Integrand size = 21, antiderivative size = 197 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^4 \left (3 a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^2 d}-\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^3(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \] Output:
2*a*b*x/(a^2+b^2)^2-(a^2-b^2)*ln(cos(d*x+c))/(a^2+b^2)^2/d+a^4*(3*a^2+5*b^ 2)*ln(a+b*tan(d*x+c))/b^4/(a^2+b^2)^2/d-a*(3*a^2+2*b^2)*tan(d*x+c)/b^3/(a^ 2+b^2)/d+1/2*(3*a^2+b^2)*tan(d*x+c)^2/b^2/(a^2+b^2)/d-a^2*tan(d*x+c)^3/b/( a^2+b^2)/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 1.98 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.92 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {b \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {b \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {2 a^4 \left (3 a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2}+\frac {6 a^5+4 a^3 b^2}{b^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {3 a \tan ^2(c+d x)}{b (a+b \tan (c+d x))}+\frac {\tan ^3(c+d x)}{a+b \tan (c+d x)}}{2 b d} \] Input:
Integrate[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^2,x]
Output:
((b*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (b*Log[I + Tan[c + d*x]])/(a - I* b)^2 + (2*a^4*(3*a^2 + 5*b^2)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2) + (6*a^5 + 4*a^3*b^2)/(b^3*(a^2 + b^2)*(a + b*Tan[c + d*x])) - (3*a*Tan[c + d*x]^2)/(b*(a + b*Tan[c + d*x])) + Tan[c + d*x]^3/(a + b*Tan[c + d*x])) /(2*b*d)
Time = 1.36 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4048, 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.
| \(\displaystyle \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^5}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {\tan ^2(c+d x) \left (3 a^2-b \tan (c+d x) a+\left (3 a^2+b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\tan (c+d x)^2 \left (3 a^2-b \tan (c+d x) a+\left (3 a^2+b^2\right ) \tan (c+d x)^2\right )}{a+b \tan (c+d x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\int -\frac {2 \tan (c+d x) \left (\tan (c+d x) b^3+a \left (3 a^2+2 b^2\right ) \tan ^2(c+d x)+a \left (3 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{2 b}+\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\int \frac {\tan (c+d x) \left (\tan (c+d x) b^3+a \left (3 a^2+2 b^2\right ) \tan ^2(c+d x)+a \left (3 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\int \frac {\tan (c+d x) \left (\tan (c+d x) b^3+a \left (3 a^2+2 b^2\right ) \tan (c+d x)^2+a \left (3 a^2+b^2\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {\int -\frac {a \tan (c+d x) b^3+\left (3 a^4+2 b^2 a^2-b^4\right ) \tan ^2(c+d x)+a^2 \left (3 a^2+2 b^2\right )}{a+b \tan (c+d x)}dx}{b}+\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b d}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b d}-\frac {\int \frac {a \tan (c+d x) b^3+\left (3 a^2-b^2\right ) \left (a^2+b^2\right ) \tan ^2(c+d x)+a^2 \left (3 a^2+2 b^2\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b d}-\frac {\int \frac {a \tan (c+d x) b^3+\left (3 a^2-b^2\right ) \left (a^2+b^2\right ) \tan (c+d x)^2+a^2 \left (3 a^2+2 b^2\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4109 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {b^3 \left (a^2-b^2\right ) \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^4 \left (3 a^2+5 b^2\right ) \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {2 a b^4 x}{a^2+b^2}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {b^3 \left (a^2-b^2\right ) \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^4 \left (3 a^2+5 b^2\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {2 a b^4 x}{a^2+b^2}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {a^4 \left (3 a^2+5 b^2\right ) \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {2 a b^4 x}{a^2+b^2}-\frac {b^3 \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4100 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {a^4 \left (3 a^2+5 b^2\right ) \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}+\frac {2 a b^4 x}{a^2+b^2}-\frac {b^3 \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\left (3 a^2+b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (3 a^2+2 b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {2 a b^4 x}{a^2+b^2}-\frac {b^3 \left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {a^4 \left (3 a^2+5 b^2\right ) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}}{b}}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \tan ^3(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}\) |
