\(\int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [481]

Optimal result

Integrand size = 21, antiderivative size = 129 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {b \left (3 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {a^2}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {2 a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.72 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.55 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {-\frac {b^2 \tan ^3(c+d x)}{(a+b \tan (c+d x))^2}+\frac {b \tan ^2(c+d x)}{a+b \tan (c+d x)}+a \left (\frac {(i a+b)^3 \log (i-\tan (c+d x))}{\left (a^2+b^2\right )^2}+\frac {i (a+i b) \log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 b \left (-3 a^2+b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}+\frac {2 a (a-b) (a+b)}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{2 a \left (a^2+b^2\right ) d} \] Input:

Integrate[Tan[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4025, 25, 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.

\(\displaystyle \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (c+d x)^2}{(a+b \tan (c+d x))^3}dx\)

\(\Big \downarrow \) 4025

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4014

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4013

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

Input:

Int[Tan[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]
 

Output:

-1/2*a^2/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (((a*(a^2 - 3*b^2)*x)/ 
(a^2 + b^2) + (b*(3*a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 
 + b^2)*d))/(a^2 + b^2) - (2*a*b)/((a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(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 4025

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

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{2}}{2 \left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {2 a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(150\)
default	\(\frac {\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{2}}{2 \left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {2 a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(150\)
norman	\(\frac {\frac {a \left (-a^{3}+a \,b^{2}\right )}{2 b \left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) d}-\frac {\left (a^{2}-3 b^{2}\right ) a^{3} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{3} \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) d}-\frac {2 b \left (a^{2}-3 b^{2}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (a^{2}-3 b^{2}\right ) a x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(331\)
risch	\(\frac {x}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {6 i a^{2} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {6 i a^{2} b c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}-\frac {2 i b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {2 i a \left (-i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-i a^{3}+2 i a \,b^{2}+a^{2} b -2 b^{3}\right )}{\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} \left (-i a +b \right )^{2} d \left (i a +b \right )^{3}}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(417\)
parallelrisch	\(\frac {-2 x \tan \left (d x +c \right )^{2} a^{3} b^{4} d +6 x \tan \left (d x +c \right )^{2} a \,b^{6} d -4 x \tan \left (d x +c \right ) a^{4} b^{3} d +12 x \tan \left (d x +c \right ) a^{2} b^{5} d -6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{2} b^{5}+6 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{3} b^{4}-2 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a \,b^{6}-12 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b^{4}+4 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{6}+3 b^{5} a^{2}-2 x \,a^{5} b^{2} d +6 x \,a^{3} b^{4} d +3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{2} b^{5}-a^{6} b +2 a^{4} b^{3}+3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{4} b^{3}-\ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{2} b^{5}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{4} b^{3}+2 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2} b^{5}-\ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} b^{7}+4 \tan \left (d x +c \right ) a^{3} b^{4}+4 \tan \left (d x +c \right ) a \,b^{6}+2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} b^{7}}{2 \left (a +b \tan \left (d x +c \right )\right )^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{2} d}\)	\(455\)

Input:

int(tan(d*x+c)^2/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/(a^2+b^2)^3*(1/2*(3*a^2*b-b^3)*ln(1+tan(d*x+c)^2)+(-a^3+3*a*b^2)*ar 
ctan(tan(d*x+c)))-1/2*a^2/(a^2+b^2)/b/(a+b*tan(d*x+c))^2+2*a*b/(a^2+b^2)^2 
/(a+b*tan(d*x+c))-b*(3*a^2-b^2)/(a^2+b^2)^3*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. 326 vs. \(2 (127) = 254\).

Time = 0.10 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.53 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {3 \, a^{4} b - 3 \, a^{2} b^{3} + 2 \, {\left (a^{5} - 3 \, a^{3} b^{2}\right )} d x - {\left (a^{4} b - 5 \, a^{2} b^{3} - 2 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{4} b - a^{2} b^{3} + {\left (3 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (a^{5} - 3 \, a^{3} b^{2} + 2 \, a b^{4} - 2 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} d\right )}} \] Input:

integrate(tan(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \] Input:

integrate(tan(d*x+c)**2/(a+b*tan(d*x+c))**3,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. 256 vs. \(2 (127) = 254\).

