\(\int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\) [295]

Optimal result

Integrand size = 29, antiderivative size = 302 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}+\frac {A \log (\sin (c+d x))}{a^4 d}-\frac {b \left (10 a^6 A b+5 a^4 A b^3+4 a^2 A b^5+A b^7-4 a^7 B+4 a^5 b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^4 d}+\frac {b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3690, 3730, 3732, 3611, 3556} \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {A \log (\sin (c+d x))}{a^4 d}+\frac {b (A b-a B)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {x \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4}+\frac {b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right )}{a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {b \left (-4 a^7 B+10 a^6 A b+4 a^5 b^2 B+5 a^4 A b^3+4 a^2 A b^5+A b^7\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^4} \]

[In]

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

[Out]

-(((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*x)/(a^2 + b^2)^4) + (A*Log[Sin[c + d*x]])/(a^4*d) - (
b*(10*a^6*A*b + 5*a^4*A*b^3 + 4*a^2*A*b^5 + A*b^7 - 4*a^7*B + 4*a^5*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]
])/(a^4*(a^2 + b^2)^4*d) + (b*(A*b - a*B))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (b*(3*a^2*A*b + A*b^3
- 2*a^3*B))/(2*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a
^3*b^2*B))/(a^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*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 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3690

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*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n
+ 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 0])))

Rule 3730

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*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3732

Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d)
)*(x/((a^2 + b^2)*(c^2 + d^2))), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*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]

