Integrand size = 19, antiderivative size = 129 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^2-b^2}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:
b*(3*a^2-b^2)*x/(a^2+b^2)^3-a*(a^2-3*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a ^2+b^2)^3/d+1/2*a/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+(a^2-b^2)/(a^2+b^2)^2/d/( a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.70 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.81 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {-\frac {i \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {i \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {4 a b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}+a \left (\frac {i \log (i-\tan (c+d x))}{(a+i b)^3}-\frac {\log (i+\tan (c+d x))}{(i a+b)^3}+\frac {b \left (\left (-6 a^2+2 b^2\right ) \log (a+b \tan (c+d x))+\frac {\left (a^2+b^2\right ) \left (5 a^2+b^2+4 a b \tan (c+d x)\right )}{(a+b \tan (c+d x))^2}\right )}{\left (a^2+b^2\right )^3}\right )}{2 b d} \] Input:
Integrate[Tan[c + d*x]/(a + b*Tan[c + d*x])^3,x]
Output:
(((-I)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (4*a*b*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^2 - (2*b)/((a^2 + b^ 2)*(a + b*Tan[c + d*x])) + a*((I*Log[I - Tan[c + d*x]])/(a + I*b)^3 - Log[ I + Tan[c + d*x]]/(I*a + b)^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))/(2*b*d)
Time = 0.70 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {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.
| \(\displaystyle \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\int \frac {b+a \tan (c+d x)}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}+\frac {a}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b+a \tan (c+d x)}{(a+b \tan (c+d x))^2}dx}{a^2+b^2}+\frac {a}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {\int \frac {2 a b+\left (a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a^2-b^2}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}+\frac {a}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 a b+\left (a^2-b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {a^2-b^2}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}+\frac {a}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {\frac {\frac {b x \left (3 a^2-b^2\right )}{a^2+b^2}-\frac {a \left (a^2-3 b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a^2+b^2}+\frac {a^2-b^2}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}+\frac {a}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {b x \left (3 a^2-b^2\right )}{a^2+b^2}-\frac {a \left (a^2-3 b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a^2+b^2}}{a^2+b^2}+\frac {a^2-b^2}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}+\frac {a}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {a}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {a^2-b^2}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {b x \left (3 a^2-b^2\right )}{a^2+b^2}-\frac {a \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}}{a^2+b^2}}{a^2+b^2}\) |
Input:
Int[Tan[c + d*x]/(a + b*Tan[c + d*x])^3,x]
Output:
a/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (((b*(3*a^2 - b^2)*x)/(a^2 + b^2) - (a*(a^2 - 3*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2) *d))/(a^2 + b^2) + (a^2 - b^2)/((a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(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 = 0.61 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2}-b^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(147\) |
| default | \(\frac {\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2}-b^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(147\) |
| norman | \(\frac {\frac {\left (b^{2} a^{2}-b^{4}\right ) \tan \left (d x +c \right )}{d b \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}+\frac {b^{3} \left (3 a^{2}-b^{2}\right ) x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {a \left (3 b^{2} a^{2}-b^{4}\right )}{2 d \,b^{2} \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}+\frac {2 b^{2} \left (3 a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(344\) |
| risch | \(\frac {i x}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {2 i a^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 i a \,b^{2} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 i a^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i a \,b^{2} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i b \left (-i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 i a^{2} b -i b^{3}+2 a^{3}-a \,b^{2}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} \left (i b +a \right )^{2} d \left (-i b +a \right )^{3}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(419\) |
| parallelrisch | \(\frac {6 x \tan \left (d x +c \right )^{2} a^{3} b^{4} d -2 x \tan \left (d x +c \right )^{2} a \,b^{6} d +12 x \tan \left (d x +c \right ) a^{4} b^{3} d -4 x \tan \left (d x +c \right ) a^{2} b^{5} d +6 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{2} b^{5}-6 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{3} b^{4}+12 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b^{4}+\ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{4} b^{3}-2 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{6} b +6 x \,a^{5} b^{2} d -2 x \,a^{3} b^{4} d -3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right )^{2} a^{2} b^{5}+\ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{6} b -2 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )^{2} a^{4} b^{3}+2 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) \tan \left (d x +c \right ) a^{5} b^{2}-4 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{5} b^{2}-\tan \left (d x +c \right )^{2} a^{4} b^{3}+2 a^{6} b +2 a^{4} b^{3}-3 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{4} b^{3}+6 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{4} b^{3}+\tan \left (d x +c \right )^{2} b^{7}}{2 \left (a +b \tan \left (d x +c \right )\right )^{2} a \left (a^{2}+b^{2}\right ) b d \left (a^{4}+2 b^{2} a^{2}+b^{4}\right )}\) | \(457\) |
Input:
int(tan(d*x+c)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a^2+b^2)^3*(1/2*(a^3-3*a*b^2)*ln(1+tan(d*x+c)^2)+(3*a^2*b-b^3)*arc tan(tan(d*x+c)))+1/2*a/(a^2+b^2)/(a+b*tan(d*x+c))^2+(a^2-b^2)/(a^2+b^2)^2/ (a+b*tan(d*x+c))-a*(a^2-3*b^2)/(a^2+b^2)^3*ln(a+b*tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (127) = 254\).
