Integrand size = 21, antiderivative size = 42 \[ \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx=-\frac {67 x}{250}-\frac {28}{125} \log (\cos (x)+3 \sin (x))-\frac {7}{10 (1+3 \tan (x))^2}-\frac {29}{50 (1+3 \tan (x))} \]
Result contains complex when optimal does not.
Time = 1.84 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38 \[ \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx=\frac {1}{500} \left ((56+67 i) \log (i-\tan (x))+(56-67 i) \log (i+\tan (x))-112 \log (1+3 \tan (x))-\frac {350}{(1+3 \tan (x))^2}-\frac {290}{1+3 \tan (x)}\right ) \]
((56 + 67*I)*Log[I - Tan[x]] + (56 - 67*I)*Log[I + Tan[x]] - 112*Log[1 + 3 *Tan[x]] - 350/(1 + 3*Tan[x])^2 - 290/(1 + 3*Tan[x]))/500
Time = 0.43 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4111, 27, 3042, 4012, 25, 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.
\(\displaystyle \int \frac {-6 \tan ^2(x)-\tan (x)+5}{(3 \tan (x)+1)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {-6 \tan (x)^2-\tan (x)+5}{(3 \tan (x)+1)^3}dx\) |
\(\Big \downarrow \) 4111 |
\(\displaystyle \frac {1}{10} \int \frac {2 (4-17 \tan (x))}{(3 \tan (x)+1)^2}dx-\frac {7}{10 (3 \tan (x)+1)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {4-17 \tan (x)}{(3 \tan (x)+1)^2}dx-\frac {7}{10 (3 \tan (x)+1)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} \int \frac {4-17 \tan (x)}{(3 \tan (x)+1)^2}dx-\frac {7}{10 (3 \tan (x)+1)^2}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{10} \int -\frac {29 \tan (x)+47}{3 \tan (x)+1}dx-\frac {29}{10 (3 \tan (x)+1)}\right )-\frac {7}{10 (3 \tan (x)+1)^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{5} \left (-\frac {1}{10} \int \frac {29 \tan (x)+47}{3 \tan (x)+1}dx-\frac {29}{10 (3 \tan (x)+1)}\right )-\frac {7}{10 (3 \tan (x)+1)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} \left (-\frac {1}{10} \int \frac {29 \tan (x)+47}{3 \tan (x)+1}dx-\frac {29}{10 (3 \tan (x)+1)}\right )-\frac {7}{10 (3 \tan (x)+1)^2}\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{10} \left (-\frac {56}{5} \int \frac {3-\tan (x)}{3 \tan (x)+1}dx-\frac {67 x}{5}\right )-\frac {29}{10 (3 \tan (x)+1)}\right )-\frac {7}{10 (3 \tan (x)+1)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{10} \left (-\frac {56}{5} \int \frac {3-\tan (x)}{3 \tan (x)+1}dx-\frac {67 x}{5}\right )-\frac {29}{10 (3 \tan (x)+1)}\right )-\frac {7}{10 (3 \tan (x)+1)^2}\) |
\(\Big \downarrow \) 4013 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{10} \left (-\frac {67 x}{5}-\frac {56}{5} \log (3 \sin (x)+\cos (x))\right )-\frac {29}{10 (3 \tan (x)+1)}\right )-\frac {7}{10 (3 \tan (x)+1)^2}\) |
-7/(10*(1 + 3*Tan[x])^2) + (((-67*x)/5 - (56*Log[Cos[x] + 3*Sin[x]])/5)/10 - 29/(10*(1 + 3*Tan[x])))/5
3.4.82.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x ] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B , C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0 ]
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {14 \ln \left (1+\tan ^{2}\left (x \right )\right )}{125}-\frac {67 \arctan \left (\tan \left (x \right )\right )}{250}-\frac {7}{10 \left (1+3 \tan \left (x \right )\right )^{2}}-\frac {29}{50 \left (1+3 \tan \left (x \right )\right )}-\frac {28 \ln \left (1+3 \tan \left (x \right )\right )}{125}\) | \(45\) |
