Integrand size = 12, antiderivative size = 165 \[ \int \frac {1}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {b}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \] Output:
(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4+4*a*b*(a^2-b^2)*ln(a*cos(d*x+c)+b*sin(d* x+c))/(a^2+b^2)^4/d-1/3*b/(a^2+b^2)/d/(a+b*tan(d*x+c))^3-a*b/(a^2+b^2)^2/d /(a+b*tan(d*x+c))^2-b*(3*a^2-b^2)/(a^2+b^2)^3/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.06 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(a+b \tan (c+d x))^4} \, dx=\frac {-\frac {3 i \log (i-\tan (c+d x))}{(a+i b)^4}+\frac {3 i \log (i+\tan (c+d x))}{(a-i b)^4}+\frac {2 b \left (12 a \left (a^2-b^2\right ) \log (a+b \tan (c+d x))-\frac {\left (a^2+b^2\right ) \left (13 a^4+2 a^2 b^2+b^4+3 a b \left (7 a^2-b^2\right ) \tan (c+d x)+\left (9 a^2 b^2-3 b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3}\right )}{\left (a^2+b^2\right )^4}}{6 d} \] Input:
Integrate[(a + b*Tan[c + d*x])^(-4),x]
Output:
(((-3*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + ((3*I)*Log[I + Tan[c + d*x]] )/(a - I*b)^4 + (2*b*(12*a*(a^2 - b^2)*Log[a + b*Tan[c + d*x]] - ((a^2 + b ^2)*(13*a^4 + 2*a^2*b^2 + b^4 + 3*a*b*(7*a^2 - b^2)*Tan[c + d*x] + (9*a^2* b^2 - 3*b^4)*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3))/(a^2 + b^2)^4)/(6*d )
Time = 0.93 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3042, 3964, 3042, 4012, 3042, 4012, 3042, 4014, 3042, 4013}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a+b \tan (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \tan (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle \frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^3}dx}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^3}dx}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}-\frac {a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\frac {\frac {\frac {4 a b \left (a^2-b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{a^2+b^2}}{a^2+b^2}-\frac {b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {4 a b \left (a^2-b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{a^2+b^2}}{a^2+b^2}-\frac {b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {\frac {\frac {\frac {4 a b \left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}+\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{a^2+b^2}}{a^2+b^2}-\frac {b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}-\frac {a b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}-\frac {b}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
Input:
Int[(a + b*Tan[c + d*x])^(-4),x]
Output:
-1/3*b/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (-((a*b)/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)) + ((((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2) + (4*a* b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d))/(a^2 + b^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(a^2 + b^ 2))/(a^2 + b^2)
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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]
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]
Time = 0.76 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.11
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {b}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (3 a^{2}-b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\) | \(183\) |
| default | \(\frac {\frac {\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {b}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (3 a^{2}-b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}}{d}\) | \(183\) |
| norman | \(\frac {\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) a^{3} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {4 b^{2} a^{3} \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {b^{3} \left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) x \tan \left (d x +c \right )^{3}}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {-10 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{3 b^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {b \left (3 a^{2} b^{3}-b^{5}\right ) \tan \left (d x +c \right )^{3}}{3 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {3 b \left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {3 b^{2} \left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) a x \tan \left (d x +c \right )^{2}}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(531\) |
