Integrand size = 21, antiderivative size = 283 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right )^3 d}-\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^4(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
-a*(a^2-3*b^2)*x/(a^2+b^2)^3-b*(3*a^2-b^2)*ln(cos(d*x+c))/(a^2+b^2)^3/d+a^ 4*(6*a^4+17*a^2*b^2+15*b^4)*ln(a+b*tan(d*x+c))/b^5/(a^2+b^2)^3/d-a*(6*a^4+ 11*a^2*b^2+3*b^4)*tan(d*x+c)/b^4/(a^2+b^2)^2/d+1/2*(6*a^4+11*a^2*b^2+b^4)* tan(d*x+c)^2/b^3/(a^2+b^2)^2/d-1/2*a^2*tan(d*x+c)^4/b/(a^2+b^2)/d/(a+b*tan (d*x+c))^2-2*a^2*(a^2+2*b^2)*tan(d*x+c)^3/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c ))
Result contains complex when optimal does not.
Time = 4.11 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.86 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {i b \log (i-\tan (c+d x))}{(a+i b)^3}-\frac {b \log (i+\tan (c+d x))}{(i a+b)^3}+\frac {2 a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3}-\frac {a^4 \left (6 a^2+5 b^2\right )}{b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {4 a \tan ^3(c+d x)}{b (a+b \tan (c+d x))^2}+\frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2}+\frac {4 a^3 \left (6 a^4+11 a^2 b^2+4 b^4\right )}{b^4 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 b d} \]
((I*b*Log[I - Tan[c + d*x]])/(a + I*b)^3 - (b*Log[I + Tan[c + d*x]])/(I*a + b)^3 + (2*a^4*(6*a^4 + 17*a^2*b^2 + 15*b^4)*Log[a + b*Tan[c + d*x]])/(b^ 4*(a^2 + b^2)^3) - (a^4*(6*a^2 + 5*b^2))/(b^4*(a^2 + b^2)*(a + b*Tan[c + d *x])^2) - (4*a*Tan[c + d*x]^3)/(b*(a + b*Tan[c + d*x])^2) + Tan[c + d*x]^4 /(a + b*Tan[c + d*x])^2 + (4*a^3*(6*a^4 + 11*a^2*b^2 + 4*b^4))/(b^4*(a^2 + b^2)^2*(a + b*Tan[c + d*x])))/(2*b*d)
Time = 1.92 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 4048, 27, 3042, 4128, 3042, 4130, 27, 3042, 4130, 25, 3042, 4109, 3042, 3956, 4100, 16}
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.
| \(\displaystyle \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^6}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {2 \tan ^3(c+d x) \left (2 a^2-b \tan (c+d x) a+\left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\tan ^3(c+d x) \left (2 a^2-b \tan (c+d x) a+\left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\tan (c+d x)^3 \left (2 a^2-b \tan (c+d x) a+\left (2 a^2+b^2\right ) \tan (c+d x)^2\right )}{(a+b \tan (c+d x))^2}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {\int \frac {\tan ^2(c+d x) \left (-2 a \tan (c+d x) b^3+\left (6 a^4+11 b^2 a^2+b^4\right ) \tan ^2(c+d x)+6 a^2 \left (a^2+2 b^2\right )\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\tan (c+d x)^2 \left (-2 a \tan (c+d x) b^3+\left (6 a^4+11 b^2 a^2+b^4\right ) \tan (c+d x)^2+6 a^2 \left (a^2+2 b^2\right )\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {2 \tan (c+d x) \left (-\left (\left (a^2-b^2\right ) \tan (c+d x) b^3\right )+a \left (6 a^4+11 b^2 a^2+3 b^4\right ) \tan ^2(c+d x)+a \left (6 a^4+11 b^2 a^2+b^4\right )\right )}{a+b \tan (c+d x)}dx}{2 b}+\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\int \frac {\tan (c+d x) \left (-\left (\left (a^2-b^2\right ) \tan (c+d x) b^3\right )+a \left (6 a^4+11 b^2 a^2+3 b^4\right ) \tan ^2(c+d x)+a \left (6 a^4+11 b^2 a^2+b^4\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\int \frac {\tan (c+d x) \left (-\left (\left (a^2-b^2\right ) \tan (c+d x) b^3\right )+a \left (6 a^4+11 b^2 a^2+3 b^4\right ) \tan (c+d x)^2+a \left (6 a^4+11 b^2 a^2+b^4\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {\int -\frac {2 a \tan (c+d x) b^5+\left (6 a^2-b^2\right ) \left (a^2+b^2\right )^2 \tan ^2(c+d x)+a^2 \left (6 a^4+11 b^2 a^2+3 b^4\right )}{a+b \tan (c+d x)}dx}{b}+\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b d}}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b d}-\frac {\int \frac {2 a \tan (c+d x) b^5+\left (6 a^2-b^2\right ) \left (a^2+b^2\right )^2 \tan ^2(c+d x)+a^2 \left (6 a^4+11 b^2 a^2+3 b^4\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b d}-\frac {\int \frac {2 a \tan (c+d x) b^5+\left (6 a^2-b^2\right ) \left (a^2+b^2\right )^2 \tan (c+d x)^2+a^2 \left (6 a^4+11 b^2 a^2+3 b^4\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {b^5 \left (3 a^2-b^2\right ) \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {a b^4 x \left (a^2-3 b^2\right )}{a^2+b^2}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {b^5 \left (3 a^2-b^2\right ) \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {a b^4 x \left (a^2-3 b^2\right )}{a^2+b^2}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^5 \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {a b^4 x \left (a^2-3 b^2\right )}{a^2+b^2}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac {b^5 \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {a b^4 x \left (a^2-3 b^2\right )}{a^2+b^2}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\frac {\left (6 a^4+11 a^2 b^2+b^4\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (6 a^4+11 a^2 b^2+3 b^4\right ) \tan (c+d x)}{b d}-\frac {-\frac {b^5 \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {a b^4 x \left (a^2-3 b^2\right )}{a^2+b^2}+\frac {a^4 \left (6 a^4+17 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \left (a^2+2 b^2\right ) \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
-1/2*(a^2*Tan[c + d*x]^4)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ((-2* a^2*(a^2 + 2*b^2)*Tan[c + d*x]^3)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + (((6*a^4 + 11*a^2*b^2 + b^4)*Tan[c + d*x]^2)/(2*b*d) - (-((-((a*b^4*(a^2 - 3*b^2)*x)/(a^2 + b^2)) - (b^5*(3*a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b ^2)*d) + (a^4*(6*a^4 + 17*a^2*b^2 + 15*b^4)*Log[a + b*Tan[c + d*x]])/(b*(a ^2 + b^2)*d))/b) + (a*(6*a^4 + 11*a^2*b^2 + 3*b^4)*Tan[c + d*x])/(b*d))/b) /(b*(a^2 + b^2)))/(b*(a^2 + b^2))
3.5.77.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/(b*f) Subst[Int[(a + x)^m, x], x, b* Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]
Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2 )/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*A + b*B - a *C)*(x/(a^2 + b^2)), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[( 1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[(A*b - a*B - b*C)/( a^2 + b^2) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] & & NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a*B - b*C , 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Time = 0.42 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(\frac {-\frac {-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+3 a \tan \left (d x +c \right )}{b^{4}}-\frac {a^{6}}{2 b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{4} \left (6 a^{4}+17 a^{2} b^{2}+15 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {2 a^{5} \left (2 a^{2}+3 b^{2}\right )}{b^{5} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(203\) |
| default | \(\frac {-\frac {-\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+3 a \tan \left (d x +c \right )}{b^{4}}-\frac {a^{6}}{2 b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{4} \left (6 a^{4}+17 a^{2} b^{2}+15 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {2 a^{5} \left (2 a^{2}+3 b^{2}\right )}{b^{5} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(203\) |
| norman | \(\frac {\frac {\tan ^{4}\left (d x +c \right )}{2 b d}-\frac {2 a \left (\tan ^{3}\left (d x +c \right )\right )}{b^{2} d}+\frac {a^{2} \left (18 a^{6}+33 a^{4} b^{2}+11 a^{2} b^{4}\right )}{2 d \,b^{5} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (a^{2}-3 b^{2}\right ) a^{3} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {2 a \left (6 a^{6}+11 a^{4} b^{2}+4 a^{2} b^{4}\right ) \tan \left (d x +c \right )}{d \,b^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b \left (a^{2}-3 b^{2}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (a^{2}-3 b^{2}\right ) a x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{4} \left (6 a^{4}+17 a^{2} b^{2}+15 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{5} d}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(410\) |
| parallelrisch | \(\frac {-4 x \tan \left (d x +c \right ) a^{4} b^{6} d +12 x \tan \left (d x +c \right ) a^{2} b^{8} d +18 a^{10}+\left (\tan ^{4}\left (d x +c \right )\right ) a^{6} b^{4}+3 \left (\tan ^{4}\left (d x +c \right )\right ) a^{2} b^{8}+3 \left (\tan ^{4}\left (d x +c \right )\right ) a^{4} b^{6}+68 \tan \left (d x +c \right ) a^{7} b^{3}+60 \tan \left (d x +c \right ) a^{5} b^{5}+16 \tan \left (d x +c \right ) a^{3} b^{7}+34 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{8} b^{2}+30 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{6} b^{4}+24 \tan \left (d x +c \right ) a^{9} b -2 x \,a^{5} b^{5} d +6 x \,a^{3} b^{7} d +3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4} b^{6}-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{8}-2 