Integrand size = 21, antiderivative size = 239 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {2 a^5 \left (2 a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right )^2 d}+\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right ) d}-\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \] Output:
-(a^2-b^2)*x/(a^2+b^2)^2-2*a*b*ln(cos(d*x+c))/(a^2+b^2)^2/d-2*a^5*(2*a^2+3 *b^2)*ln(a+b*tan(d*x+c))/b^5/(a^2+b^2)^2/d+(4*a^4+2*a^2*b^2-b^4)*tan(d*x+c )/b^4/(a^2+b^2)/d-a*(2*a^2+b^2)*tan(d*x+c)^2/b^3/(a^2+b^2)/d+1/3*(4*a^2+b^ 2)*tan(d*x+c)^3/b^2/(a^2+b^2)/d-a^2*tan(d*x+c)^4/b/(a^2+b^2)/d/(a+b*tan(d* x+c))
Result contains complex when optimal does not.
Time = 4.97 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.87 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {-\frac {3 i b \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {3 i b \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {12 a^5 \left (2 a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^2}+\frac {6 \left (-2 a^2+b^2\right ) \tan (c+d x)}{b^3}+\frac {6 a^4 \left (2 a^2+b^2\right )}{b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {4 a \tan ^3(c+d x)}{b (a+b \tan (c+d x))}-\frac {2 \tan ^4(c+d x)}{a+b \tan (c+d x)}}{6 b d} \] Input:
Integrate[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]
Output:
-1/6*(((-3*I)*b*Log[I - Tan[c + d*x]])/(a + I*b)^2 + ((3*I)*b*Log[I + Tan[ c + d*x]])/(a - I*b)^2 + (12*a^5*(2*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/ (b^4*(a^2 + b^2)^2) + (6*(-2*a^2 + b^2)*Tan[c + d*x])/b^3 + (6*a^4*(2*a^2 + b^2))/(b^4*(a^2 + b^2)*(a + b*Tan[c + d*x])) + (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]))/(b*d)
Time = 1.79 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 4048, 3042, 4130, 27, 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.
| \(\displaystyle \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^6}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {\tan ^3(c+d x) \left (4 a^2-b \tan (c+d x) a+\left (4 a^2+b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\tan (c+d x)^3 \left (4 a^2-b \tan (c+d x) a+\left (4 a^2+b^2\right ) \tan (c+d x)^2\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\int -\frac {3 \tan ^2(c+d x) \left (\tan (c+d x) b^3+2 a \left (2 a^2+b^2\right ) \tan ^2(c+d x)+a \left (4 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{3 b}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\int \frac {\tan ^2(c+d x) \left (\tan (c+d x) b^3+2 a \left (2 a^2+b^2\right ) \tan ^2(c+d x)+a \left (4 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\int \frac {\tan (c+d x)^2 \left (\tan (c+d x) b^3+2 a \left (2 a^2+b^2\right ) \tan (c+d x)^2+a \left (4 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {\int -\frac {2 \tan (c+d x) \left (a \tan (c+d x) b^3+\left (4 a^4+2 b^2 a^2-b^4\right ) \tan ^2(c+d x)+2 a^2 \left (2 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{2 b}+\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\int \frac {\tan (c+d x) \left (a \tan (c+d x) b^3+\left (4 a^4+2 b^2 a^2-b^4\right ) \tan ^2(c+d x)+2 a^2 \left (2 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\int \frac {\tan (c+d x) \left (a \tan (c+d x) b^3+\left (4 a^4+2 b^2 a^2-b^4\right ) \tan (c+d x)^2+2 a^2 \left (2 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\frac {\int -\frac {-\tan (c+d x) b^5+2 a \left (2 a^4+b^2 a^2-b^4\right ) \tan ^2(c+d x)+a \left (4 a^4+2 b^2 a^2-b^4\right )}{a+b \tan (c+d x)}dx}{b}+\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b d}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b d}-\frac {\int \frac {-\tan (c+d x) b^5+2 a \left (2 a^4+b^2 a^2-b^4\right ) \tan ^2(c+d x)+a \left (4 a^4+2 b^2 a^2-b^4\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b d}-\frac {\int \frac {-\tan (c+d x) b^5+2 a \left (2 a^4+b^2 a^2-b^4\right ) \tan (c+d x)^2+a \left (4 a^4+2 b^2 a^2-b^4\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b d}-\frac {-\frac {2 a b^5 \int \tan (c+d x)dx}{a^2+b^2}+\frac {2 a^5 \left (2 a^2+3 b^2\right ) \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b^4 