Rubi steps \begin{align*} \text {integral}& = \frac {b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {\cot (c+d x) \left (3 A \left (a^2+b^2\right )-3 a (A b-a B) \tan (c+d x)+3 b (A b-a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a \left (a^2+b^2\right )} \\ & = \frac {b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (6 A \left (a^2+b^2\right )^2-6 a^2 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+6 b \left (3 a^2 A b+A b^3-2 a^3 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^2 \left (a^2+b^2\right )^2} \\ & = \frac {b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (6 A \left (a^2+b^2\right )^3-6 a^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan (c+d x)+6 b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^3 \left (a^2+b^2\right )^3} \\ & = -\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 ) x}{\left (a^2+b^2\right )^4}+\frac {b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {A \int \cot (c+d x) \, dx}{a^4}-\frac {\left (b \left (10 a^6 A b+5 a^4 A b^3+4 a^2 A b^5+A b^7-4 a^7 B+4 a^5 b^2 B\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \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 ) x}{\left (a^2+b^2\right )^4}+\frac {A \log (\sin (c+d x))}{a^4 d}-\frac {b \left (10 a^6 A b+5 a^4 A b^3+4 a^2 A b^5+A b^7-4 a^7 B+4 a^5 b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^4 d}+\frac {b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right )}{a^3 \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 = 3.54 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.02 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \left (-a^4 (a-i b)^4 (A+i B) \log (i-\tan (c+d x))+2 A \left (a^2+b^2\right )^4 \log (\tan (c+d x))-a^4 (a+i b)^4 (A-i B) \log (i+\tan (c+d x))-2 b \left (10 a^6 A b+5 a^4 A b^3+4 a^2 A b^5+A b^7-4 a^7 B+4 a^5 b^2 B\right ) \log (a+b \tan (c+d x))\right )}{a^2 \left (a^2+b^2\right )^2}+\frac {2 a b \left (a^2+b^2\right ) (A b-a B)}{(a+b \tan (c+d x))^3}+\frac {3 b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{(a+b \tan (c+d x))^2}+\frac {6 b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right )}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{6 a^2 \left (a^2+b^2\right )^2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 A \,a^{3} b +4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right )}{2 \left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (6 A \,a^{4} b +3 A \,a^{2} b^{3}+A \,b^{5}-3 B \,a^{5}+B \,a^{3} b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (10 A \,a^{6} b +5 A \,a^{4} b^{3}+4 A \,a^{2} b^{5}+A \,b^{7}-4 B \,a^{7}+4 B \,a^{5} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} a^{4}}+\frac {\left (A b -B a \right ) b}{3 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\)	\(335\)
default	\(\frac {\frac {\frac {\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 A \,a^{3} b +4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right )}{2 \left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (6 A \,a^{4} b +3 A \,a^{2} b^{3}+A \,b^{5}-3 B \,a^{5}+B \,a^{3} b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (10 A \,a^{6} b +5 A \,a^{4} b^{3}+4 A \,a^{2} b^{5}+A \,b^{7}-4 B \,a^{7}+4 B \,a^{5} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} a^{4}}+\frac {\left (A b -B a \right ) b}{3 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\)	\(335\)
parallelrisch	\(\frac {-20 \left (A \,a^{6} b +\frac {1}{2} A \,a^{4} b^{3}+\frac {2}{5} A \,a^{2} b^{5}+\frac {1}{10} A \,b^{7}-\frac {2}{5} B \,a^{7}+\frac {2}{5} B \,a^{5} b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (a +b \tan \left (d x +c \right )\right )-a^{4} \left (a +b \tan \left (d x +c \right )\right )^{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 ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+2 A \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\tan \left (d x +c \right )\right )-8 \left (-\frac {B \,a^{8} d x}{4}+b \left (A d x -\frac {13 B}{12}\right ) a^{7}+\frac {47 \left (\frac {36 B d x}{47}+A \right ) b^{2} a^{6}}{24}-b^{3} \left (A d x +\frac {5 B}{4}\right ) a^{5}+\frac {27 b^{4} \left (-\frac {2 B d x}{27}+A \right ) a^{4}}{8}-\frac {B \,a^{3} b^{5}}{4}+\frac {15 A \,a^{2} b^{6}}{8}-\frac {B a \,b^{7}}{12}+\frac {11 A \,b^{8}}{24}\right ) b^{3} \left (\tan ^{3}\left (d x +c \right )\right )-24 a \,b^{2} \left (-\frac {B \,a^{8} d x}{4}+b \left (A d x -\frac {5 B}{6}\right ) a^{7}+\frac {35 \left (\frac {36 B d x}{35}+A \right ) b^{2} a^{6}}{24}-\left (A d x +\frac {13 B}{12}\right ) b^{3} a^{5}+\frac {21 \left (-\frac {2 B d x}{21}+A \right ) b^{4} a^{4}}{8}-\frac {B \,a^{3} b^{5}}{3}+\frac {37 A \,a^{2} b^{6}}{24}-\frac {B a \,b^{7}}{12}+\frac {3 A \,b^{8}}{8}\right ) \left (\tan ^{2}\left (d x +c \right )\right )-24 a^{2} \left (-\frac {B \,a^{8} d x}{4}+b \left (A d x -\frac {B}{2}\right ) a^{7}+\frac {5 b^{2} \left (\frac {9 B d x}{5}+A \right ) a^{6}}{6}-b^{3} \left (A d x +\frac {3 B}{4}\right ) a^{5}+\frac {19 \left (-\frac {3 B d x}{19}+A \right ) b^{4} a^{4}}{12}-\frac {B \,a^{3} b^{5}}{3}+A \,a^{2} b^{6}-\frac {B a \,b^{7}}{12}+\frac {A \,b^{8}}{4}\right ) b \tan \left (d x +c \right )-8 d \,a^{7} x \left (A \,a^{3} b -A a \,b^{3}-\frac {1}{4} B \,a^{4}+\frac {3}{2} B \,a^{2} b^{2}-\frac {1}{4} B \,b^{4}\right )}{2 \left (a^{2}+b^{2}\right )^{4} d \,a^{4} \left (a +b \tan \left (d x +c \right )\right )^{3}}\)	\(571\)
norman	\(\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 ) 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 (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\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 {b \left (10 A \,a^{4} b^{2}+9 A \,a^{2} b^{4}+3 A \,b^{6}-6 B \,a^{5} b -3 B \,a^{3} b^{3}-B a \,b^{5}\right ) \tan \left (d x +c \right )}{d \,a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{2} \left (35 A \,a^{4} b^{2}+28 A \,a^{2} b^{4}+9 A \,b^{6}-20 B \,a^{5} b -6 B \,a^{3} b^{3}-2 B a \,b^{5}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{3} \left (47 A \,a^{4} b^{2}+34 A \,a^{2} b^{4}+11 A \,b^{6}-26 B \,a^{5} b -4 B \,a^{3} b^{3}-2 B a \,b^{5}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{6 d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 b \left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\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 (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\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 \ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}-\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 ) \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 )}-\frac {b \left (10 A \,a^{6} b +5 A \,a^{4} b^{3}+4 A \,a^{2} b^{5}+A \,b^{7}-4 B \,a^{7}+4 B \,a^{5} b^{2}\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 ) a^{4} d}\)	\(813\)
risch	\(\text {Expression too large to display}\)	\(1616\)

[In]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1126 vs. \(2 (299) = 598\).