Input:
Int[Tan[c + d*x]^5/(a + b*Tan[c + d*x])^2,x]
Output:
-((a^2*Tan[c + d*x]^3)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))) + (((3*a^2 + b^2)*Tan[c + d*x]^2)/(2*b*d) - (-(((2*a*b^4*x)/(a^2 + b^2) - (b^3*(a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)*d) + (a^4*(3*a^2 + 5*b^2)*Log[a + b* Tan[c + d*x]])/(b*(a^2 + b^2)*d))/b) + (a*(3*a^2 + 2*b^2)*Tan[c + d*x])/(b *d))/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 = 0.78 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(\frac {-\frac {-\frac {b \tan \left (d x +c \right )^{2}}{2}+2 a \tan \left (d x +c \right )}{b^{3}}+\frac {a^{4} \left (3 a^{2}+5 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2}}+\frac {a^{5}}{b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(142\) |
| default | \(\frac {-\frac {-\frac {b \tan \left (d x +c \right )^{2}}{2}+2 a \tan \left (d x +c \right )}{b^{3}}+\frac {a^{4} \left (3 a^{2}+5 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{4} \left (a^{2}+b^{2}\right )^{2}}+\frac {a^{5}}{b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(142\) |
| norman | \(\frac {\frac {\left (3 a^{4}+2 b^{2} a^{2}\right ) a}{d \,b^{4} \left (a^{2}+b^{2}\right )}+\frac {\tan \left (d x +c \right )^{3}}{2 b d}-\frac {3 a \tan \left (d x +c \right )^{2}}{2 b^{2} d}+\frac {2 b \,a^{2} x}{a^{4}+2 b^{2} a^{2}+b^{4}}+\frac {2 b^{2} a x \tan \left (d x +c \right )}{a^{4}+2 b^{2} a^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {a^{4} \left (3 a^{2}+5 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b^{4} d}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}\) | \(225\) |
| parallelrisch | \(\frac {6 a^{7}+4 b^{6} a x \tan \left (d x +c \right ) d +4 b^{5} a^{2} x d +4 a^{3} b^{4}+10 a^{5} b^{2}+\ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{2} b^{5}+6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{6} b +10 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{4} b^{3}+\tan \left (d x +c \right )^{3} b^{7}+6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{7}-\ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) b^{7}+2 \tan \left (d x +c \right )^{3} a^{2} b^{5}-6 \tan \left (d x +c \right )^{2} a^{3} b^{4}+\ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{3} b^{4}-3 \tan \left (d x +c \right )^{2} a \,b^{6}-\ln \left (1+\tan \left (d x +c \right )^{2}\right ) a \,b^{6}-3 \tan \left (d x +c \right )^{2} a^{5} b^{2}+\tan \left (d x +c \right )^{3} a^{4} b^{3}+10 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{5} b^{2}}{2 \left (a +b \tan \left (d x +c \right )\right ) b^{4} d \left (a^{2}+b^{2}\right )^{2}}\) | \(329\) |
| risch | \(\frac {i x}{2 i a b -a^{2}+b^{2}}+\frac {6 i a^{2} x}{b^{4}}+\frac {6 i a^{2} c}{b^{4} d}-\frac {2 i x}{b^{2}}-\frac {2 i c}{b^{2} d}-\frac {6 i a^{6} x}{b^{4} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {6 i a^{6} c}{b^{4} d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {10 i a^{4} x}{b^{2} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {10 i a^{4} c}{b^{2} d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}-\frac {2 i \left (2 a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+i b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-i b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{5} {\mathrm e}^{4 i \left (d x +c \right )}-4 i a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-2 i a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 a^{3} b^{2}+2 a \,b^{4}+6 a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{5}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left (i b +a \right ) \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) b^{3} d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{b^{4} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}+\frac {3 a^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{4} d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}+\frac {5 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{2} d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}\) | \(582\) |
