Integrand size = 19, antiderivative size = 172 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {a}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {a^2-b^2}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \] Output:
4*a*b*(a^2-b^2)*x/(a^2+b^2)^4-(a^4-6*a^2*b^2+b^4)*ln(a*cos(d*x+c)+b*sin(d* x+c))/(a^2+b^2)^4/d+1/3*a/(a^2+b^2)/d/(a+b*tan(d*x+c))^3+1/2*(a^2-b^2)/(a^ 2+b^2)^2/d/(a+b*tan(d*x+c))^2+a*(a^2-3*b^2)/(a^2+b^2)^3/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 4.08 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.76 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \log (i-\tan (c+d x))}{(-i a+b)^3}+\frac {3 \log (i+\tan (c+d x))}{(i a+b)^3}-\frac {3 b \left (\left (-6 a^2+2 b^2\right ) \log (a+b \tan (c+d x))+\frac {\left (a^2+b^2\right ) \left (5 a^2+b^2+4 a b \tan (c+d x)\right )}{(a+b \tan (c+d x))^2}\right )}{\left (a^2+b^2\right )^3}+a \left (\frac {3 i \log (i-\tan (c+d x))}{(a+i b)^4}-\frac {3 i \log (i+\tan (c+d x))}{(a-i b)^4}+\frac {2 b \left (-12 a \left (a^2-b^2\right ) \log (a+b \tan (c+d x))+\frac {\left (a^2+b^2\right ) \left (13 a^4+2 a^2 b^2+b^4+3 a b \left (7 a^2-b^2\right ) \tan (c+d x)+\left (9 a^2 b^2-3 b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3}\right )}{\left (a^2+b^2\right )^4}\right )}{6 b d} \] Input:
Integrate[Tan[c + d*x]/(a + b*Tan[c + d*x])^4,x]
Output:
((3*Log[I - Tan[c + d*x]])/((-I)*a + b)^3 + (3*Log[I + Tan[c + d*x]])/(I*a + b)^3 - (3*b*((-6*a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]] + ((a^2 + b^2)*(5 *a^2 + b^2 + 4*a*b*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2))/(a^2 + b^2)^3 + a*(((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*b*d)
Time = 0.96 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3042, 4012, 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.
| \(\displaystyle \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\int \frac {b+a \tan (c+d x)}{(a+b \tan (c+d x))^3}dx}{a^2+b^2}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b+a \tan (c+d x)}{(a+b \tan (c+d x))^3}dx}{a^2+b^2}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int \frac {2 a b+\left (a^2-b^2\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}+\frac {a^2-b^2}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 a b+\left (a^2-b^2\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}+\frac {a^2-b^2}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}+\frac {a^2-b^2}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}+\frac {a^2-b^2}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\frac {\frac {\frac {4 a b x \left (a^2-b^2\right )}{a^2+b^2}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a^2+b^2}+\frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}+\frac {a^2-b^2}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {4 a b x \left (a^2-b^2\right )}{a^2+b^2}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a^2+b^2}+\frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}+\frac {a^2-b^2}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}+\frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {a}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {\frac {a^2-b^2}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {4 a b x \left (a^2-b^2\right )}{a^2+b^2}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}}{a^2+b^2}}{a^2+b^2}}{a^2+b^2}\) |
Input:
Int[Tan[c + d*x]/(a + b*Tan[c + d*x])^4,x]
Output:
a/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + ((a^2 - b^2)/(2*(a^2 + b^2)*d *(a + b*Tan[c + d*x])^2) + (((4*a*b*(a^2 - b^2)*x)/(a^2 + b^2) - ((a^4 - 6 *a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d))/(a^ 2 + b^2) + (a*(a^2 - 3*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[(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 = 0.88 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.11
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{2}-b^{2}}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(191\) |
| default | \(\frac {\frac {\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a^{2}-b^{2}}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(191\) |