Time = 0.38 (sec) , antiderivative size = 1126, normalized size of antiderivative = 3.73 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

integrate(cot(d*x+c)*(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 = 0.30 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.92 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (4 \, B a^{7} b - 10 \, A a^{6} b^{2} - 4 \, B a^{5} b^{3} - 5 \, A a^{4} b^{4} - 4 \, A a^{2} b^{6} - A b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}} - \frac {3 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {26 \, B a^{7} b - 47 \, A a^{6} b^{2} + 4 \, B a^{5} b^{3} - 34 \, A a^{4} b^{4} + 2 \, B a^{3} b^{5} - 11 \, A a^{2} b^{6} + 6 \, {\left (3 \, B a^{5} b^{3} - 6 \, A a^{4} b^{4} - B a^{3} b^{5} - 3 \, A a^{2} b^{6} - A b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (14 \, B a^{6} b^{2} - 27 \, A a^{5} b^{3} - 2 \, B a^{4} b^{4} - 16 \, A a^{3} b^{5} - 5 \, A a b^{7}\right )} \tan \left (d x + c\right )}{a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6} + {\left (a^{9} b^{3} + 3 \, a^{7} b^{5} + 3 \, a^{5} b^{7} + a^{3} b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} \tan \left (d x + c\right )} + \frac {6 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \]

[In]

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

[Out]

1/6*(6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*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*(4*B*a^7*b - 10*A*a^6*b^2 - 4*B*a^5*b^3 - 5*A*a^4*b^4 - 4*A*a^2*b^6 - A*b^8)*log(b*tan(d*x + c) +
a)/(a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4*b^8) - 3*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A
*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) - (26*B*a^7*b - 47*A*a^6*b^2 + 4
*B*a^5*b^3 - 34*A*a^4*b^4 + 2*B*a^3*b^5 - 11*A*a^2*b^6 + 6*(3*B*a^5*b^3 - 6*A*a^4*b^4 - B*a^3*b^5 - 3*A*a^2*b^
6 - A*b^8)*tan(d*x + c)^2 + 3*(14*B*a^6*b^2 - 27*A*a^5*b^3 - 2*B*a^4*b^4 - 16*A*a^3*b^5 - 5*A*a*b^7)*tan(d*x +
c))/(a^12 + 3*a^10*b^2 + 3*a^8*b^4 + a^6*b^6 + (a^9*b^3 + 3*a^7*b^5 + 3*a^5*b^7 + a^3*b^9)*tan(d*x + c)^3 + 3
*(a^10*b^2 + 3*a^8*b^4 + 3*a^6*b^6 + a^4*b^8)*tan(d*x + c)^2 + 3*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*ta
n(d*x + c)) + 6*A*log(tan(d*x + c))/a^4)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (299) = 598\).

Time = 0.99 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.39 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B 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 (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A 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 (4 \, B a^{7} b^{2} - 10 \, A a^{6} b^{3} - 4 \, B a^{5} b^{4} - 5 \, A a^{4} b^{5} - 4 \, A a^{2} b^{7} - A b^{9}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{12} b + 4 \, a^{10} b^{3} + 6 \, a^{8} b^{5} + 4 \, a^{6} b^{7} + a^{4} b^{9}} + \frac {6 \, A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {44 \, B a^{7} b^{4} \tan \left (d x + c\right )^{3} - 110 \, A a^{6} b^{5} \tan \left (d x + c\right )^{3} - 44 \, B a^{5} b^{6} \tan \left (d x + c\right )^{3} - 55 \, A a^{4} b^{7} \tan \left (d x + c\right )^{3} - 44 \, A a^{2} b^{9} \tan \left (d x + c\right )^{3} - 11 \, A b^{11} \tan \left (d x + c\right )^{3} + 150 \, B a^{8} b^{3} \tan \left (d x + c\right )^{2} - 366 \, A a^{7} b^{4} \tan \left (d x + c\right )^{2} - 120 \, B a^{6} b^{5} \tan \left (d x + c\right )^{2} - 219 \, A a^{5} b^{6} \tan \left (d x + c\right )^{2} - 6 \, B a^{4} b^{7} \tan \left (d x + c\right )^{2} - 156 \, A a^{3} b^{8} \tan \left (d x + c\right )^{2} - 39 \, A a b^{10} \tan \left (d x + c\right )^{2} + 174 \, B a^{9} b^{2} \tan \left (d x + c\right ) - 411 \, A a^{8} b^{3} \tan \left (d x + c\right ) - 96 \, B a^{7} b^{4} \tan \left (d x + c\right ) - 294 \, A a^{6} b^{5} \tan \left (d x + c\right ) - 6 \, B a^{5} b^{6} \tan \left (d x + c\right ) - 195 \, A a^{4} b^{7} \tan \left (d x + c\right ) - 48 \, A a^{2} b^{9} \tan \left (d x + c\right ) + 70 \, B a^{10} b - 157 \, A a^{9} b^{2} - 14 \, B a^{8} b^{3} - 136 \, A a^{7} b^{4} + 6 \, B a^{6} b^{5} - 89 \, A a^{5} b^{6} + 2 \, B a^{4} b^{7} - 22 \, A a^{3} b^{8}}{{\left (a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \]