x \left (\tan ^{2}\left (d x +c \right )\right ) a^{3} b^{7} d +6 x \left (\tan ^{2}\left (d x +c \right )\right ) a \,b^{9} d +51 a^{8} b^{2}-4 \left (\tan ^{3}\left (d x +c \right )\right ) a \,b^{9}-4 \left (\tan ^{3}\left (d x +c \right )\right ) a^{7} b^{3}-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) b^{10}-12 \left (\tan ^{3}\left (d x +c \right )\right ) a^{3} b^{7}-12 \left (\tan ^{3}\left (d x +c \right )\right ) a^{5} b^{5}+12 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{10}+44 a^{6} b^{4}+11 a^{4} b^{6}+\left (\tan ^{4}\left (d x +c \right )\right ) b^{10}+3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{8}+12 \ln \left (a +b \tan \left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a^{8} b^{2}+34 \ln \left (a +b \tan \left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a^{6} b^{4}+30 \ln \left (a +b \tan \left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right ) a^{4} b^{6}+6 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b^{7}-2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{9}+24 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{9} b +68 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{7} b^{3}+60 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{5} b^{5}}{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^{5} d}\) | \(682\) |
| risch | \(\frac {x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 a \,b^{2}}-\frac {12 i a^{8} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{5}}-\frac {34 i a^{6} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{3}}-\frac {2 i x}{b^{3}}-\frac {34 i a^{6} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{3} d}-\frac {2 i c}{d \,b^{3}}-\frac {30 i a^{4} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b}-\frac {12 i a^{8} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b^{5} d}-\frac {30 i a^{4} c}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b d}+\frac {12 i a^{2} c}{b^{5} d}+\frac {12 i a^{2} x}{b^{5}}+\frac {-6 b^{6} a^{2}-6 i a \,b^{7}-18 a^{4} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-48 a^{4} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-66 a^{6} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+16 a^{2} b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-10 a^{6} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-6 a^{2} b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-86 a^{6} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 a^{2} b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+20 a^{4} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+30 i a^{3} b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+12 i a^{3} b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-12 a^{8}+16 i a^{3} b^{5} {\mathrm e}^{6 i \left (d x +c \right )}+36 i a^{7} b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 i a \,b^{7} {\mathrm e}^{4 i \left (d x +c \right )}+44 i a^{5} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+66 i a^{5} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-30 a^{6} b^{2}-18 a^{4} b^{4}-12 i a^{7} b -30 i a^{5} b^{3}-18 i a^{3} b^{5}-12 a^{8} {\mathrm e}^{6 i \left (d x +c \right )}+2 b^{8} {\mathrm e}^{6 i \left (d x +c \right )}-36 a^{8} {\mathrm e}^{2 i \left (d x +c \right )}-36 a^{8} {\mathrm e}^{4 i \left (d x +c \right )}-4 b^{8} {\mathrm e}^{4 i \left (d x +c \right )}+2 b^{8} {\mathrm e}^{2 i \left (d x +c \right )}+4 i a \,b^{7} {\mathrm e}^{2 i \left (d x +c \right )}+24 i a^{7} b \,{\mathrm e}^{6 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \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} \left (i a +b \right )^{3} d \,b^{4}}+\frac {6 a^{8} \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 ) b^{5} d}+\frac {17 a^{6} \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 ) b^{3} d}+\frac {15 a^{4} \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 ) b d}-\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{b^{5} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}\) | \(1049\) |
1/d*(-1/b^4*(-1/2*b*tan(d*x+c)^2+3*a*tan(d*x+c))-1/2/b^5*a^6/(a^2+b^2)/(a+ b*tan(d*x+c))^2+1/b^5*a^4*(6*a^4+17*a^2*b^2+15*b^4)/(a^2+b^2)^3*ln(a+b*tan (d*x+c))+2/b^5*a^5*(2*a^2+3*b^2)/(a^2+b^2)^2/(a+b*tan(d*x+c))+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)*arctan(tan(d*x+c))) )
Leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (279) = 558\).
