^7 - 4*B*a^3*b^8 + A*a^2*b^9)*d*x)*tan(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*tan(d*x + c)^3 + 3*(a^9*b^6 + 4*a^7*b^8 + 6*a^5*b^10 + 4*a^3*b^12 + a*b^14)*d*ta
n(d*x + c)^2 + 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*tan(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)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(tan(d*x+c)**4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.66 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (B a^{8} + 4 \, B a^{6} b^{2} + 5 \, B a^{4} b^{4} + 4 \, A a^{3} b^{5} + 10 \, B a^{2} b^{6} - 4 \, A a b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} b^{4} + 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {11 \, B a^{9} - 2 \, A a^{8} b + 34 \, B a^{7} b^{2} - 4 \, A a^{6} b^{3} + 47 \, B a^{5} b^{4} - 26 \, A a^{4} b^{5} + 6 \, {\left (3 \, B a^{7} b^{2} - A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 3 \, A a^{4} b^{5} + 10 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (9 \, B a^{8} b - 2 \, A a^{7} b^{2} + 28 \, B a^{6} b^{3} - 6 \, A a^{5} b^{4} + 35 \, B a^{4} b^{5} - 20 \, A a^{3} b^{6}\right )} \tan \left (d x + c\right )}{a^{9} b^{4} + 3 \, a^{7} b^{6} + 3 \, a^{5} b^{8} + a^{3} b^{10} + {\left (a^{6} b^{7} + 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} + b^{13}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{6} + 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} + a b^{12}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{5} + 3 \, a^{6} b^{7} + 3 \, a^{4} b^{9} + a^{2} b^{11}\right )} \tan \left (d x + c\right )}}{6 \, d} \]

[In]

integrate(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/6*(6*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8) + 6*(B*a^8 + 4*B*a^6*b^2 + 5*B*a^4*b^4 + 4*A*a^3*b^5 + 10*B*a^2*b^6 - 4*A*a*b^7)*log(b*tan(d*x + c) +
a)/(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12) + 3*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B
*b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (11*B*a^9 - 2*A*a^8*b + 34*B*a
^7*b^2 - 4*A*a^6*b^3 + 47*B*a^5*b^4 - 26*A*a^4*b^5 + 6*(3*B*a^7*b^2 - A*a^6*b^3 + 9*B*a^5*b^4 - 3*A*a^4*b^5 +
10*B*a^3*b^6 - 6*A*a^2*b^7)*tan(d*x + c)^2 + 3*(9*B*a^8*b - 2*A*a^7*b^2 + 28*B*a^6*b^3 - 6*A*a^5*b^4 + 35*B*a^
4*b^5 - 20*A*a^3*b^6)*tan(d*x + c))/(a^9*b^4 + 3*a^7*b^6 + 3*a^5*b^8 + a^3*b^10 + (a^6*b^7 + 3*a^4*b^9 + 3*a^2
*b^11 + b^13)*tan(d*x + c)^3 + 3*(a^7*b^6 + 3*a^5*b^8 + 3*a^3*b^10 + a*b^12)*tan(d*x + c)^2 + 3*(a^8*b^5 + 3*a
^6*b^7 + 3*a^4*b^9 + a^2*b^11)*tan(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 719 vs. \(2 (346) = 692\).

Time = 1.26 (sec) , antiderivative size = 719, normalized size of antiderivative = 2.05 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (B a^{8} + 4 \, B a^{6} b^{2} + 5 \, B a^{4} b^{4} + 4 \, A a^{3} b^{5} + 10 \, B a^{2} b^{6} - 4 \, A a b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b^{4} + 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}} - \frac {11 \, B a^{8} b^{2} \tan \left (d x + c\right )^{3} + 44 \, B a^{6} b^{4} \tan \left (d x + c\right )^{3} + 55 \, B a^{4} b^{6} \tan \left (d x + c\right )^{3} + 44 \, A a^{3} b^{7} \tan \left (d x + c\right )^{3} + 110 \, B a^{2} b^{8} \tan \left (d x + c\right )^{3} - 44 \, A a b^{9} \tan \left (d x + c\right )^{3} + 15 \, B a^{9} b \tan \left (d x + c\right )^{2} + 6 \, A a^{8} b^{2} \tan \left (d x + c\right )^{2} + 60 \, B a^{7} b^{3} \tan \left (d x + c\right )^{2} + 24 \, A a^{6} b^{4} \tan \left (d x + c\right )^{2} + 51 \, B a^{5} b^{5} \tan \left (d x + c\right )^{2} + 186 \, A a^{4} b^{6} \tan \left (d x + c\right )^{2} + 270 \, B a^{3} b^{7} \tan \left (d x + c\right )^{2} - 96 \, A a^{2} b^{8} \tan \left (d x + c\right )^{2} + 6 \, B a^{10} \tan \left (d x + c\right ) + 6 \, A a^{9} b \tan \left (d x + c\right ) + 21 \, B a^{8} b^{2} \tan \left (d x + c\right ) + 24 \, A a^{7} b^{3} \tan \left (d x + c\right ) - 24 \, B a^{6} b^{4} \tan \left (d x + c\right ) + 210 \, A a^{5} b^{5} \tan \left (d x + c\right ) + 225 \, B a^{4} b^{6} \tan \left (d x + c\right ) - 72 \, A a^{3} b^{7} \tan \left (d x + c\right ) + 2 \, A a^{10} - B a^{9} b + 6 \, A a^{8} b^{2} - 26 \, B a^{7} b^{3} + 74 \, A a^{6} b^{4} + 63 \, B a^{5} b^{5} - 18 \, A a^{4} b^{6}}{{\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \]