[In]

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

[Out]

1/6*(6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*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*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*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*(4*B*a^7*b^2 - 10*A*a^6*b^3 - 4*B*a^5*b^4 - 5*A*a^4*b^5 - 4*A*a^2*b^7 - A*b^9
)*log(abs(b*tan(d*x + c) + a))/(a^12*b + 4*a^10*b^3 + 6*a^8*b^5 + 4*a^6*b^7 + a^4*b^9) + 6*A*log(abs(tan(d*x +
c)))/a^4 - (44*B*a^7*b^4*tan(d*x + c)^3 - 110*A*a^6*b^5*tan(d*x + c)^3 - 44*B*a^5*b^6*tan(d*x + c)^3 - 55*A*a
^4*b^7*tan(d*x + c)^3 - 44*A*a^2*b^9*tan(d*x + c)^3 - 11*A*b^11*tan(d*x + c)^3 + 150*B*a^8*b^3*tan(d*x + c)^2
- 366*A*a^7*b^4*tan(d*x + c)^2 - 120*B*a^6*b^5*tan(d*x + c)^2 - 219*A*a^5*b^6*tan(d*x + c)^2 - 6*B*a^4*b^7*tan
(d*x + c)^2 - 156*A*a^3*b^8*tan(d*x + c)^2 - 39*A*a*b^10*tan(d*x + c)^2 + 174*B*a^9*b^2*tan(d*x + c) - 411*A*a
^8*b^3*tan(d*x + c) - 96*B*a^7*b^4*tan(d*x + c) - 294*A*a^6*b^5*tan(d*x + c) - 6*B*a^5*b^6*tan(d*x + c) - 195*
A*a^4*b^7*tan(d*x + c) - 48*A*a^2*b^9*tan(d*x + c) + 70*B*a^10*b - 157*A*a^9*b^2 - 14*B*a^8*b^3 - 136*A*a^7*b^
4 + 6*B*a^6*b^5 - 89*A*a^5*b^6 + 2*B*a^4*b^7 - 22*A*a^3*b^8)/((a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4
*b^8)*(b*tan(d*x + c) + a)^3))/d

Mupad [B] (verification not implemented)

Time = 11.61 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.60 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {-26\,B\,a^5\,b+47\,A\,a^4\,b^2-4\,B\,a^3\,b^3+34\,A\,a^2\,b^4-2\,B\,a\,b^5+11\,A\,b^6}{6\,a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-3\,B\,a^5\,b^3+6\,A\,a^4\,b^4+B\,a^3\,b^5+3\,A\,a^2\,b^6+A\,b^8\right )}{a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-14\,B\,a^5\,b^2+27\,A\,a^4\,b^3+2\,B\,a^3\,b^4+16\,A\,a^2\,b^5+5\,A\,b^7\right )}{2\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {A\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-4\,B\,a^7+10\,A\,a^6\,b+4\,B\,a^5\,b^2+5\,A\,a^4\,b^3+4\,A\,a^2\,b^5+A\,b^7\right )}{a^4\,d\,{\left (a^2+b^2\right )}^4} \]

[In]

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

[Out]

((11*A*b^6 + 34*A*a^2*b^4 + 47*A*a^4*b^2 - 4*B*a^3*b^3 - 2*B*a*b^5 - 26*B*a^5*b)/(6*a*(a^6 + b^6 + 3*a^2*b^4 +
3*a^4*b^2)) + (tan(c + d*x)^2*(A*b^8 + 3*A*a^2*b^6 + 6*A*a^4*b^4 + B*a^3*b^5 - 3*B*a^5*b^3))/(a^3*(a^6 + b^6
+ 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(5*A*b^7 + 16*A*a^2*b^5 + 27*A*a^4*b^3 + 2*B*a^3*b^4 - 14*B*a^5*b^2)
)/(2*a^2*(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))) + (A*log(tan(c + d*x)))/(a^4*d) - (log(tan(c + d*x) + 1i)*(A*1i + B))/(2*d*(4*a^3*b - 4*a*b^3
+ a^4*1i + b^4*1i - a^2*b^2*6i)) - (log(tan(c + d*x) - 1i)*(A + B*1i))/(2*d*(a^3*b*4i - a*b^3*4i + a^4 + b^4 -
6*a^2*b^2)) - (b*log(a + b*tan(c + d*x))*(A*b^7 - 4*B*a^7 + 4*A*a^2*b^5 + 5*A*a^4*b^3 + 4*B*a^5*b^2 + 10*A*a^
6*b))/(a^4*d*(a^2 + b^2)^4)