Input:
int(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/b^3*(-1/2*b*tan(d*x+c)^2+2*a*tan(d*x+c))+1/b^4*a^4*(3*a^2+5*b^2)/( a^2+b^2)^2*ln(a+b*tan(d*x+c))+1/b^4*a^5/(a^2+b^2)/(a+b*tan(d*x+c))+1/(a^2+ b^2)^2*(1/2*(a^2-b^2)*ln(1+tan(d*x+c)^2)+2*a*b*arctan(tan(d*x+c))))
Time = 0.13 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.73 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4 \, a^{2} b^{5} d x + 3 \, a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} - 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{7} + 5 \, a^{5} b^{2} + {\left (3 \, a^{6} b + 5 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (3 \, a^{7} + 5 \, a^{5} b^{2} + a^{3} b^{4} - a b^{6} + {\left (3 \, a^{6} b + 5 \, a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (4 \, a b^{6} d x - 6 \, a^{6} b - 7 \, a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} d\right )}} \] Input:
integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
1/2*(4*a^2*b^5*d*x + 3*a^5*b^2 + 2*a^3*b^4 + a*b^6 + (a^4*b^3 + 2*a^2*b^5 + b^7)*tan(d*x + c)^3 - 3*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*tan(d*x + c)^2 + ( 3*a^7 + 5*a^5*b^2 + (3*a^6*b + 5*a^4*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*a^7 + 5*a^5*b ^2 + a^3*b^4 - a*b^6 + (3*a^6*b + 5*a^4*b^3 + a^2*b^5 - b^7)*tan(d*x + c)) *log(1/(tan(d*x + c)^2 + 1)) + (4*a*b^6*d*x - 6*a^6*b - 7*a^4*b^3 - 2*a^2* b^5 + b^7)*tan(d*x + c))/((a^4*b^5 + 2*a^2*b^7 + b^9)*d*tan(d*x + c) + (a^ 5*b^4 + 2*a^3*b^6 + a*b^8)*d)
Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 2837, normalized size of antiderivative = 14.40 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(tan(d*x+c)**5/(a+b*tan(d*x+c))**2,x)
Output:
Piecewise((zoo*x*tan(c)**3, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((log(tan(c + d*x)**2 + 1)/(2*d) + tan(c + d*x)**4/(4*d) - tan(c + d*x)**2/(2*d))/a**2, Eq(b, 0)), (-15*I*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b** 2*d*tan(c + d*x) - 4*b**2*d) - 30*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)* *2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 15*I*d*x/(4*b**2*d*tan(c + d*x) **2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 8*log(tan(c + d*x)**2 + 1)*tan (c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2* d) + 16*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*log(tan(c + d*x)**2 + 1)/(4*b**2 *d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*tan(c + d*x)* *4/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*I*t an(c + d*x)**3/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b** 2*d) + 29*I*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d* x) - 4*b**2*d) + 22/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, -I*b)), (15*I*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)* *2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 30*d*x*tan(c + d*x)/(4*b**2*d*t an(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 15*I*d*x/(4*b**2*d* tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 8*log(tan(c + d*x) **2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d* x) - 4*b**2*d) - 16*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*t...
Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.91 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, a^{5}}{a^{3} b^{4} + a b^{6} + {\left (a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )} + \frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (3 \, a^{6} + 5 \, a^{4} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {b \tan \left (d x + c\right )^{2} - 4 \, a \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \] Input:
integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
1/2*(2*a^5/(a^3*b^4 + a*b^6 + (a^2*b^5 + b^7)*tan(d*x + c)) + 4*(d*x + c)* a*b/(a^4 + 2*a^2*b^2 + b^4) + 2*(3*a^6 + 5*a^4*b^2)*log(b*tan(d*x + c) + a )/(a^4*b^4 + 2*a^2*b^6 + b^8) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + (b*tan(d*x + c)^2 - 4*a*tan(d*x + c))/b^3)/d
Time = 0.24 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.02 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {2 \, {\left (d x + c\right )} a b}{a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} + \frac {{\left (3 \, a^{6} + 5 \, a^{4} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{4} d + 2 \, a^{2} b^{6} d + b^{8} d} + \frac {b^{2} d \tan \left (d x + c\right )^{2} - 4 \, a b d \tan \left (d x + c\right )}{2 \, b^{4} d^{2}} + \frac {a^{7} + a^{5} b^{2}}{{\left (a^{2} + b^{2}\right )}^{2} {\left (b \tan \left (d x + c\right ) + a\right )} b^{4} d} \] Input:
integrate(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
2*(d*x + c)*a*b/(a^4*d + 2*a^2*b^2*d + b^4*d) + 1/2*(a^2 - b^2)*log(tan(d* x + c)^2 + 1)/(a^4*d + 2*a^2*b^2*d + b^4*d) + (3*a^6 + 5*a^4*b^2)*log(abs( b*tan(d*x + c) + a))/(a^4*b^4*d + 2*a^2*b^6*d + b^8*d) + 1/2*(b^2*d*tan(d* x + c)^2 - 4*a*b*d*tan(d*x + c))/(b^4*d^2) + (a^7 + a^5*b^2)/((a^2 + b^2)^ 2*(b*tan(d*x + c) + a)*b^4*d)
Time = 1.37 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b^2\,d}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^6+5\,a^4\,b^2\right )}{d\,\left (a^4\,b^4+2\,a^2\,b^6+b^8\right )}-\frac {2\,a\,\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}+\frac {a^5}{b\,d\,\left (\mathrm {tan}\left (c+d\,x\right )\,b^4+a\,b^3\right )\,\left (a^2+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \] Input:
int(tan(c + d*x)^5/(a + b*tan(c + d*x))^2,x)
Output:
log(tan(c + d*x) - 1i)/(2*d*(a*b*2i + a^2 - b^2)) + (log(tan(c + d*x) + 1i )*1i)/(2*d*(2*a*b + a^2*1i - b^2*1i)) + tan(c + d*x)^2/(2*b^2*d) + (log(a + b*tan(c + d*x))*(3*a^6 + 5*a^4*b^2))/(d*(b^8 + 2*a^2*b^6 + a^4*b^4)) - ( 2*a*tan(c + d*x))/(b^3*d) + a^5/(b*d*(a*b^3 + b^4*tan(c + d*x))*(a^2 + b^2 ))
Time = 0.15 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.93 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{2} b^{5}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) b^{7}+\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{3} b^{4}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a \,b^{6}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{6} b +10 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{4} b^{3}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{7}+10 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{5} b^{2}+\tan \left (d x +c \right )^{3} a^{4} b^{3}+2 \tan \left (d x +c \right )^{3} a^{2} b^{5}+\tan \left (d x +c \right )^{3} b^{7}-3 \tan \left (d x +c \right )^{2} a^{5} b^{2}-6 \tan \left (d x +c \right )^{2} a^{3} b^{4}-3 \tan \left (d x +c \right )^{2} a \,b^{6}-6 \tan \left (d x +c \right ) a^{6} b -10 \tan \left (d x +c \right ) a^{4} b^{3}-4 \tan \left (d x +c \right ) a^{2} b^{5}+4 \tan \left (d x +c \right ) a \,b^{6} d x +4 a^{2} b^{5} d x}{2 b^{4} d \left (\tan \left (d x +c \right ) a^{4} b +2 \tan \left (d x +c \right ) a^{2} b^{3}+\tan \left (d x +c \right ) b^{5}+a^{5}+2 a^{3} b^{2}+a \,b^{4}\right )} \] Input:
int(tan(d*x+c)^5/(a+b*tan(d*x+c))^2,x)
Output:
(log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a**2*b**5 - log(tan(c + d*x)**2 + 1 )*tan(c + d*x)*b**7 + log(tan(c + d*x)**2 + 1)*a**3*b**4 - log(tan(c + d*x )**2 + 1)*a*b**6 + 6*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**6*b + 10*log( tan(c + d*x)*b + a)*tan(c + d*x)*a**4*b**3 + 6*log(tan(c + d*x)*b + a)*a** 7 + 10*log(tan(c + d*x)*b + a)*a**5*b**2 + tan(c + d*x)**3*a**4*b**3 + 2*t an(c + d*x)**3*a**2*b**5 + tan(c + d*x)**3*b**7 - 3*tan(c + d*x)**2*a**5*b **2 - 6*tan(c + d*x)**2*a**3*b**4 - 3*tan(c + d*x)**2*a*b**6 - 6*tan(c + d *x)*a**6*b - 10*tan(c + d*x)*a**4*b**3 - 4*tan(c + d*x)*a**2*b**5 + 4*tan( c + d*x)*a*b**6*d*x + 4*a**2*b**5*d*x)/(2*b**4*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))