\(\int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\) [293]

Optimal result

Integrand size = 29, antiderivative size = 250 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^2 A-A b^2+2 a b B}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a^3 A-3 a A b^2+3 a^2 b B-b^3 B}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.97 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.99 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 (A+i B) \log (i-\tan (c+d x))}{(a+i b)^4}+\frac {3 (A-i B) \log (i+\tan (c+d x))}{(a-i b)^4}-\frac {6 \left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac {2 a (A b-a B)}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {3 \left (a^2 A-A b^2+2 a b B\right )}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {6 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}}{6 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 4074, 3042, 4012, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4074

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4014

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4013

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

Input:

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

Output:

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

Defintions of rubi rules used

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 4074

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

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a \left (A b -B a \right )}{3 \left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {A \,a^{2}-A \,b^{2}+2 B a b}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(287\)
default	\(\frac {\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a \left (A b -B a \right )}{3 \left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {A \,a^{2}-A \,b^{2}+2 B a b}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(287\)
norman	\(\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) a^{3} x}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{3} \left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) x \tan \left (d x +c \right )^{3}}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {a \left (9 A \,a^{4} b^{2}-8 A \,a^{2} b^{4}-A \,b^{6}-2 B \,a^{5} b +14 B \,a^{3} b^{3}\right )}{6 b^{2} \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) d}+\frac {\left (3 A \,a^{4} b^{2}-6 A \,a^{2} b^{4}-A \,b^{6}+8 B \,a^{3} b^{3}\right ) \tan \left (d x +c \right )}{2 b d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {b \left (A \,a^{3} b^{2}-3 A a \,b^{4}+3 B \,a^{2} b^{3}-B \,b^{5}\right ) \tan \left (d x +c \right )^{3}}{3 a d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {3 b \left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {3 b^{2} \left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) a x \tan \left (d x +c \right )^{2}}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(729\)
risch	\(\text {Expression too large to display}\)	\(1422\)
parallelrisch	\(\text {Expression too large to display}\)	\(1648\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 838 vs. \(2 (245) = 490\).

Time = 0.14 (sec) , antiderivative size = 838, normalized size of antiderivative = 3.35 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(12*B*a^6*b - 27*A*a^5*b^2 - 30*B*a^4*b^3 + 18*A*a^3*b^4 + 2*B*a^2*b^ 
5 + A*a*b^6 - (2*B*a^5*b^2 - 11*A*a^4*b^3 - 30*B*a^3*b^4 + 30*A*a^2*b^5 + 
12*B*a*b^6 - 3*A*b^7 - 6*(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)*d*x)*tan(d*x + c)^3 + 6*(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)*d*x - 3*(2*B*a^6*b - 9*A*a^5*b^2 - 24*B*a^4*b^3 + 26* 
A*a^3*b^4 + 16*B*a^2*b^5 - 9*A*a*b^6 - 2*B*b^7 - 6*(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)*d*x)*tan(d*x + c)^2 + 3*(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 + (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)*tan(d*x + c)^3 + 3*(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)*tan(d*x + c)^2 + 3*(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)*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*B*a^7 - 6*A*a^6*b - 16*B*a^5*b^2 + 23*A*a^4*b^3 + 24*B*a^3*b^4 - 16*A*a 
^2*b^5 - 2*B*a*b^6 - A*b^7 - 6*(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)*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. \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \] Input:

integrate(tan(d*x+c)*(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. 523 vs. \(2 (245) = 490\).

Time = 0.12 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.09 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {2 \, B a^{6} - 11 \, A a^{5} b - 20 \, B a^{4} b^{2} + 14 \, A a^{3} b^{3} + 2 \, B a^{2} b^{4} + A a b^{5} - 6 \, {\left (A a^{3} b^{3} + 3 \, B a^{2} b^{4} - 3 \, A a b^{5} - B b^{6}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (5 \, A a^{4} b^{2} + 14 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4} - 2 \, B a b^{5} - A b^{6}\right )} \tan \left (d x + c\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7} + {\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{3} + 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{2} + 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} + a^{2} b^{8}\right )} \tan \left (d x + c\right )}}{6 \, d} \] Input:

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

Output:

-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*(A*a^4 + 4*B*a^3*b - 6*A 
*a^2*b^2 - 4*B*a*b^3 + A*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*(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) + (2*B*a^6 - 11*A*a^5*b - 20*B*a^4*b^2 + 14*A*a^3*b^3 + 2*B*a^2* 
b^4 + A*a*b^5 - 6*(A*a^3*b^3 + 3*B*a^2*b^4 - 3*A*a*b^5 - B*b^6)*tan(d*x + 
c)^2 - 3*(5*A*a^4*b^2 + 14*B*a^3*b^3 - 12*A*a^2*b^4 - 2*B*a*b^5 - A*b^6)*t 
an(d*x + c))/(a^9*b + 3*a^7*b^3 + 3*a^5*b^5 + a^3*b^7 + (a^6*b^4 + 3*a^4*b 
^6 + 3*a^2*b^8 + b^10)*tan(d*x + c)^3 + 3*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 
 + a*b^9)*tan(d*x + c)^2 + 3*(a^8*b^2 + 3*a^6*b^4 + 3*a^4*b^6 + a^2*b^8)*t 
an(d*x + c)))/d
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.92 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {{\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )}}{a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d} + \frac {{\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d\right )}} - \frac {{\left (A a^{4} b + 4 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3} - 4 \, B a b^{4} + A b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b d + 4 \, a^{6} b^{3} d + 6 \, a^{4} b^{5} d + 4 \, a^{2} b^{7} d + b^{9} d} - \frac {2 \, B a^{8} - 11 \, A a^{7} b - 18 \, B a^{6} b^{2} + 3 \, A a^{5} b^{3} - 18 \, B a^{4} b^{4} + 15 \, A a^{3} b^{5} + 2 \, B a^{2} b^{6} + A a b^{7} - 6 \, {\left (A a^{5} b^{3} + 3 \, B a^{4} b^{4} - 2 \, A a^{3} b^{5} + 2 \, B a^{2} b^{6} - 3 \, A a b^{7} - B b^{8}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (5 \, A a^{6} b^{2} + 14 \, B a^{5} b^{3} - 7 \, A a^{4} b^{4} + 12 \, B a^{3} b^{5} - 13 \, A a^{2} b^{6} - 2 \, B a b^{7} - A b^{8}\right )} \tan \left (d x + c\right )}{6 \, {\left (a^{2} + b^{2}\right )}^{4} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} b d} \] Input:

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

Output:

-(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*d + 
4*a^6*b^2*d + 6*a^4*b^4*d + 4*a^2*b^6*d + b^8*d) + 1/2*(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*d + 4*a^6* 
b^2*d + 6*a^4*b^4*d + 4*a^2*b^6*d + b^8*d) - (A*a^4*b + 4*B*a^3*b^2 - 6*A* 
a^2*b^3 - 4*B*a*b^4 + A*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*(2*B*a^8 - 11*A*a^7*b - 
18*B*a^6*b^2 + 3*A*a^5*b^3 - 18*B*a^4*b^4 + 15*A*a^3*b^5 + 2*B*a^2*b^6 + A 
*a*b^7 - 6*(A*a^5*b^3 + 3*B*a^4*b^4 - 2*A*a^3*b^5 + 2*B*a^2*b^6 - 3*A*a*b^ 
7 - B*b^8)*tan(d*x + c)^2 - 3*(5*A*a^6*b^2 + 14*B*a^5*b^3 - 7*A*a^4*b^4 + 
12*B*a^3*b^5 - 13*A*a^2*b^6 - 2*B*a*b^7 - A*b^8)*tan(d*x + c))/((a^2 + b^2 
)^4*(b*tan(d*x + c) + a)^3*b*d)
 

Mupad [B] (verification not implemented)

Time = 4.02 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.79 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-5\,A\,a^4\,b-14\,B\,a^3\,b^2+12\,A\,a^2\,b^3+2\,B\,a\,b^4+A\,b^5\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,B\,a^6-11\,A\,a^5\,b-20\,B\,a^4\,b^2+14\,A\,a^3\,b^3+2\,B\,a^2\,b^4+A\,a\,b^5}{6\,b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-A\,a^3\,b^2-3\,B\,a^2\,b^3+3\,A\,a\,b^4+B\,b^5\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {A}{{\left (a^2+b^2\right )}^2}-\frac {4\,b\,\left (2\,A\,b-B\,a\right )}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^3\,\left (A\,b-B\,a\right )}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )} \] Input:

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

Output:

(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(a + b*tan(c + d*x))*(A/(a^2 + b^2)^2 - (4*b*(2*A 
*b - B*a))/(a^2 + b^2)^3 + (8*b^3*(A*b - B*a))/(a^2 + b^2)^4))/d - ((tan(c 
 + d*x)*(A*b^5 + 12*A*a^2*b^3 - 14*B*a^3*b^2 - 5*A*a^4*b + 2*B*a*b^4))/(2* 
(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*B*a^6 + 14*A*a^3*b^3 + 2*B*a^2*b 
^4 - 20*B*a^4*b^2 + A*a*b^5 - 11*A*a^5*b)/(6*b*(a^6 + b^6 + 3*a^2*b^4 + 3* 
a^4*b^2)) + (tan(c + d*x)^2*(B*b^5 - A*a^3*b^2 - 3*B*a^2*b^3 + 3*A*a*b^4)) 
/(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*(a^3*b*4i - a*b^3*4i + a^4 + b^4 - 6*a^2*b^2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.18 \[ \int \frac {\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a^{4} b^{2}-3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a^{2} b^{4}+2 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{5} b -6 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{3} b^{3}+\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{6}-3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{4} b^{2}-2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a^{4} b^{2}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a^{2} b^{4}-4 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{5} b +12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{3} b^{3}-2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{6}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{4} b^{2}-\tan \left (d x +c \right )^{2} a^{4} b^{2}+6 \tan \left (d x +c \right )^{2} a^{3} b^{3} d x -2 \tan \left (d x +c \right )^{2} a \,b^{5} d x +\tan \left (d x +c \right )^{2} b^{6}+12 \tan \left (d x +c \right ) a^{4} b^{2} d x -4 \tan \left (d x +c \right ) a^{2} b^{4} d x +2 a^{6}+6 a^{5} b d x +2 a^{4} b^{2}-2 a^{3} b^{3} d x}{2 a d \left (\tan \left (d x +c \right )^{2} a^{6} b^{2}+3 \tan \left (d x +c \right )^{2} a^{4} b^{4}+3 \tan \left (d x +c \right )^{2} a^{2} b^{6}+\tan \left (d x +c \right )^{2} b^{8}+2 \tan \left (d x +c \right ) a^{7} b +6 \tan \left (d x +c \right ) a^{5} b^{3}+6 \tan \left (d x +c \right ) a^{3} b^{5}+2 \tan \left (d x +c \right ) a \,b^{7}+a^{8}+3 a^{6} b^{2}+3 a^{4} b^{4}+a^{2} b^{6}\right )} \] Input:

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

Output:

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