Time = 0.32 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.22 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} + 3 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{4} - 4 \, {\left (a^{7} b^{3} + 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (a^{5} b^{5} - 3 \, a^{3} b^{7}\right )} d x - {\left (18 \, a^{8} b^{2} + 45 \, a^{6} b^{4} + 30 \, a^{4} b^{6} + 8 \, a^{2} b^{8} - b^{10} + 2 \, {\left (a^{3} b^{7} - 3 \, a b^{9}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (6 \, a^{10} + 17 \, a^{8} b^{2} + 15 \, a^{6} b^{4} + {\left (6 \, a^{8} b^{2} + 17 \, a^{6} b^{4} + 15 \, a^{4} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{9} b + 17 \, a^{7} b^{3} + 15 \, a^{5} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (6 \, a^{10} + 17 \, a^{8} b^{2} + 15 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8} + {\left (6 \, a^{8} b^{2} + 17 \, a^{6} b^{4} + 15 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{9} b + 17 \, a^{7} b^{3} + 15 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (6 \, a^{9} b + 11 \, a^{7} b^{3} - a b^{9} + 2 \, {\left (a^{4} b^{6} - 3 \, a^{2} b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{7} + 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} + b^{13}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{6} + 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} + a b^{12}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{5} + 3 \, a^{6} b^{7} + 3 \, a^{4} b^{9} + a^{2} b^{11}\right )} d\right )}} \]
1/2*(6*a^8*b^2 + 14*a^6*b^4 + 3*a^4*b^6 + a^2*b^8 + (a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10)*tan(d*x + c)^4 - 4*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 + a *b^9)*tan(d*x + c)^3 - 2*(a^5*b^5 - 3*a^3*b^7)*d*x - (18*a^8*b^2 + 45*a^6* b^4 + 30*a^4*b^6 + 8*a^2*b^8 - b^10 + 2*(a^3*b^7 - 3*a*b^9)*d*x)*tan(d*x + c)^2 + (6*a^10 + 17*a^8*b^2 + 15*a^6*b^4 + (6*a^8*b^2 + 17*a^6*b^4 + 15*a ^4*b^6)*tan(d*x + c)^2 + 2*(6*a^9*b + 17*a^7*b^3 + 15*a^5*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 )) - (6*a^10 + 17*a^8*b^2 + 15*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + (6*a^8*b^2 + 17*a^6*b^4 + 15*a^4*b^6 + 3*a^2*b^8 - b^10)*tan(d*x + c)^2 + 2*(6*a^9*b + 17*a^7*b^3 + 15*a^5*b^5 + 3*a^3*b^7 - a*b^9)*tan(d*x + c))*log(1/(tan(d* x + c)^2 + 1)) - 2*(6*a^9*b + 11*a^7*b^3 - a*b^9 + 2*(a^4*b^6 - 3*a^2*b^8) *d*x)*tan(d*x + c))/((a^6*b^7 + 3*a^4*b^9 + 3*a^2*b^11 + b^13)*d*tan(d*x + c)^2 + 2*(a^7*b^6 + 3*a^5*b^8 + 3*a^3*b^10 + a*b^12)*d*tan(d*x + c) + (a^ 8*b^5 + 3*a^6*b^7 + 3*a^4*b^9 + a^2*b^11)*d)
Exception generated. \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
Time = 0.32 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (6 \, a^{8} + 17 \, a^{6} b^{2} + 15 \, a^{4} b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {7 \, a^{8} + 11 \, a^{6} b^{2} + 4 \, {\left (2 \, a^{7} b + 3 \, a^{5} b^{3}\right )} \tan \left (d x + c\right )}{a^{6} b^{5} + 2 \, a^{4} b^{7} + a^{2} b^{9} + {\left (a^{4} b^{7} + 2 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{6} + 2 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )} - \frac {b \tan \left (d x + c\right )^{2} - 6 \, a \tan \left (d x + c\right )}{b^{4}}}{2 \, d} \]