| risch | \(-\frac {x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 b^{2} a^{2}-b^{4}}-\frac {8 i a^{3} b x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {8 i a \,b^{3} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {8 i a^{3} b c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {8 i a \,b^{3} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {4 i b^{2} \left (-12 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+18 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+15 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+18 i a^{3} b -4 i a \,b^{3}+9 a^{4}-11 b^{2} a^{2}+2 b^{4}\right )}{3 \left (-i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} d \left (i a +b \right )^{4}}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right ) d}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(565\) |
| parallelrisch | \(-\frac {-3 x \tan \left (d x +c \right )^{3} a^{4} b^{6} d +18 x \tan \left (d x +c \right )^{3} a^{2} b^{8} d -9 x \tan \left (d x +c \right )^{2} a^{5} b^{5} d -9 x \tan \left (d x +c \right ) a^{6} b^{4} d +54 x \tan \left (d x +c \right )^{2} a^{3} b^{7} d -9 x \tan \left (d x +c \right )^{2} a \,b^{9} d +b^{10}+54 x \tan \left (d x +c \right ) a^{4} b^{6} d -9 x \tan \left (d x +c \right ) a^{2} b^{8} d -3 x \tan \left (d x +c \right )^{3} b^{10} d -3 x \,a^{7} b^{3} d +6 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{3} a^{3} b^{7}-6 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{3} a \,b^{9}-12 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{3} a^{3} b^{7}+12 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{3} a \,b^{9}+18 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{4} b^{6}+36 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{2} b^{8}+18 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{5} b^{5}+36 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b^{7}+15 a^{4} b^{6}+6 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{6} b^{4}-3 \tan \left (d x +c \right )^{2} b^{10}-6 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{4} b^{6}-12 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{6} b^{4}+21 \tan \left (d x +c \right ) a^{5} b^{5}+18 \tan \left (d x +c \right ) a^{3} b^{7}+6 \tan \left (d x +c \right )^{2} a^{2} b^{8}+9 \tan \left (d x +c \right )^{2} a^{4} b^{6}+12 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{4} b^{6}+13 a^{6} b^{4}+18 x \,a^{5} b^{5} d -3 x \,a^{3} b^{7} d -18 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{2} b^{8}-36 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{4} b^{6}-18 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{3} b^{7}-36 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{5} b^{5}+3 a^{2} b^{8}-3 \tan \left (d x +c \right ) a \,b^{9}}{3 \left (a +b \tan \left (d x +c \right )\right )^{3} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right ) b^{3} d}\) | \(736\) |
Input:
int(1/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a^2+b^2)^4*(1/2*(-4*a^3*b+4*a*b^3)*ln(1+tan(d*x+c)^2)+(a^4-6*a^2*b ^2+b^4)*arctan(tan(d*x+c)))-1/3*b/(a^2+b^2)/(a+b*tan(d*x+c))^3-b*(3*a^2-b^ 2)/(a^2+b^2)^3/(a+b*tan(d*x+c))-a*b/(a^2+b^2)^2/(a+b*tan(d*x+c))^2+4*a*b*( a^2-b^2)/(a^2+b^2)^4*ln(a+b*tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (163) = 326\).
Time = 0.13 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.10 \[ \int \frac {1}{(a+b \tan (c+d x))^4} \, dx=-\frac {24 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7} - {\left (13 \, a^{3} b^{4} - 9 \, a b^{6} + 3 \, {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4}\right )} d x - 3 \, {\left (10 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + b^{7} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{2} - 6 \, {\left (a^{6} b - a^{4} b^{3} + {\left (a^{3} b^{4} - a b^{6}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (6 \, a^{5} b^{2} - 15 \, a^{3} b^{4} + a b^{6} + 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )}{3 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \] Input:
integrate(1/(a+b*tan(d*x+c))^4,x, algorithm="fricas")
Output:
-1/3*(24*a^4*b^3 + 3*a^2*b^5 + b^7 - (13*a^3*b^4 - 9*a*b^6 + 3*(a^4*b^3 - 6*a^2*b^5 + b^7)*d*x)*tan(d*x + c)^3 - 3*(a^7 - 6*a^5*b^2 + a^3*b^4)*d*x - 3*(10*a^4*b^3 - 11*a^2*b^5 + b^7 + 3*(a^5*b^2 - 6*a^3*b^4 + a*b^6)*d*x)*t an(d*x + c)^2 - 6*(a^6*b - a^4*b^3 + (a^3*b^4 - a*b^6)*tan(d*x + c)^3 + 3* (a^4*b^3 - a^2*b^5)*tan(d*x + c)^2 + 3*(a^5*b^2 - a^3*b^4)*tan(d*x + c))*l og((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - 3*(6*a^5*b^2 - 15*a^3*b^4 + a*b^6 + 3*(a^6*b - 6*a^4*b^3 + a^2*b^5)*d*x)* tan(d*x + c))/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*tan( d*x + c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d*ta n(d*x + c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6*a^6*b^5 + 4*a^4*b^7 + a^2*b^9)*d* tan(d*x + c) + (a^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4*a^5*b^6 + a^3*b^8)*d)
Exception generated. \[ \int \frac {1}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate(1/(a+b*tan(d*x+c))**4,x)
Output:
Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr imitive'
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (163) = 326\).