| norman | \(\frac {\frac {a \left (a^{2} b^{3}-3 b^{5}\right ) \tan \left (d x +c \right )^{2}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) b}+\frac {a \left (11 a^{4} b^{3}-14 b^{5} a^{2}-b^{7}\right )}{6 b^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) d}+\frac {\left (5 a^{4} b^{3}-12 b^{5} a^{2}-b^{7}\right ) \tan \left (d x +c \right )}{2 d \,b^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {4 \left (a^{2}-b^{2}\right ) a^{4} b x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \tan \left (d x +c \right )^{2}}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )^{3}}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {\left (a^{4}-6 b^{2} a^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(541\) |
| risch | \(\frac {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 b^{2} a^{2}-b^{4}}+\frac {2 i a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {12 i a^{2} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {2 i b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {2 i a^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {12 i a^{2} b^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {2 i b^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {2 b \left (6 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+22 i a^{3} b^{2}-9 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}+18 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+24 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+9 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-9 i a^{5}-13 i a \,b^{4}-9 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+18 a^{4} b -26 a^{2} b^{3}\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 {a^{4} \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 )}+\frac {6 a^{2} b^{2} \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 )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(792\) |
| parallelrisch | \(\frac {-a \,b^{9}+24 x \tan \left (d x +c \right )^{3} a^{3} b^{7} d -24 x \tan \left (d x +c \right )^{3} a \,b^{9} d +72 x \tan \left (d x +c \right )^{2} a^{4} b^{6} d -72 x \tan \left (d x +c \right )^{2} a^{2} b^{8} d +72 x \tan \left (d x +c \right ) a^{5} b^{5} d -72 x \tan \left (d x +c \right ) a^{3} b^{7} d -24 x \,a^{4} b^{6} d +11 a^{7} b^{3}-3 a^{5} b^{5}-15 a^{3} b^{7}+3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{3} a^{4} b^{6}-18 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{3} a^{2} b^{8}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{3} a^{4} b^{6}+36 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{3} a^{2} b^{8}+9 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{5} b^{5}-54 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{3} b^{7}+9 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a \,b^{9}-18 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{5} b^{5}+108 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{3} b^{7}-18 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a \,b^{9}+9 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{6} b^{4}-54 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{4} b^{6}+9 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{2} b^{8}-18 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{6} b^{4}+108 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{4} b^{6}-18 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{8}+24 x \,a^{6} b^{4} d +3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{7} b^{3}-18 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{5} b^{5}-3 \tan \left (d x +c \right ) b^{10}+3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{3} b^{7}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{7} b^{3}+36 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{5} b^{5}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{3} b^{7}+3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{3} b^{10}+6 \tan \left (d x +c \right )^{2} a^{5} b^{5}-12 \tan \left (d x +c \right )^{2} a^{3} b^{7}-18 \tan \left (d x +c \right )^{2} a \,b^{9}+15 \tan \left (d x +c \right ) a^{6} b^{4}-21 \tan \left (d x +c \right ) a^{4} b^{6}-39 \tan \left (d x +c \right ) a^{2} b^{8}-6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{3} b^{10}}{6 \left (a +b \tan \left (d x +c \right )\right )^{3} b^{3} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\) | \(884\) |
Input:
int(tan(d*x+c)/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a^2+b^2)^4*(1/2*(a^4-6*a^2*b^2+b^4)*ln(1+tan(d*x+c)^2)+(4*a^3*b-4* a*b^3)*arctan(tan(d*x+c)))+1/2*(a^2-b^2)/(a^2+b^2)^2/(a+b*tan(d*x+c))^2+1/ 3*a/(a^2+b^2)/(a+b*tan(d*x+c))^3-(a^4-6*a^2*b^2+b^4)/(a^2+b^2)^4*ln(a+b*ta n(d*x+c))+a*(a^2-3*b^2)/(a^2+b^2)^3/(a+b*tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (168) = 336\).