-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* (6*a^8 + 17*a^6*b^2 + 15*a^4*b^4)*log(b*tan(d*x + c) + a)/(a^6*b^5 + 3*a^4 *b^7 + 3*a^2*b^9 + b^11) - (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) - (7*a^8 + 11*a^6*b^2 + 4*(2*a^7*b + 3*a^5*b^ 3)*tan(d*x + c))/(a^6*b^5 + 2*a^4*b^7 + a^2*b^9 + (a^4*b^7 + 2*a^2*b^9 + b ^11)*tan(d*x + c)^2 + 2*(a^5*b^6 + 2*a^3*b^8 + a*b^10)*tan(d*x + c)) - (b* tan(d*x + c)^2 - 6*a*tan(d*x + c))/b^4)/d
Time = 2.36 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.22 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (6 \, a^{8} + 17 \, a^{6} b^{2} + 15 \, a^{4} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}} + \frac {18 \, a^{8} b^{2} \tan \left (d x + c\right )^{2} + 51 \, a^{6} b^{4} \tan \left (d x + c\right )^{2} + 45 \, a^{4} b^{6} \tan \left (d x + c\right )^{2} + 28 \, a^{9} b \tan \left (d x + c\right ) + 82 \, a^{7} b^{3} \tan \left (d x + c\right ) + 78 \, a^{5} b^{5} \tan \left (d x + c\right ) + 11 \, a^{10} + 33 \, a^{8} b^{2} + 34 \, a^{6} b^{4}}{{\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} - \frac {b^{3} \tan \left (d x + c\right )^{2} - 6 \, a b^{2} \tan \left (d x + c\right )}{b^{6}}}{2 \, d} \]
-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) - (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) - 2*(6*a^8 + 17*a^6*b^2 + 15*a^4*b^4)*log(abs(b*tan(d*x + c) + a))/(a^6*b^ 5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11) + (18*a^8*b^2*tan(d*x + c)^2 + 51*a^6*b^ 4*tan(d*x + c)^2 + 45*a^4*b^6*tan(d*x + c)^2 + 28*a^9*b*tan(d*x + c) + 82* a^7*b^3*tan(d*x + c) + 78*a^5*b^5*tan(d*x + c) + 11*a^10 + 33*a^8*b^2 + 34 *a^6*b^4)/((a^6*b^5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11)*(b*tan(d*x + c) + a)^2 ) - (b^3*tan(d*x + c)^2 - 6*a*b^2*tan(d*x + c))/b^6)/d
Time = 5.71 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2\,\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a^7+3\,a^5\,b^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {7\,a^8+11\,a^6\,b^2}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,b^4+2\,a\,b^5\,\mathrm {tan}\left (c+d\,x\right )+b^6\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {b}{{\left (a^2+b^2\right )}^2}-\frac {1}{b^3}+\frac {6\,a^2}{b^5}-\frac {4\,a^2\,b}{{\left (a^2+b^2\right )}^3}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b^3\,d}-\frac {3\,a\,\mathrm {tan}\left (c+d\,x\right )}{b^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \]
((2*tan(c + d*x)*(2*a^7 + 3*a^5*b^2))/(a^4 + b^4 + 2*a^2*b^2) + (7*a^8 + 1 1*a^6*b^2)/(2*b*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2*b^4 + b^6*tan(c + d*x)^2 + 2*a*b^5*tan(c + d*x))) - log(tan(c + d*x) + 1i)/(2*d*(a*b^2*3i - 3*a^2* b - a^3*1i + b^3)) - (log(tan(c + d*x) - 1i)*1i)/(2*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) + (log(a + b*tan(c + d*x))*(b/(a^2 + b^2)^2 - 1/b^3 + (6* a^2)/b^5 - (4*a^2*b)/(a^2 + b^2)^3))/d + tan(c + d*x)^2/(2*b^3*d) - (3*a*t an(c + d*x))/(b^4*d)