Time = 0.12 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.33 \[ \int \frac {1}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, {\left (a^{3} b - a b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {13 \, a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 3 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (7 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (d x + c\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} + {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )}}{3 \, d} \] Input:
integrate(1/(a+b*tan(d*x+c))^4,x, algorithm="maxima")
Output:
1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4* a^2*b^6 + b^8) + 12*(a^3*b - a*b^3)*log(b*tan(d*x + c) + a)/(a^8 + 4*a^6*b ^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^3*b - a*b^3)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - (13*a^4*b + 2*a^2*b^ 3 + b^5 + 3*(3*a^2*b^3 - b^5)*tan(d*x + c)^2 + 3*(7*a^3*b^2 - a*b^4)*tan(d *x + c))/(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6 + (a^6*b^3 + 3*a^4*b^5 + 3 *a^2*b^7 + b^9)*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^ 8)*tan(d*x + c)^2 + 3*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*tan(d*x + c)))/d
Time = 0.24 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(a+b \tan (c+d x))^4} \, dx=\frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d} - \frac {2 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d} + \frac {4 \, {\left (a^{3} b^{2} - a b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b d + 4 \, a^{6} b^{3} d + 6 \, a^{4} b^{5} d + 4 \, a^{2} b^{7} d + b^{9} d} - \frac {13 \, a^{6} b + 15 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7} + 3 \, {\left (3 \, a^{4} b^{3} + 2 \, a^{2} b^{5} - b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (7 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - a b^{6}\right )} \tan \left (d x + c\right )}{3 \, {\left (a^{2} + b^{2}\right )}^{4} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} d} \] Input:
integrate(1/(a+b*tan(d*x+c))^4,x, algorithm="giac")
Output:
(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8*d + 4*a^6*b^2*d + 6*a^4*b^4*d + 4*a ^2*b^6*d + b^8*d) - 2*(a^3*b - a*b^3)*log(tan(d*x + c)^2 + 1)/(a^8*d + 4*a ^6*b^2*d + 6*a^4*b^4*d + 4*a^2*b^6*d + b^8*d) + 4*(a^3*b^2 - a*b^4)*log(ab s(b*tan(d*x + c) + a))/(a^8*b*d + 4*a^6*b^3*d + 6*a^4*b^5*d + 4*a^2*b^7*d + b^9*d) - 1/3*(13*a^6*b + 15*a^4*b^3 + 3*a^2*b^5 + b^7 + 3*(3*a^4*b^3 + 2 *a^2*b^5 - b^7)*tan(d*x + c)^2 + 3*(7*a^5*b^2 + 6*a^3*b^4 - a*b^6)*tan(d*x + c))/((a^2 + b^2)^4*(b*tan(d*x + c) + a)^3*d)