Time = 0.11 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.07 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {27 \, a^{5} b^{2} - 18 \, a^{3} b^{4} - a b^{6} - {\left (11 \, a^{4} b^{3} - 30 \, a^{2} b^{5} + 3 \, b^{7} - 24 \, {\left (a^{3} b^{4} - a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 24 \, {\left (a^{6} b - a^{4} b^{3}\right )} d x - 3 \, {\left (9 \, a^{5} b^{2} - 26 \, a^{3} b^{4} + 9 \, a b^{6} - 24 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (6 \, a^{6} b - 23 \, a^{4} b^{3} + 16 \, a^{2} b^{5} + b^{7} - 24 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \] Input:
integrate(tan(d*x+c)/(a+b*tan(d*x+c))^4,x, algorithm="fricas")
Output:
1/6*(27*a^5*b^2 - 18*a^3*b^4 - a*b^6 - (11*a^4*b^3 - 30*a^2*b^5 + 3*b^7 - 24*(a^3*b^4 - a*b^6)*d*x)*tan(d*x + c)^3 + 24*(a^6*b - a^4*b^3)*d*x - 3*(9 *a^5*b^2 - 26*a^3*b^4 + 9*a*b^6 - 24*(a^4*b^3 - a^2*b^5)*d*x)*tan(d*x + c) ^2 - 3*(a^7 - 6*a^5*b^2 + a^3*b^4 + (a^4*b^3 - 6*a^2*b^5 + b^7)*tan(d*x + c)^3 + 3*(a^5*b^2 - 6*a^3*b^4 + a*b^6)*tan(d*x + c)^2 + 3*(a^6*b - 6*a^4*b ^3 + a^2*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)) - 3*(6*a^6*b - 23*a^4*b^3 + 16*a^2*b^5 + b^7 - 24*(a^5*b^2 - a^3*b^4)*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*tan(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. \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate(tan(d*x+c)/(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. 394 vs. \(2 (168) = 336\).
Time = 0.11 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.29 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {11 \, a^{5} - 14 \, a^{3} b^{2} - a b^{4} + 6 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{4} b - 12 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} + {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )}}{6 \, d} \] Input:
integrate(tan(d*x+c)/(a+b*tan(d*x+c))^4,x, algorithm="maxima")
Output:
1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^4 - 6*a^2*b^2 + b^4)*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) + 3*(a^4 - 6*a^2*b^2 + b^4)*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) + (11*a^5 - 14 *a^3*b^2 - a*b^4 + 6*(a^3*b^2 - 3*a*b^4)*tan(d*x + c)^2 + 3*(5*a^4*b - 12* a^2*b^3 - b^5)*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 = 0.29 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.87 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {4 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{8} d + 4 \, a^{6} b^{2} d + 6 \, a^{4} b^{4} d + 4 \, a^{2} b^{6} d + b^{8} d\right )}} - \frac {{\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b d + 4 \, a^{6} b^{3} d + 6 \, a^{4} b^{5} d + 4 \, a^{2} b^{7} d + b^{9} d} + \frac {11 \, a^{7} - 3 \, a^{5} b^{2} - 15 \, a^{3} b^{4} - a b^{6} + 6 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{6} b - 7 \, a^{4} b^{3} - 13 \, a^{2} b^{5} - b^{7}\right )} \tan \left (d x + c\right )}{6 \, {\left (a^{2} + b^{2}\right )}^{4} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} d} \] Input:
integrate(tan(d*x+c)/(a+b*tan(d*x+c))^4,x, algorithm="giac")
Output:
4*(a^3*b - a*b^3)*(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) + 1/2*(a^4 - 6*a^2*b^2 + b^4)*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) - (a^4*b - 6*a^2*b^3 + b ^5)*log(abs(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) + 1/6*(11*a^7 - 3*a^5*b^2 - 15*a^3*b^4 - a*b^6 + 6*(a^5 *b^2 - 2*a^3*b^4 - 3*a*b^6)*tan(d*x + c)^2 + 3*(5*a^6*b - 7*a^4*b^3 - 13*a ^2*b^5 - b^7)*tan(d*x + c))/((a^2 + b^2)^4*(b*tan(d*x + c) + a)^3*d)
Time = 1.32 (sec) , antiderivative size = 355, normalized size of antiderivative = 2.06 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\frac {-11\,a^5+14\,a^3\,b^2+a\,b^4}{6\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-5\,a^4\,b+12\,a^2\,b^3+b^5\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a\,b^4-a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {1}{{\left (a^2+b^2\right )}^2}-\frac {8\,b^2}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^4}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \] Input:
int(tan(c + d*x)/(a + b*tan(c + d*x))^4,x)
Output:
(log(tan(c + d*x) + 1i)*1i)/(2*d*(4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^ 2*b^2*6i)) + log(tan(c + d*x) - 1i)/(2*d*(a^3*b*4i - a*b^3*4i + a^4 + b^4 - 6*a^2*b^2)) - ((a*b^4 - 11*a^5 + 14*a^3*b^2)/(6*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(b^5 - 5*a^4*b + 12*a^2*b^3))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^2*(3*a*b^4 - 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))) - (log(a + b*tan(c + d*x))*(1/(a^2 + b^2)^ 2 - (8*b^2)/(a^2 + b^2)^3 + (8*b^4)/(a^2 + b^2)^4))/d
Time = 0.18 (sec) , antiderivative size = 1074, normalized size of antiderivative = 6.24 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(tan(d*x+c)/(a+b*tan(d*x+c))^4,x)
Output:
(3*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3*a**4*b**3 - 18*log(tan(c + d*x )**2 + 1)*tan(c + d*x)**3*a**2*b**5 + 3*log(tan(c + d*x)**2 + 1)*tan(c + d *x)**3*b**7 + 9*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a**5*b**2 - 54*lo g(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a**3*b**4 + 9*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a*b**6 + 9*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a**6* b - 54*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a**4*b**3 + 9*log(tan(c + d*x )**2 + 1)*tan(c + d*x)*a**2*b**5 + 3*log(tan(c + d*x)**2 + 1)*a**7 - 18*lo g(tan(c + d*x)**2 + 1)*a**5*b**2 + 3*log(tan(c + d*x)**2 + 1)*a**3*b**4 - 6*log(tan(c + d*x)*b + a)*tan(c + d*x)**3*a**4*b**3 + 36*log(tan(c + d*x)* b + a)*tan(c + d*x)**3*a**2*b**5 - 6*log(tan(c + d*x)*b + a)*tan(c + d*x)* *3*b**7 - 18*log(tan(c + d*x)*b + a)*tan(c + d*x)**2*a**5*b**2 + 108*log(t an(c + d*x)*b + a)*tan(c + d*x)**2*a**3*b**4 - 18*log(tan(c + d*x)*b + a)* tan(c + d*x)**2*a*b**6 - 18*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**6*b + 108*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**4*b**3 - 18*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**2*b**5 - 6*log(tan(c + d*x)*b + a)*a**7 + 36*log(tan (c + d*x)*b + a)*a**5*b**2 - 6*log(tan(c + d*x)*b + a)*a**3*b**4 - 2*tan(c + d*x)**3*a**4*b**3 + 24*tan(c + d*x)**3*a**3*b**4*d*x + 4*tan(c + d*x)** 3*a**2*b**5 - 24*tan(c + d*x)**3*a*b**6*d*x + 6*tan(c + d*x)**3*b**7 + 72* tan(c + d*x)**2*a**4*b**3*d*x - 72*tan(c + d*x)**2*a**2*b**5*d*x + 9*tan(c + d*x)*a**6*b + 72*tan(c + d*x)*a**5*b**2*d*x - 9*tan(c + d*x)*a**4*b*...