x \left (a^2-b^2\right )}{a^2+b^2}}{b}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b d}-\frac {-\frac {2 a b^5 \int \tan (c+d x)dx}{a^2+b^2}+\frac {2 a^5 \left (2 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {b^4 x \left (a^2-b^2\right )}{a^2+b^2}}{b}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {2 a^5 \left (2 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {2 a b^5 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^4 x \left (a^2-b^2\right )}{a^2+b^2}}{b}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {2 a^5 \left (2 a^2+3 b^2\right ) \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}+\frac {2 a b^5 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^4 x \left (a^2-b^2\right )}{a^2+b^2}}{b}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b d}-\frac {\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b d}-\frac {\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b d}-\frac {\frac {2 a b^5 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {b^4 x \left (a^2-b^2\right )}{a^2+b^2}+\frac {2 a^5 \left (2 a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}}{b}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
Input:
Int[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]
Output:
-((a^2*Tan[c + d*x]^4)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))) + (((4*a^2 + b^2)*Tan[c + d*x]^3)/(3*b*d) - ((a*(2*a^2 + b^2)*Tan[c + d*x]^2)/(b*d) - (-(((b^4*(a^2 - b^2)*x)/(a^2 + b^2) + (2*a*b^5*Log[Cos[c + d*x]])/((a^2 + b^2)*d) + (2*a^5*(2*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(b*(a^2 + b^2)* d))/b) + ((4*a^4 + 2*a^2*b^2 - b^4)*Tan[c + d*x])/(b*d))/b)/b)/(b*(a^2 + b ^2))
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[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 = 0.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {b^{2} \tan \left (d x +c \right )^{3}}{3}-a b \tan \left (d x +c \right )^{2}+3 a^{2} \tan \left (d x +c \right )-b^{2} \tan \left (d x +c \right )}{b^{4}}-\frac {a^{6}}{b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a^{5} \left (2 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )^{2}}+\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(168\) |
| default | \(\frac {\frac {\frac {b^{2} \tan \left (d x +c \right )^{3}}{3}-a b \tan \left (d x +c \right )^{2}+3 a^{2} \tan \left (d x +c \right )-b^{2} \tan \left (d x +c \right )}{b^{4}}-\frac {a^{6}}{b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a^{5} \left (2 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )^{2}}+\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(168\) |
| norman | \(\frac {\frac {\left (2 a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{2}}{b^{3} d}+\frac {\tan \left (d x +c \right )^{4}}{3 b d}-\frac {2 a \tan \left (d x +c \right )^{3}}{3 b^{2} d}-\frac {\left (a^{2}-b^{2}\right ) a x}{a^{4}+2 b^{2} a^{2}+b^{4}}-\frac {b \left (a^{2}-b^{2}\right ) x \tan \left (d x +c \right )}{a^{4}+2 b^{2} a^{2}+b^{4}}-\frac {\left (4 a^{5}+2 a^{3} b^{2}-a \,b^{4}\right ) a}{d \,b^{5} \left (a^{2}+b^{2}\right )}}{a +b \tan \left (d x +c \right )}+\frac {a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {2 a^{5} \left (2 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{5} d}\) | \(263\) |
| parallelrisch | \(\frac {-12 a^{8}-3 x \tan \left (d x +c \right ) a^{2} b^{6} d +3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a \,b^{7}-12 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{7} b -18 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{5} b^{3}+3 x \tan \left (d x +c \right ) b^{8} d -3 x \,a^{3} b^{5} d +3 x a \,b^{7} d -18 a^{6} b^{2}-3 a^{4} b^{4}+3 a^{2} b^{6}-12 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{8}+\tan \left (d x +c \right )^{4} b^{8}-3 \tan \left (d x +c \right )^{2} b^{8}+2 \tan \left (d x +c \right )^{4} a^{2} b^{6}-2 \tan \left (d x +c \right )^{3} a \,b^{7}-2 \tan \left (d x +c \right )^{3} a^{5} b^{3}-18 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{6} b^{2}+\tan \left (d x +c \right )^{4} a^{4} b^{4}+9 \tan \left (d x +c \right )^{2} a^{4} b^{4}-4 \tan \left (d x +c \right )^{3} a^{3} b^{5}+6 \tan \left (d x +c \right )^{2} a^{6} b^{2}+3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{2} b^{6}}{3 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{2}+b^{2}\right )^{2} b^{5} d}\) | \(366\) |
