\(\int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\) [292]

Optimal result

Integrand size = 31, antiderivative size = 261 \[ \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 (A b-a B)}{3 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {3 a^2 A b-A b^3-a^3 B+3 a b^2 B}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.22 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.57 \[ \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {B \tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac {\frac {2 A b+a B}{3 b d (a+b \tan (c+d x))^3}+\frac {\frac {\left (6 A b^3-6 a b^2 B\right ) \left (-\frac {i \log (i-\tan (c+d x))}{2 (a+i b)^4}+\frac {i \log (i+\tan (c+d x))}{2 (a-i b)^4}+\frac {4 a (a-b) b (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}-\frac {b}{3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}\right )}{b}+6 b B \left (-\frac {\log (i-\tan (c+d x))}{2 (i a-b)^3}+\frac {\log (i+\tan (c+d x))}{2 (i a+b)^3}+\frac {b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac {b}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {2 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )}{3 b d}}{2 b} \] Input:

Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]
 

Output:

-1/2*(B*Tan[c + d*x])/(b*d*(a + b*Tan[c + d*x])^3) - ((2*A*b + a*B)/(3*b*d 
*(a + b*Tan[c + d*x])^3) + (((6*A*b^3 - 6*a*b^2*B)*(((-1/2*I)*Log[I - Tan[ 
c + d*x]])/(a + I*b)^4 + ((I/2)*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (4*a* 
(a - b)*b*(a + b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 - b/(3*(a^2 + b^2 
)*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) - 
 (b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/b + 6*b*B*(-1/2* 
Log[I - Tan[c + d*x]]/(I*a - b)^3 + Log[I + Tan[c + d*x]]/(2*(I*a + b)^3) 
+ (b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 - b/(2*(a^2 + b^ 
2)*(a + b*Tan[c + d*x])^2) - (2*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x]))) 
)/(3*b*d))/(2*b)
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {3042, 4087, 25, 3042, 4111, 3042, 4012, 3042, 4014, 3042, 4013}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (c+d x)^2 (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4}dx\)

\(\Big \downarrow \) 4087

\(\displaystyle \frac {\int -\frac {-\left (\left (a^2+b^2\right ) B \tan ^2(c+d x)\right )-b (A b-a B) \tan (c+d x)+a (A b-a B)}{(a+b \tan (c+d x))^3}dx}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {-\left (\left (a^2+b^2\right ) B \tan ^2(c+d x)\right )-b (A b-a B) \tan (c+d x)+a (A b-a B)}{(a+b \tan (c+d x))^3}dx}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {-\left (\left (a^2+b^2\right ) B \tan (c+d x)^2\right )-b (A b-a B) \tan (c+d x)+a (A b-a B)}{(a+b \tan (c+d x))^3}dx}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4111

\(\displaystyle -\frac {\frac {\int \frac {b \left (A a^2+2 b B a-A b^2\right )-b \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {b \left (A a^2+2 b B a-A b^2\right )-b \left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4012

\(\displaystyle -\frac {\frac {\frac {\int \frac {b \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )-b \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\int \frac {b \left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right )-b \left (-B a^3+3 A b a^2+3 b^2 B a-A b^3\right ) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4014

\(\displaystyle -\frac {\frac {\frac {\frac {b \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{a^2+b^2}}{a^2+b^2}-\frac {b \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\frac {b \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{a^2+b^2}}{a^2+b^2}-\frac {b \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{b \left (a^2+b^2\right )}-\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\)

\(\Big \downarrow \) 4013

\(\displaystyle -\frac {a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {\frac {\frac {\frac {b \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}+\frac {b x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{a^2+b^2}}{a^2+b^2}-\frac {b \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{b \left (a^2+b^2\right )}\)

Input:

Int[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]
 

Output:

-1/3*(a^2*(A*b - a*B))/(b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (-1/2* 
(a*(2*A*b^3 - a*(a^2 + 3*b^2)*B))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) 
 + (((b*(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) + (b*(4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*Log[a*Cos[c 
+ d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d))/(a^2 + b^2) - (b*(3*a^2*A*b - A 
*b^3 - a^3*B + 3*a*b^2*B))/((a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(a^2 + b^ 
2))/(b*(a^2 + b^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4012

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

rule 4013

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*Si 
n[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 4014

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a 
*d)/(a^2 + b^2)   Int[(b - a*Tan[e + f*x])/(a + b*Tan[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] && N 
eQ[a*c + b*d, 0]
 

rule 4087

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

rule 4111

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

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a^{2} \left (A b -B a \right )}{3 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a \left (2 A \,b^{3}-B \,a^{3}-3 B a \,b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(301\)
default	\(\frac {\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a^{2} \left (A b -B a \right )}{3 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a \left (2 A \,b^{3}-B \,a^{3}-3 B a \,b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(301\)
norman	\(\frac {-\frac {\left (8 A \,a^{3} b^{3}-B \,a^{6}-6 B \,a^{4} b^{2}+3 B \,a^{2} b^{4}\right ) \tan \left (d x +c \right )^{2}}{2 a d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {a \left (A \,a^{5}-3 A \,a^{3} b^{2}+3 B \,a^{4} b -B \,a^{2} b^{3}\right )}{3 b d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a^{3} x}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{3} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) x \tan \left (d x +c \right )^{3}}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {b \left (14 A \,a^{3} b^{3}-2 A a \,b^{5}-B \,a^{6}-8 B \,a^{4} b^{2}+9 B \,a^{2} b^{4}\right ) \tan \left (d x +c \right )^{3}}{6 d \,a^{2} \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {3 b \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {3 b^{2} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a x \tan \left (d x +c \right )^{2}}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(729\)
risch	\(\text {Expression too large to display}\)	\(1422\)
parallelrisch	\(\text {Expression too large to display}\)	\(1655\)

Input:

int(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBO 
SE)
 

Output:

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)))-1/3*a^2*(A*b-B*a)/b^2/(a^2+b^2)/(a+b*tan(d*x+c))^3+(3*A*a^2*b-A*b^ 
3-B*a^3+3*B*a*b^2)/(a^2+b^2)^3/(a+b*tan(d*x+c))-(4*A*a^3*b-4*A*a*b^3-B*a^4 
+6*B*a^2*b^2-B*b^4)/(a^2+b^2)^4*ln(a+b*tan(d*x+c))+1/2*a*(2*A*b^3-B*a^3-3* 
B*a*b^2)/(a^2+b^2)^2/b^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. 836 vs. \(2 (253) = 506\).

Time = 0.12 (sec) , antiderivative size = 836, normalized size of antiderivative = 3.20 \[ \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:

integrate(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="f 
ricas")
 

Output:

1/6*(3*B*a^7 - 12*A*a^6*b - 30*B*a^5*b^2 + 30*A*a^4*b^3 + 11*B*a^3*b^4 - 2 
*A*a^2*b^5 + (B*a^6*b + 2*A*a^5*b^2 + 18*B*a^4*b^3 - 30*A*a^3*b^4 - 27*B*a 
^2*b^5 + 12*A*a*b^6 - 6*(A*a^4*b^3 + 4*B*a^3*b^4 - 6*A*a^2*b^5 - 4*B*a*b^6 
 + A*b^7)*d*x)*tan(d*x + c)^3 - 6*(A*a^7 + 4*B*a^6*b - 6*A*a^5*b^2 - 4*B*a 
^4*b^3 + A*a^3*b^4)*d*x + 3*(B*a^7 + 2*A*a^6*b + 16*B*a^5*b^2 - 24*A*a^4*b 
^3 - 23*B*a^3*b^4 + 16*A*a^2*b^5 + 6*B*a*b^6 - 2*A*b^7 - 6*(A*a^5*b^2 + 4* 
B*a^4*b^3 - 6*A*a^3*b^4 - 4*B*a^2*b^5 + A*a*b^6)*d*x)*tan(d*x + c)^2 + 3*( 
B*a^7 - 4*A*a^6*b - 6*B*a^5*b^2 + 4*A*a^4*b^3 + B*a^3*b^4 + (B*a^4*b^3 - 4 
*A*a^3*b^4 - 6*B*a^2*b^5 + 4*A*a*b^6 + B*b^7)*tan(d*x + c)^3 + 3*(B*a^5*b^ 
2 - 4*A*a^4*b^3 - 6*B*a^3*b^4 + 4*A*a^2*b^5 + B*a*b^6)*tan(d*x + c)^2 + 3* 
(B*a^6*b - 4*A*a^5*b^2 - 6*B*a^4*b^3 + 4*A*a^3*b^4 + B*a^2*b^5)*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*(2*A*a^7 + 9*B*a^6*b - 16*A*a^5*b^2 - 26*B*a^4*b^3 + 24*A*a^3*b^4 
+ 9*B*a^2*b^5 - 2*A*a*b^6 - 6*(A*a^6*b + 4*B*a^5*b^2 - 6*A*a^4*b^3 - 4*B*a 
^3*b^4 + A*a^2*b^5)*d*x)*tan(d*x + c))/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 
 4*a^2*b^9 + b^11)*d*tan(d*x + c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 
 4*a^3*b^8 + a*b^10)*d*tan(d*x + c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6*a^6*b^5 
+ 4*a^4*b^7 + a^2*b^9)*d*tan(d*x + c) + (a^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4* 
a^5*b^6 + a^3*b^8)*d)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \] Input:

integrate(tan(d*x+c)**2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**4,x)
 

Output:

Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr 
imitive'
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (253) = 506\).

Time = 0.13 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.02 \[ \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {B a^{7} + 2 \, A a^{6} b + 14 \, B a^{5} b^{2} - 20 \, A a^{4} b^{3} - 11 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + 6 \, {\left (B a^{3} b^{4} - 3 \, A a^{2} b^{5} - 3 \, B a b^{6} + A b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{6} b + 8 \, B a^{4} b^{3} - 14 \, A a^{3} b^{4} - 9 \, B a^{2} b^{5} + 2 \, A a b^{6}\right )} \tan \left (d x + c\right )}{a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8} + {\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )}}{6 \, d} \] Input:

integrate(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="m 
axima")
 

Output:

-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^4 - 4*A*a^3*b - 6*B 
*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(b*tan(d*x + c) + a)/(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) + (B*a^7 + 2*A*a^6*b + 14*B*a^5*b^2 - 20*A*a^4*b^3 - 11*B*a^3*b^ 
4 + 2*A*a^2*b^5 + 6*(B*a^3*b^4 - 3*A*a^2*b^5 - 3*B*a*b^6 + A*b^7)*tan(d*x 
+ c)^2 + 3*(B*a^6*b + 8*B*a^4*b^3 - 14*A*a^3*b^4 - 9*B*a^2*b^5 + 2*A*a*b^6 
)*tan(d*x + c))/(a^9*b^2 + 3*a^7*b^4 + 3*a^5*b^6 + a^3*b^8 + (a^6*b^5 + 3* 
a^4*b^7 + 3*a^2*b^9 + b^11)*tan(d*x + c)^3 + 3*(a^7*b^4 + 3*a^5*b^6 + 3*a^ 
3*b^8 + a*b^10)*tan(d*x + c)^2 + 3*(a^8*b^3 + 3*a^6*b^5 + 3*a^4*b^7 + a^2* 
b^9)*tan(d*x + c)))/d
 

