\(\int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [480]

Optimal result

Integrand size = 21, antiderivative size = 149 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {a \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {a^2 \tan (c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (a^2+5 b^2\right )}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:

-b*(3*a^2-b^2)*x/(a^2+b^2)^3+a*(a^2-3*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/( 
a^2+b^2)^3/d-1/2*a^2*tan(d*x+c)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-1/2*a^2*( 
a^2+5*b^2)/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.05 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.81 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {i \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {a \log (i-\tan (c+d x))}{(-i a+b)^3}-\frac {i \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {a \log (i+\tan (c+d x))}{(i a+b)^3}-\frac {4 a b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}-\frac {a}{b (a+b \tan (c+d x))^2}-\frac {2 \tan (c+d x)}{(a+b \tan (c+d x))^2}+\frac {2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {a b \left (\left (-6 a^2+2 b^2\right ) \log (a+b \tan (c+d x))+\frac {\left (a^2+b^2\right ) \left (5 a^2+b^2+4 a b \tan (c+d x)\right )}{(a+b \tan (c+d x))^2}\right )}{\left (a^2+b^2\right )^3}}{2 b d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4048, 3042, 4111, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4048

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4111

\(\displaystyle \frac {\frac {\int -\frac {2 \left (2 a b^2+\left (a^2-b^2\right ) \tan (c+d x) b\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {a^2 \left (a^2+5 b^2\right )}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {2 \int \frac {2 a b^2+\left (a^2-b^2\right ) \tan (c+d x) b}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {a^2 \left (a^2+5 b^2\right )}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 \int \frac {2 a b^2+\left (a^2-b^2\right ) \tan (c+d x) b}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {a^2 \left (a^2+5 b^2\right )}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 4014

\(\displaystyle \frac {-\frac {2 \left (\frac {b^2 x \left (3 a^2-b^2\right )}{a^2+b^2}-\frac {a b \left (a^2-3 b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a^2 \left (a^2+5 b^2\right )}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 \left (\frac {b^2 x \left (3 a^2-b^2\right )}{a^2+b^2}-\frac {a b \left (a^2-3 b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}\right )}{a^2+b^2}-\frac {a^2 \left (a^2+5 b^2\right )}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\)

\(\Big \downarrow \) 4013

\(\displaystyle \frac {-\frac {a^2 \left (a^2+5 b^2\right )}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {2 \left (\frac {b^2 x \left (3 a^2-b^2\right )}{a^2+b^2}-\frac {a b \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}\right )}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan (c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\)

Input:

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

Output:

-1/2*(a^2*Tan[c + d*x])/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ((-2*(( 
b^2*(3*a^2 - b^2)*x)/(a^2 + b^2) - (a*b*(a^2 - 3*b^2)*Log[a*Cos[c + d*x] + 
 b*Sin[c + d*x]])/((a^2 + b^2)*d)))/(a^2 + b^2) - (a^2*(a^2 + 5*b^2))/(b*( 
a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(2*b*(a^2 + b^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

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

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 4048

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m 
 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 
/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + 
f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c 
*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) 
*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( 
n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ 
[n, -1] && IntegerQ[2*m]
 

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 = 0.61 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(\frac {\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{3}}{2 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 a^{2} b +b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(158\)
default	\(\frac {\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{3}}{2 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 a^{2} b +b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(158\)
norman	\(\frac {\frac {\left (a^{4}+3 b^{2} a^{2}\right ) \tan \left (d x +c \right )^{2}}{2 a \left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) d}-\frac {a^{3}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) d}-\frac {b^{3} \left (3 a^{2}-b^{2}\right ) x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 b^{2} \left (3 a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(332\)
risch	\(-\frac {i x}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}-\frac {2 i a^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {6 i a \,b^{2} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i a^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {6 i a \,b^{2} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {2 a^{2} \left (2 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a b -3 b^{2}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} \left (i b +a \right )^{2} d \left (-i b +a \right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(401\)
parallelrisch	\(-\frac {a^{7}+6 x \tan \left (d x +c \right )^{2} a^{2} b^{5} d +12 x \tan \left (d x +c \right ) a^{3} b^{4} d -4 b^{6} a x \tan \left (d x +c \right ) d -2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{3} b^{4}+6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a \,b^{6}-2 b^{5} a^{2} x d +5 a^{3} b^{4}+6 a^{5} b^{2}+2 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{4} b^{3}-2 x \tan \left (d x +c \right )^{2} b^{7} d +6 \tan \left (d x +c \right ) a^{2} b^{5}+6 x \,a^{4} b^{3} d +\ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{3} b^{4}-3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a \,b^{6}-6 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{2} b^{5}-4 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{4} b^{3}+12 \tan \left (d x +c \right ) \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2} b^{5}+2 \tan \left (d x +c \right ) a^{6} b +6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{3} b^{4}+\ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{5} b^{2}-3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{3} b^{4}-2 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{5} b^{2}+8 \tan \left (d x +c \right ) a^{4} b^{3}}{2 \left (a +b \tan \left (d x +c \right )\right )^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{2} d}\)	\(467\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (145) = 290\).