| risch | \(\frac {x}{2 i a b -a^{2}+b^{2}}+\frac {8 i a^{7} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{5}}+\frac {8 i a^{7} c}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{5} d}+\frac {12 i a^{5} x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3}}+\frac {12 i a^{5} c}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3} d}-\frac {8 i a^{3} x}{b^{5}}-\frac {8 i a^{3} c}{b^{5} d}+\frac {4 i a x}{b^{3}}+\frac {4 i a c}{b^{3} d}-\frac {2 i \left (-12 i a^{3} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a \,b^{5} {\mathrm e}^{6 i \left (d x +c \right )}-24 i a^{5} b \,{\mathrm e}^{4 i \left (d x +c \right )}+12 a^{6} {\mathrm e}^{6 i \left (d x +c \right )}+6 a^{4} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+3 a^{2} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+6 b^{6} {\mathrm e}^{6 i \left (d x +c \right )}+14 i a \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-6 i a^{3} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+12 i a \,b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+36 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+18 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-12 i a^{5} b \,{\mathrm e}^{2 i \left (d x +c \right )}-12 i a^{5} b \,{\mathrm e}^{6 i \left (d x +c \right )}+2 i a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+36 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}+26 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+12 a^{6}+14 a^{4} b^{2}+a^{2} b^{4}-4 b^{6}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left (-i a +b \right ) \left (i a +b \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 ) b^{4} d}-\frac {4 a^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{5} d}-\frac {6 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{3} d}+\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{5} d}-\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}\) | \(754\) |
Input:
int(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/b^4*(1/3*b^2*tan(d*x+c)^3-a*b*tan(d*x+c)^2+3*a^2*tan(d*x+c)-b^2*tan (d*x+c))-1/b^5*a^6/(a^2+b^2)/(a+b*tan(d*x+c))-2/b^5*a^5*(2*a^2+3*b^2)/(a^2 +b^2)^2*ln(a+b*tan(d*x+c))+1/(a^2+b^2)^2*(a*b*ln(1+tan(d*x+c)^2)+(-a^2+b^2 )*arctan(tan(d*x+c))))
Time = 0.13 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.62 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {6 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{3} b^{5} - a b^{7}\right )} d x - 3 \, {\left (2 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (2 \, a^{8} + 3 \, a^{6} b^{2} + {\left (2 \, a^{7} b + 3 \, a^{5} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (2 \, a^{8} + 3 \, a^{6} b^{2} - a^{2} b^{6} + {\left (2 \, a^{7} b + 3 \, a^{5} b^{3} - a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (4 \, a^{7} b + 4 \, a^{5} b^{3} - a^{3} b^{5} - 2 \, a b^{7} - {\left (a^{2} b^{6} - b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{3 \, {\left ({\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} d\right )}} \] Input:
integrate(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
-1/3*(6*a^6*b^2 + 6*a^4*b^4 + 3*a^2*b^6 - (a^4*b^4 + 2*a^2*b^6 + b^8)*tan( d*x + c)^4 + 2*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*tan(d*x + c)^3 + 3*(a^3*b^5 - a*b^7)*d*x - 3*(2*a^6*b^2 + 3*a^4*b^4 - b^8)*tan(d*x + c)^2 + 3*(2*a^8 + 3*a^6*b^2 + (2*a^7*b + 3*a^5*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)) - 3*(2*a^8 + 3*a^6*b^2 - a ^2*b^6 + (2*a^7*b + 3*a^5*b^3 - a*b^7)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) - 3*(4*a^7*b + 4*a^5*b^3 - a^3*b^5 - 2*a*b^7 - (a^2*b^6 - b^8)*d*x) *tan(d*x + c))/((a^4*b^6 + 2*a^2*b^8 + b^10)*d*tan(d*x + c) + (a^5*b^5 + 2 *a^3*b^7 + a*b^9)*d)
Result contains complex when optimal does not.