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.84 \[ \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {{\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d} - \frac {{\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d\right )}} + \frac {{\left (B a^{4} b - 4 \, A a^{3} b^{2} - 6 \, B a^{2} b^{3} + 4 \, A a b^{4} + B b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b d + 4 \, a^{6} b^{3} d + 6 \, a^{4} b^{5} d + 4 \, a^{2} b^{7} d + b^{9} d} - \frac {B a^{9} + 2 \, A a^{8} b + 15 \, B a^{7} b^{2} - 18 \, A a^{6} b^{3} + 3 \, B a^{5} b^{4} - 18 \, A a^{4} b^{5} - 11 \, B a^{3} b^{6} + 2 \, A a^{2} b^{7} + 6 \, {\left (B a^{5} b^{4} - 3 \, A a^{4} b^{5} - 2 \, B a^{3} b^{6} - 2 \, A a^{2} b^{7} - 3 \, B a b^{8} + A b^{9}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{8} b + 9 \, B a^{6} b^{3} - 14 \, A a^{5} b^{4} - B a^{4} b^{5} - 12 \, A a^{3} b^{6} - 9 \, B a^{2} b^{7} + 2 \, A a b^{8}\right )} \tan \left (d x + c\right )}{6 \, {\left (a^{2} + b^{2}\right )}^{4} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} b^{2} d} \] Input:

integrate(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="g 
iac")
 

Output:

-(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*d + 
4*a^6*b^2*d + 6*a^4*b^4*d + 4*a^2*b^6*d + b^8*d) - 1/2*(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*d + 4*a^6* 
b^2*d + 6*a^4*b^4*d + 4*a^2*b^6*d + b^8*d) + (B*a^4*b - 4*A*a^3*b^2 - 6*B* 
a^2*b^3 + 4*A*a*b^4 + B*b^5)*log(abs(b*tan(d*x + c) + a))/(a^8*b*d + 4*a^6 
*b^3*d + 6*a^4*b^5*d + 4*a^2*b^7*d + b^9*d) - 1/6*(B*a^9 + 2*A*a^8*b + 15* 
B*a^7*b^2 - 18*A*a^6*b^3 + 3*B*a^5*b^4 - 18*A*a^4*b^5 - 11*B*a^3*b^6 + 2*A 
*a^2*b^7 + 6*(B*a^5*b^4 - 3*A*a^4*b^5 - 2*B*a^3*b^6 - 2*A*a^2*b^7 - 3*B*a* 
b^8 + A*b^9)*tan(d*x + c)^2 + 3*(B*a^8*b + 9*B*a^6*b^3 - 14*A*a^5*b^4 - B* 
a^4*b^5 - 12*A*a^3*b^6 - 9*B*a^2*b^7 + 2*A*a*b^8)*tan(d*x + c))/((a^2 + b^ 
2)^4*(b*tan(d*x + c) + a)^3*b^2*d)
 

Mupad [B] (verification not implemented)

Time = 4.34 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.71 \[ \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {B}{{\left (a^2+b^2\right )}^2}-\frac {4\,b\,\left (A\,a+2\,B\,b\right )}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^3\,\left (A\,a+B\,b\right )}{{\left (a^2+b^2\right )}^4}\right )}{d}-\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (B\,a^3\,b^2-3\,A\,a^2\,b^3-3\,B\,a\,b^4+A\,b^5\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {a\,\left (B\,a^6+2\,A\,a^5\,b+14\,B\,a^4\,b^2-20\,A\,a^3\,b^3-11\,B\,a^2\,b^4+2\,A\,a\,b^5\right )}{6\,b^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^6+8\,B\,a^4\,b^2-14\,A\,a^3\,b^3-9\,B\,a^2\,b^4+2\,A\,a\,b^5\right )}{2\,b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \] Input:

int((tan(c + d*x)^2*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4,x)
 

Output:

(log(a + b*tan(c + d*x))*(B/(a^2 + b^2)^2 - (4*b*(A*a + 2*B*b))/(a^2 + b^2 
)^3 + (8*b^3*(A*a + B*b))/(a^2 + b^2)^4))/d - ((tan(c + d*x)^2*(A*b^5 - 3* 
A*a^2*b^3 + B*a^3*b^2 - 3*B*a*b^4))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + 
(a*(B*a^6 - 20*A*a^3*b^3 - 11*B*a^2*b^4 + 14*B*a^4*b^2 + 2*A*a*b^5 + 2*A*a 
^5*b))/(6*b^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(B*a^6 
- 14*A*a^3*b^3 - 9*B*a^2*b^4 + 8*B*a^4*b^2 + 2*A*a*b^5))/(2*b*(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))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.10 \[ \int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a^{2} b^{4}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} b^{6}+6 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{3} b^{3}-2 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a \,b^{5}+3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{4} b^{2}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{2} b^{4}-6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a^{2} b^{4}+2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} b^{6}-12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{3} b^{3}+4 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a \,b^{5}-6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{4} b^{2}+2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{2} b^{4}-2 \tan \left (d x +c \right )^{2} a^{3} b^{3} d x -2 \tan \left (d x +c \right )^{2} a^{2} b^{4}+6 \tan \left (d x +c \right )^{2} a \,b^{5} d x -2 \tan \left (d x +c \right )^{2} b^{6}-4 \tan \left (d x +c \right ) a^{4} b^{2} d x +12 \tan \left (d x +c \right ) a^{2} b^{4} d x -a^{6}-2 a^{5} b d x +6 a^{3} b^{3} d x +a^{2} b^{4}}{2 b d \left (\tan \left (d x +c \right )^{2} a^{6} b^{2}+3 \tan \left (d x +c \right )^{2} a^{4} b^{4}+3 \tan \left (d x +c \right )^{2} a^{2} b^{6}+\tan \left (d x +c \right )^{2} b^{8}+2 \tan \left (d x +c \right ) a^{7} b +6 \tan \left (d x +c \right ) a^{5} b^{3}+6 \tan \left (d x +c \right ) a^{3} b^{5}+2 \tan \left (d x +c \right ) a \,b^{7}+a^{8}+3 a^{6} b^{2}+3 a^{4} b^{4}+a^{2} b^{6}\right )} \] Input:

int(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x)
 

Output:

(3*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a**2*b**4 - log(tan(c + d*x)** 
2 + 1)*tan(c + d*x)**2*b**6 + 6*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a**3 
*b**3 - 2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a*b**5 + 3*log(tan(c + d*x 
)**2 + 1)*a**4*b**2 - log(tan(c + d*x)**2 + 1)*a**2*b**4 - 6*log(tan(c + d 
*x)*b + a)*tan(c + d*x)**2*a**2*b**4 + 2*log(tan(c + d*x)*b + a)*tan(c + d 
*x)**2*b**6 - 12*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**3*b**3 + 4*log(ta 
n(c + d*x)*b + a)*tan(c + d*x)*a*b**5 - 6*log(tan(c + d*x)*b + a)*a**4*b** 
2 + 2*log(tan(c + d*x)*b + a)*a**2*b**4 - 2*tan(c + d*x)**2*a**3*b**3*d*x 
- 2*tan(c + d*x)**2*a**2*b**4 + 6*tan(c + d*x)**2*a*b**5*d*x - 2*tan(c + d 
*x)**2*b**6 - 4*tan(c + d*x)*a**4*b**2*d*x + 12*tan(c + d*x)*a**2*b**4*d*x 
 - a**6 - 2*a**5*b*d*x + 6*a**3*b**3*d*x + a**2*b**4)/(2*b*d*(tan(c + d*x) 
**2*a**6*b**2 + 3*tan(c + d*x)**2*a**4*b**4 + 3*tan(c + d*x)**2*a**2*b**6 
+ tan(c + d*x)**2*b**8 + 2*tan(c + d*x)*a**7*b + 6*tan(c + d*x)*a**5*b**3 
+ 6*tan(c + d*x)*a**3*b**5 + 2*tan(c + d*x)*a*b**7 + a**8 + 3*a**6*b**2 + 
3*a**4*b**4 + a**2*b**6))