[In]

integrate(tan(d*x+c)^4*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/6*(6*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8) + 3*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8) + 6*(B*a^8 + 4*B*a^6*b^2 + 5*B*a^4*b^4 + 4*A*a^3*b^5 + 10*B*a^2*b^6 - 4*A*a*b^7)*
log(abs(b*tan(d*x + c) + a))/(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12) - (11*B*a^8*b^2*tan(d*x + c
)^3 + 44*B*a^6*b^4*tan(d*x + c)^3 + 55*B*a^4*b^6*tan(d*x + c)^3 + 44*A*a^3*b^7*tan(d*x + c)^3 + 110*B*a^2*b^8*
tan(d*x + c)^3 - 44*A*a*b^9*tan(d*x + c)^3 + 15*B*a^9*b*tan(d*x + c)^2 + 6*A*a^8*b^2*tan(d*x + c)^2 + 60*B*a^7
*b^3*tan(d*x + c)^2 + 24*A*a^6*b^4*tan(d*x + c)^2 + 51*B*a^5*b^5*tan(d*x + c)^2 + 186*A*a^4*b^6*tan(d*x + c)^2
+ 270*B*a^3*b^7*tan(d*x + c)^2 - 96*A*a^2*b^8*tan(d*x + c)^2 + 6*B*a^10*tan(d*x + c) + 6*A*a^9*b*tan(d*x + c)
+ 21*B*a^8*b^2*tan(d*x + c) + 24*A*a^7*b^3*tan(d*x + c) - 24*B*a^6*b^4*tan(d*x + c) + 210*A*a^5*b^5*tan(d*x +
c) + 225*B*a^4*b^6*tan(d*x + c) - 72*A*a^3*b^7*tan(d*x + c) + 2*A*a^10 - B*a^9*b + 6*A*a^8*b^2 - 26*B*a^7*b^3
+ 74*A*a^6*b^4 + 63*B*a^5*b^5 - 18*A*a^4*b^6)/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*(b*tan(d*
x + c) + a)^3))/d

Mupad [B] (verification not implemented)

Time = 8.81 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.38 \[ \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {11\,B\,a^9-2\,A\,a^8\,b+34\,B\,a^7\,b^2-4\,A\,a^6\,b^3+47\,B\,a^5\,b^4-26\,A\,a^4\,b^5}{6\,b^4\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,B\,a^7-A\,a^6\,b+9\,B\,a^5\,b^2-3\,A\,a^4\,b^3+10\,B\,a^3\,b^4-6\,A\,a^2\,b^5\right )}{b^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (9\,B\,a^8-2\,A\,a^7\,b+28\,B\,a^6\,b^2-6\,A\,a^5\,b^3+35\,B\,a^4\,b^4-20\,A\,a^3\,b^5\right )}{2\,b^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )}+\frac {a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^7+4\,B\,a^5\,b^2+5\,B\,a^3\,b^4+4\,A\,a^2\,b^5+10\,B\,a\,b^6-4\,A\,b^7\right )}{b^4\,d\,{\left (a^2+b^2\right )}^4} \]

[In]

int((tan(c + d*x)^4*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4,x)

[Out]

((11*B*a^9 - 26*A*a^4*b^5 - 4*A*a^6*b^3 + 47*B*a^5*b^4 + 34*B*a^7*b^2 - 2*A*a^8*b)/(6*b^4*(a^6 + b^6 + 3*a^2*b
^4 + 3*a^4*b^2)) + (tan(c + d*x)^2*(3*B*a^7 - 6*A*a^2*b^5 - 3*A*a^4*b^3 + 10*B*a^3*b^4 + 9*B*a^5*b^2 - A*a^6*b
))/(b^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(9*B*a^8 - 20*A*a^3*b^5 - 6*A*a^5*b^3 + 35*B*a^4*
b^4 + 28*B*a^6*b^2 - 2*A*a^7*b))/(2*b^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3 + b^3*tan(c + d*x)^3 + 3
*a*b^2*tan(c + d*x)^2 + 3*a^2*b*tan(c + d*x))) + (log(tan(c + d*x) - 1i)*(A + B*1i))/(2*d*(4*a*b^3 - 4*a^3*b +
a^4*1i + b^4*1i - a^2*b^2*6i)) + (log(tan(c + d*x) + 1i)*(A*1i + B))/(2*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 -
6*a^2*b^2)) + (a*log(a + b*tan(c + d*x))*(B*a^7 - 4*A*b^7 + 4*A*a^2*b^5 + 5*B*a^3*b^4 + 4*B*a^5*b^2 + 10*B*a*b
^6))/(b^4*d*(a^2 + b^2)^4)