Time = 0.12 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.13 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {a^{5} - 5 \, a^{3} b^{2} - 2 \, {\left (3 \, a^{4} b - a^{2} b^{3}\right )} d x + {\left (a^{5} + 7 \, a^{3} b^{2} - 2 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (a^{5} - 3 \, a^{3} b^{2} + {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (3 \, a^{4} b - 3 \, a^{2} b^{3} - 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} d\right )}} \] Input:

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

Output:

1/2*(a^5 - 5*a^3*b^2 - 2*(3*a^4*b - a^2*b^3)*d*x + (a^5 + 7*a^3*b^2 - 2*(3 
*a^2*b^3 - b^5)*d*x)*tan(d*x + c)^2 + (a^5 - 3*a^3*b^2 + (a^3*b^2 - 3*a*b^ 
4)*tan(d*x + c)^2 + 2*(a^4*b - 3*a^2*b^3)*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)) + 2*(3*a^4*b - 3*a 
^2*b^3 - 2*(3*a^3*b^2 - a*b^4)*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. \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \] Input:

integrate(tan(d*x+c)**3/(a+b*tan(d*x+c))**3,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.76 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {a^{5} + 5 \, a^{3} b^{2} + 2 \, {\left (a^{4} b + 3 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )}}{2 \, d} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.62 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {{\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} - \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} + \frac {{\left (a^{3} b - 3 \, a b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b d + 3 \, a^{4} b^{3} d + 3 \, a^{2} b^{5} d + b^{7} d} - \frac {a^{7} + 6 \, a^{5} b^{2} + 5 \, a^{3} b^{4} + 2 \, {\left (a^{6} b + 4 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} b^{2} d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.34 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.58 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {a}{{\left (a^2+b^2\right )}^2}-\frac {4\,a\,b^2}{{\left (a^2+b^2\right )}^3}\right )}{d}-\frac {\frac {a\,\left (a^4+5\,a^2\,b^2\right )}{2\,b^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^4+3\,a^2\,b^2\right )}{b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )} \] Input:

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

Output:

(log(tan(c + d*x) - 1i)*1i)/(2*d*(a*b^2*3i + 3*a^2*b - a^3*1i - b^3)) + lo 
g(tan(c + d*x) + 1i)/(2*d*(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)) + (log(a + 
b*tan(c + d*x))*(a/(a^2 + b^2)^2 - (4*a*b^2)/(a^2 + b^2)^3))/d - ((a*(a^4 
+ 5*a^2*b^2))/(2*b^2*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)*(a^4 + 3*a^2 
*b^2))/(b*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*t 
an(c + d*x)))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.70 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a^{3} b^{2}+3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a \,b^{4}-2 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{4} b +6 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{2} b^{3}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{5}+3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{3} b^{2}+2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a^{3} b^{2}-6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a \,b^{4}+4 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{4} b -12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{2} b^{3}+2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{5}-6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{3} b^{2}+\tan \left (d x +c \right )^{2} a^{5}+4 \tan \left (d x +c \right )^{2} a^{3} b^{2}-6 \tan \left (d x +c \right )^{2} a^{2} b^{3} d x +3 \tan \left (d x +c \right )^{2} a \,b^{4}+2 \tan \left (d x +c \right )^{2} b^{5} d x -12 \tan \left (d x +c \right ) a^{3} b^{2} d x +4 \tan \left (d x +c \right ) a \,b^{4} d x -2 a^{5}-6 a^{4} b d x -2 a^{3} b^{2}+2 a^{2} b^{3} d x}{2 d \left (\tan \left (d x +c \right )^{2} a^{6} b^{2}+3 \tan \left (d x +c \right )^{2} a^{4} b^{4}+3 \tan \left (d x +c \right )^{2} a^{2} b^{6}+\tan \left (d x +c \right )^{2} b^{8}+2 \tan \left (d x +c \right ) a^{7} b +6 \tan \left (d x +c \right ) a^{5} b^{3}+6 \tan \left (d x +c \right ) a^{3} b^{5}+2 \tan \left (d x +c \right ) a \,b^{7}+a^{8}+3 a^{6} b^{2}+3 a^{4} b^{4}+a^{2} b^{6}\right )} \] Input:

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

Output:

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