Time = 0.11 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.98 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, a b^{3} \tan \left (d x + c\right ) - a^{4} + 3 \, a^{2} b^{2}}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )}}{2 \, d} \] Input:

integrate(tan(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.82 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {{\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} + \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} - \frac {{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b d + 3 \, a^{4} b^{3} d + 3 \, a^{2} b^{5} d + b^{7} d} - \frac {a^{6} - 2 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 4 \, {\left (a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} b d} \] Input:

integrate(tan(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="giac")
 

Output:

-(a^3 - 3*a*b^2)*(d*x + c)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4*d + b^6*d) + 1 
/2*(3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^ 
4*d + b^6*d) - (3*a^2*b^2 - b^4)*log(abs(b*tan(d*x + c) + a))/(a^6*b*d + 3 
*a^4*b^3*d + 3*a^2*b^5*d + b^7*d) - 1/2*(a^6 - 2*a^4*b^2 - 3*a^2*b^4 - 4*( 
a^3*b^3 + a*b^5)*tan(d*x + c))/((a^2 + b^2)^3*(b*tan(d*x + c) + a)^2*b*d)
 

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.74 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {3\,b}{{\left (a^2+b^2\right )}^2}-\frac {4\,b^3}{{\left (a^2+b^2\right )}^3}\right )}{d}-\frac {\frac {a^4-3\,a^2\,b^2}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {2\,a\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \] Input:

int(tan(c + d*x)^2/(a + b*tan(c + d*x))^3,x)
                                                                                    
                                                                                    
 

Output:

- log(tan(c + d*x) + 1i)/(2*d*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) - (log( 
a + b*tan(c + d*x))*((3*b)/(a^2 + b^2)^2 - (4*b^3)/(a^2 + b^2)^3))/d - (lo 
g(tan(c + d*x) - 1i)*1i)/(2*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) - ((a^4 
 - 3*a^2*b^2)/(2*b*(a^4 + b^4 + 2*a^2*b^2)) - (2*a*b^2*tan(c + d*x))/(a^4 
+ b^4 + 2*a^2*b^2))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*tan(c + d*x)))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 548, normalized size of antiderivative = 4.25 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a^{2} b^{4}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} b^{6}+6 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{3} b^{3}-2 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a \,b^{5}+3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{4} b^{2}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{2} b^{4}-6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a^{2} b^{4}+2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} b^{6}-12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{3} b^{3}+4 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a \,b^{5}-6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{4} b^{2}+2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{2} b^{4}-2 \tan \left (d x +c \right )^{2} a^{3} b^{3} d x -2 \tan \left (d x +c \right )^{2} a^{2} b^{4}+6 \tan \left (d x +c \right )^{2} a \,b^{5} d x -2 \tan \left (d x +c \right )^{2} b^{6}-4 \tan \left (d x +c \right ) a^{4} b^{2} d x +12 \tan \left (d x +c \right ) a^{2} b^{4} d x -a^{6}-2 a^{5} b d x +6 a^{3} b^{3} d x +a^{2} b^{4}}{2 b d \left (\tan \left (d x +c \right )^{2} a^{6} b^{2}+3 \tan \left (d x +c \right )^{2} a^{4} b^{4}+3 \tan \left (d x +c \right )^{2} a^{2} b^{6}+\tan \left (d x +c \right )^{2} b^{8}+2 \tan \left (d x +c \right ) a^{7} b +6 \tan \left (d x +c \right ) a^{5} b^{3}+6 \tan \left (d x +c \right ) a^{3} b^{5}+2 \tan \left (d x +c \right ) a \,b^{7}+a^{8}+3 a^{6} b^{2}+3 a^{4} b^{4}+a^{2} b^{6}\right )} \] Input:

int(tan(d*x+c)^2/(a+b*tan(d*x+c))^3,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))