Time = 1.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.02 \[ \int \frac {1}{(a+b \tan (c+d x))^4} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {4\,a\,b}{{\left (a^2+b^2\right )}^3}-\frac {8\,a\,b^3}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}+\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (b^5-3\,a^2\,b^3\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}-\frac {13\,a^4\,b+2\,a^2\,b^3+b^5}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a\,b^4-7\,a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \] Input:
int(1/(a + b*tan(c + d*x))^4,x)
Output:
(log(a + b*tan(c + d*x))*((4*a*b)/(a^2 + b^2)^3 - (8*a*b^3)/(a^2 + b^2)^4) )/d + log(tan(c + d*x) - 1i)/(2*d*(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a ^2*b^2*6i)) + (log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)) + ((tan(c + d*x)^2*(b^5 - 3*a^2*b^3))/(a^6 + b^6 + 3*a ^2*b^4 + 3*a^4*b^2) - (13*a^4*b + b^5 + 2*a^2*b^3)/(3*(a^6 + b^6 + 3*a^2*b ^4 + 3*a^4*b^2)) + (tan(c + d*x)*(a*b^4 - 7*a^3*b^2))/(a^6 + b^6 + 3*a^2*b ^4 + 3*a^4*b^2))/(d*(a^3 + b^3*tan(c + d*x)^3 + 3*a*b^2*tan(c + d*x)^2 + 3 *a^2*b*tan(c + d*x)))
Time = 0.20 (sec) , antiderivative size = 935, normalized size of antiderivative = 5.67 \[ \int \frac {1}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(1/(a+b*tan(d*x+c))^4,x)
Output:
( - 6*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3*a**4*b**4 + 6*log(tan(c + d *x)**2 + 1)*tan(c + d*x)**3*a**2*b**6 - 18*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a**5*b**3 + 18*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a**3*b** 5 - 18*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a**6*b**2 + 18*log(tan(c + d* x)**2 + 1)*tan(c + d*x)*a**4*b**4 - 6*log(tan(c + d*x)**2 + 1)*a**7*b + 6* log(tan(c + d*x)**2 + 1)*a**5*b**3 + 12*log(tan(c + d*x)*b + a)*tan(c + d* x)**3*a**4*b**4 - 12*log(tan(c + d*x)*b + a)*tan(c + d*x)**3*a**2*b**6 + 3 6*log(tan(c + d*x)*b + a)*tan(c + d*x)**2*a**5*b**3 - 36*log(tan(c + d*x)* b + a)*tan(c + d*x)**2*a**3*b**5 + 36*log(tan(c + d*x)*b + a)*tan(c + d*x) *a**6*b**2 - 36*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**4*b**4 + 12*log(ta n(c + d*x)*b + a)*a**7*b - 12*log(tan(c + d*x)*b + a)*a**5*b**3 + 3*tan(c + d*x)**3*a**5*b**3*d*x + 3*tan(c + d*x)**3*a**4*b**4 - 18*tan(c + d*x)**3 *a**3*b**5*d*x + 2*tan(c + d*x)**3*a**2*b**6 + 3*tan(c + d*x)**3*a*b**7*d* x - tan(c + d*x)**3*b**8 + 9*tan(c + d*x)**2*a**6*b**2*d*x - 54*tan(c + d* x)**2*a**4*b**4*d*x + 9*tan(c + d*x)**2*a**2*b**6*d*x + 9*tan(c + d*x)*a** 7*b*d*x - 12*tan(c + d*x)*a**6*b**2 - 54*tan(c + d*x)*a**5*b**3*d*x - 12*t an(c + d*x)*a**4*b**4 + 9*tan(c + d*x)*a**3*b**5*d*x + 3*a**8*d*x - 10*a** 7*b - 18*a**6*b**2*d*x - 13*a**5*b**3 + 3*a**4*b**4*d*x - 4*a**3*b**5 - a* b**7)/(3*a*d*(tan(c + d*x)**3*a**8*b**3 + 4*tan(c + d*x)**3*a**6*b**5 + 6* tan(c + d*x)**3*a**4*b**7 + 4*tan(c + d*x)**3*a**2*b**9 + tan(c + d*x)*...