Time = 1.43 (sec) , antiderivative size = 3279, normalized size of antiderivative = 13.72 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(tan(d*x+c)**6/(a+b*tan(d*x+c))**2,x)
Output:
Piecewise((zoo*x*tan(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-x + tan(c + d*x)**5/(5*d) - tan(c + d*x)**3/(3*d) + tan(c + d*x)/d)/a**2, Eq(b, 0)), (75*d*x*tan(c + d*x)**2/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) - 150*I*d*x*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 - 24 *I*b**2*d*tan(c + d*x) - 12*b**2*d) - 75*d*x/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) - 36*I*log(tan(c + d*x)**2 + 1)*tan( c + d*x)**2/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b** 2*d) - 72*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 36*I*log(tan(c + d*x)**2 + 1)/( 12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 4*tan( c + d*x)**5/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b** 2*d) + 4*I*tan(c + d*x)**4/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) - 28*tan(c + d*x)**3/(12*b**2*d*tan(c + d*x)**2 - 24*I *b**2*d*tan(c + d*x) - 12*b**2*d) - 153*tan(c + d*x)/(12*b**2*d*tan(c + d* x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 114*I/(12*b**2*d*tan(c + d *x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d), Eq(a, -I*b)), (75*d*x*tan( c + d*x)**2/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b** 2*d) + 150*I*d*x*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan (c + d*x) - 12*b**2*d) - 75*d*x/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*t an(c + d*x) - 12*b**2*d) + 36*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**...
Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.86 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {3 \, a^{6}}{a^{3} b^{5} + a b^{7} + {\left (a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )} - \frac {3 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {3 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {6 \, {\left (2 \, a^{7} + 3 \, a^{5} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}} - \frac {b^{2} \tan \left (d x + c\right )^{3} - 3 \, a b \tan \left (d x + c\right )^{2} + 3 \, {\left (3 \, a^{2} - b^{2}\right )} \tan \left (d x + c\right )}{b^{4}}}{3 \, d} \] Input:
integrate(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-1/3*(3*a^6/(a^3*b^5 + a*b^7 + (a^2*b^6 + b^8)*tan(d*x + c)) - 3*a*b*log(t an(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 3*(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 6*(2*a^7 + 3*a^5*b^2)*log(b*tan(d*x + c) + a)/(a^4*b^ 5 + 2*a^2*b^7 + b^9) - (b^2*tan(d*x + c)^3 - 3*a*b*tan(d*x + c)^2 + 3*(3*a ^2 - b^2)*tan(d*x + c))/b^4)/d
Time = 0.24 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} - \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} - \frac {2 \, {\left (2 \, a^{7} + 3 \, a^{5} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{5} d + 2 \, a^{2} b^{7} d + b^{9} d} + \frac {b^{4} d^{2} \tan \left (d x + c\right )^{3} - 3 \, a b^{3} d^{2} \tan \left (d x + c\right )^{2} + 9 \, a^{2} b^{2} d^{2} \tan \left (d x + c\right ) - 3 \, b^{4} d^{2} \tan \left (d x + c\right )}{3 \, b^{6} d^{3}} - \frac {a^{8} + a^{6} b^{2}}{{\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x + c\right ) + a\right )} b^{5} d} \] Input:
integrate(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