Time = 0.09 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.54 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {5 \, a^{3} b^{2} - a b^{4} + 2 \, {\left (3 \, a^{4} b - a^{2} b^{3}\right )} d x - {\left (3 \, a^{3} b^{2} - 3 \, a b^{4} - 2 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left (a^{5} - 3 \, a^{3} b^{2} + {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (2 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5} - 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} d\right )}} \] Input:
integrate(tan(d*x+c)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
1/2*(5*a^3*b^2 - a*b^4 + 2*(3*a^4*b - a^2*b^3)*d*x - (3*a^3*b^2 - 3*a*b^4 - 2*(3*a^2*b^3 - b^5)*d*x)*tan(d*x + c)^2 - (a^5 - 3*a^3*b^2 + (a^3*b^2 - 3*a*b^4)*tan(d*x + c)^2 + 2*(a^4*b - 3*a^2*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)) - 2*(2*a^4*b - 3*a^2*b^3 + b^5 - 2*(3*a^3*b^2 - a*b^4)*d*x)*tan(d*x + c))/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*d*tan(d*x + c)^2 + 2*(a^7*b + 3*a^5*b^3 + 3*a ^3*b^5 + a*b^7)*d*tan(d*x + c) + (a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d )
Exception generated. \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate(tan(d*x+c)/(a+b*tan(d*x+c))**3,x)
Output:
Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr imitive'
Time = 0.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.96 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {3 \, a^{3} - a b^{2} + 2 \, {\left (a^{2} b - b^{3}\right )} \tan \left (d x + c\right )}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )}}{2 \, d} \] Input:
integrate(tan(d*x+c)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*(2*(3*a^2*b - b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*( a^3 - 3*a*b^2)*log(b*tan(d*x + c) + a)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (a^3 - 3*a*b^2)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (3*a^3 - a*b^2 + 2*(a^2*b - b^3)*tan(d*x + c))/(a^6 + 2*a^4*b^2 + a ^2*b^4 + (a^4*b^2 + 2*a^2*b^4 + b^6)*tan(d*x + c)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*tan(d*x + c)))/d
Time = 0.23 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.76 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {{\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} - \frac {{\left (a^{3} b - 3 \, a b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b d + 3 \, a^{4} b^{3} d + 3 \, a^{2} b^{5} d + b^{7} d} + \frac {3 \, a^{5} + 2 \, a^{3} b^{2} - a b^{4} + 2 \, {\left (a^{4} b - b^{5}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} d} \] Input:
integrate(tan(d*x+c)/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
(3*a^2*b - b^3)*(d*x + c)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4*d + b^6*d) + 1/ 2*(a^3 - 3*a*b^2)*log(tan(d*x + c)^2 + 1)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4 *d + b^6*d) - (a^3*b - 3*a*b^3)*log(abs(b*tan(d*x + c) + a))/(a^6*b*d + 3* a^4*b^3*d + 3*a^2*b^5*d + b^7*d) + 1/2*(3*a^5 + 2*a^3*b^2 - a*b^4 + 2*(a^4 *b - b^5)*tan(d*x + c))/((a^2 + b^2)^3*(b*tan(d*x + c) + a)^2*d)
Time = 1.22 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.74 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {a\,b^2-3\,a^3}{2\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^2\,b-b^3\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}-\frac {a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-3\,b^2\right )}{d\,{\left (a^2+b^2\right )}^3}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )} \] Input:
int(tan(c + d*x)/(a + b*tan(c + d*x))^3,x)
Output:
- ((a*b^2 - 3*a^3)/(2*(a^4 + b^4 + 2*a^2*b^2)) - (tan(c + d*x)*(a^2*b - b^ 3))/(a^4 + b^4 + 2*a^2*b^2))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*tan(c + d*x))) - (log(tan(c + d*x) - 1i)*1i)/(2*d*(a*b^2*3i + 3*a^2*b - a^3*1i - b ^3)) - log(tan(c + d*x) + 1i)/(2*d*(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)) - (a*log(a + b*tan(c + d*x))*(a^2 - 3*b^2))/(d*(a^2 + b^2)^3)
Time = 0.24 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.23 \[ \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a^{4} b^{2}-3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right )^{2} a^{2} b^{4}+2 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{5} b -6 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{3} b^{3}+\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{6}-3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{4} b^{2}-2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a^{4} b^{2}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right )^{2} a^{2} b^{4}-4 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{5} b +12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{3} b^{3}-2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{6}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{4} b^{2}-\tan \left (d x +c \right )^{2} a^{4} b^{2}+6 \tan \left (d x +c \right )^{2} a^{3} b^{3} d x -2 \tan \left (d x +c \right )^{2} a \,b^{5} d x +\tan \left (d x +c \right )^{2} b^{6}+12 \tan \left (d x +c \right ) a^{4} b^{2} d x -4 \tan \left (d x +c \right ) a^{2} b^{4} d x +2 a^{6}+6 a^{5} b d x +2 a^{4} b^{2}-2 a^{3} b^{3} d x}{2 a d \left (\tan \left (d x +c \right )^{2} a^{6} b^{2}+3 \tan \left (d x +c \right )^{2} a^{4} b^{4}+3 \tan \left (d x +c \right )^{2} a^{2} b^{6}+\tan \left (d x +c \right )^{2} b^{8}+2 \tan \left (d x +c \right ) a^{7} b +6 \tan \left (d x +c \right ) a^{5} b^{3}+6 \tan \left (d x +c \right ) a^{3} b^{5}+2 \tan \left (d x +c \right ) a \,b^{7}+a^{8}+3 a^{6} b^{2}+3 a^{4} b^{4}+a^{2} b^{6}\right )} \] Input:
int(tan(d*x+c)/(a+b*tan(d*x+c))^3,x)
Output:
(log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2*a**4*b**2 - 3*log(tan(c + d*x)** 2 + 1)*tan(c + d*x)**2*a**2*b**4 + 2*log(tan(c + d*x)**2 + 1)*tan(c + d*x) *a**5*b - 6*log(tan(c + d*x)**2 + 1)*tan(c + d*x)*a**3*b**3 + log(tan(c + d*x)**2 + 1)*a**6 - 3*log(tan(c + d*x)**2 + 1)*a**4*b**2 - 2*log(tan(c + d *x)*b + a)*tan(c + d*x)**2*a**4*b**2 + 6*log(tan(c + d*x)*b + a)*tan(c + d *x)**2*a**2*b**4 - 4*log(tan(c + d*x)*b + a)*tan(c + d*x)*a**5*b + 12*log( tan(c + d*x)*b + a)*tan(c + d*x)*a**3*b**3 - 2*log(tan(c + d*x)*b + a)*a** 6 + 6*log(tan(c + d*x)*b + a)*a**4*b**2 - tan(c + d*x)**2*a**4*b**2 + 6*ta n(c + d*x)**2*a**3*b**3*d*x - 2*tan(c + d*x)**2*a*b**5*d*x + tan(c + d*x)* *2*b**6 + 12*tan(c + d*x)*a**4*b**2*d*x - 4*tan(c + d*x)*a**2*b**4*d*x + 2 *a**6 + 6*a**5*b*d*x + 2*a**4*b**2 - 2*a**3*b**3*d*x)/(2*a*d*(tan(c + d*x) **2*a**6*b**2 + 3*tan(c + d*x)**2*a**4*b**4 + 3*tan(c + d*x)**2*a**2*b**6 + tan(c + d*x)**2*b**8 + 2*tan(c + d*x)*a**7*b + 6*tan(c + d*x)*a**5*b**3 + 6*tan(c + d*x)*a**3*b**5 + 2*tan(c + d*x)*a*b**7 + a**8 + 3*a**6*b**2 + 3*a**4*b**4 + a**2*b**6))