a*b*log(tan(d*x + c)^2 + 1)/(a^4*d + 2*a^2*b^2*d + b^4*d) - (a^2 - b^2)*(d *x + c)/(a^4*d + 2*a^2*b^2*d + b^4*d) - 2*(2*a^7 + 3*a^5*b^2)*log(abs(b*ta n(d*x + c) + a))/(a^4*b^5*d + 2*a^2*b^7*d + b^9*d) + 1/3*(b^4*d^2*tan(d*x + c)^3 - 3*a*b^3*d^2*tan(d*x + c)^2 + 9*a^2*b^2*d^2*tan(d*x + c) - 3*b^4*d ^2*tan(d*x + c))/(b^6*d^3) - (a^8 + a^6*b^2)/((a^2 + b^2)^2*(b*tan(d*x + c ) + a)*b^5*d)
Time = 1.44 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.89 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {4\,a^3}{b^5}-\frac {2\,a}{b^3}+\frac {2\,a\,b}{{\left (a^2+b^2\right )}^2}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,b^2\,d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {a^2+b^2}{b^4}-\frac {4\,a^2}{b^4}\right )}{d}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{b^3\,d}-\frac {a^6}{b\,d\,\left (\mathrm {tan}\left (c+d\,x\right )\,b^5+a\,b^4\right )\,\left (a^2+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \] Input:
int(tan(c + d*x)^6/(a + b*tan(c + d*x))^2,x)
Output:
(log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b*2i - a^2 + b^2)) + log(tan(c + d*x) - 1i)/(2*d*(2*a*b - a^2*1i + b^2*1i)) - (log(a + b*tan(c + d*x))*((4*a^3)/ b^5 - (2*a)/b^3 + (2*a*b)/(a^2 + b^2)^2))/d + tan(c + d*x)^3/(3*b^2*d) - ( tan(c + d*x)*((a^2 + b^2)/b^4 - (4*a^2)/b^4))/d - (a*tan(c + d*x)^2)/(b^3* d) - a^6/(b*d*(a*b^4 + b^5*tan(c + d*x))*(a^2 + b^2))
Time = 0.19 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.76 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a \,b^{7}+3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{2} b^{6}-12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{7} b -18 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{5} b^{3}-12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{8}-18 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{6} b^{2}+\tan \left (d x +c \right )^{4} a^{4} b^{4}+2 \tan \left (d x +c \right )^{4} a^{2} b^{6}+\tan \left (d x +c \right )^{4} b^{8}-2 \tan \left (d x +c \right )^{3} a^{5} b^{3}-4 \tan \left (d x +c \right )^{3} a^{3} b^{5}-2 \tan \left (d x +c \right )^{3} a \,b^{7}+6 \tan \left (d x +c \right )^{2} a^{6} b^{2}+9 \tan \left (d x +c \right )^{2} a^{4} b^{4}-3 \tan \left (d x +c \right )^{2} b^{8}+12 \tan \left (d x +c \right ) a^{7} b +18 \tan \left (d x +c \right ) a^{5} b^{3}+3 \tan \left (d x +c \right ) a^{3} b^{5}-3 \tan \left (d x +c \right ) a^{2} b^{6} d x -3 \tan \left (d x +c \right ) a \,b^{7}+3 \tan \left (d x +c \right ) b^{8} d x -3 a^{3} b^{5} d x +3 a \,b^{7} d x}{3 b^{5} d \left (\tan \left (d x +c \right ) a^{4} b +2 \tan \left (d x +c \right ) a^{2} b^{3}+\tan \left (d x +c \right ) b^{5}+a^{5}+2 a^{3} b^{2}+a \,b^{4}\right )} \] Input:
int(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x)
Output:
(3*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a*b**7 + 3*log(tan(c + d*x)**2 + 1)*a**2*b**6 - 12*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**7*b - 18*log(tan (c + d*x)*b + a)*tan(c + d*x)*a**5*b**3 - 12*log(tan(c + d*x)*b + a)*a**8 - 18*log(tan(c + d*x)*b + a)*a**6*b**2 + tan(c + d*x)**4*a**4*b**4 + 2*tan (c + d*x)**4*a**2*b**6 + tan(c + d*x)**4*b**8 - 2*tan(c + d*x)**3*a**5*b** 3 - 4*tan(c + d*x)**3*a**3*b**5 - 2*tan(c + d*x)**3*a*b**7 + 6*tan(c + d*x )**2*a**6*b**2 + 9*tan(c + d*x)**2*a**4*b**4 - 3*tan(c + d*x)**2*b**8 + 12 *tan(c + d*x)*a**7*b + 18*tan(c + d*x)*a**5*b**3 + 3*tan(c + d*x)*a**3*b** 5 - 3*tan(c + d*x)*a**2*b**6*d*x - 3*tan(c + d*x)*a*b**7 + 3*tan(c + d*x)* b**8*d*x - 3*a**3*b**5*d*x + 3*a*b**7*d*x)/(3*b**5*d*(tan(c + d*x)*a**4*b + 2*tan(c + d*x)*a**2*b**3 + tan(c + d*x)*b**5 + a**5 + 2*a**3*b**2 + a*b* *4))