default | \(\frac {14 \ln \left (1+\tan ^{2}\left (x \right )\right )}{125}-\frac {67 \arctan \left (\tan \left (x \right )\right )}{250}-\frac {7}{10 \left (1+3 \tan \left (x \right )\right )^{2}}-\frac {29}{50 \left (1+3 \tan \left (x \right )\right )}-\frac {28 \ln \left (1+3 \tan \left (x \right )\right )}{125}\) | \(45\) |
risch | \(-\frac {67 x}{250}+\frac {28 i x}{125}+\frac {\left (-\frac {36}{24125}-\frac {621 i}{48250}\right ) \left (965 \,{\mathrm e}^{2 i x}-324+768 i\right )}{\left (5 \,{\mathrm e}^{2 i x}-4+3 i\right )^{2}}-\frac {28 \ln \left ({\mathrm e}^{2 i x}-\frac {4}{5}+\frac {3 i}{5}\right )}{125}\) | \(49\) |
norman | \(\frac {-\frac {87 \tan \left (x \right )}{50}-\frac {67 x}{250}-\frac {201 x \tan \left (x \right )}{125}-\frac {603 x \left (\tan ^{2}\left (x \right )\right )}{250}-\frac {32}{25}}{\left (1+3 \tan \left (x \right )\right )^{2}}-\frac {28 \ln \left (1+3 \tan \left (x \right )\right )}{125}+\frac {14 \ln \left (1+\tan ^{2}\left (x \right )\right )}{125}\) | \(50\) |
parallelrisch | \(-\frac {4536 \ln \left (\frac {1}{3}+\tan \left (x \right )\right ) \left (\tan ^{2}\left (x \right )\right )-2268 \ln \left (1+\tan ^{2}\left (x \right )\right ) \left (\tan ^{2}\left (x \right )\right )+5427 x \left (\tan ^{2}\left (x \right )\right )+2880+3024 \ln \left (\frac {1}{3}+\tan \left (x \right )\right ) \tan \left (x \right )-1512 \ln \left (1+\tan ^{2}\left (x \right )\right ) \tan \left (x \right )+3618 x \tan \left (x \right )+504 \ln \left (\frac {1}{3}+\tan \left (x \right )\right )-252 \ln \left (1+\tan ^{2}\left (x \right )\right )+603 x +3915 \tan \left (x \right )}{2250 \left (1+3 \tan \left (x \right )\right )^{2}}\) | \(92\) |
14/125*ln(1+tan(x)^2)-67/250*arctan(tan(x))-7/10/(1+3*tan(x))^2-29/50/(1+3 *tan(x))-28/125*ln(1+3*tan(x))
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (34) = 68\).
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.83 \[ \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx=-\frac {9 \, {\left (134 \, x - 1\right )} \tan \left (x\right )^{2} + 56 \, {\left (9 \, \tan \left (x\right )^{2} + 6 \, \tan \left (x\right ) + 1\right )} \log \left (\frac {9 \, \tan \left (x\right )^{2} + 6 \, \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) + 12 \, {\left (67 \, x + 72\right )} \tan \left (x\right ) + 134 \, x + 639}{500 \, {\left (9 \, \tan \left (x\right )^{2} + 6 \, \tan \left (x\right ) + 1\right )}} \]
-1/500*(9*(134*x - 1)*tan(x)^2 + 56*(9*tan(x)^2 + 6*tan(x) + 1)*log((9*tan (x)^2 + 6*tan(x) + 1)/(tan(x)^2 + 1)) + 12*(67*x + 72)*tan(x) + 134*x + 63 9)/(9*tan(x)^2 + 6*tan(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (39) = 78\).
Time = 0.24 (sec) , antiderivative size = 252, normalized size of antiderivative = 6.00 \[ \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx=- \frac {603 x \tan ^{2}{\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {402 x \tan {\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {67 x}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {504 \log {\left (3 \tan {\left (x \right )} + 1 \right )} \tan ^{2}{\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {336 \log {\left (3 \tan {\left (x \right )} + 1 \right )} \tan {\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {56 \log {\left (3 \tan {\left (x \right )} + 1 \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} + \frac {252 \log {\left (\tan ^{2}{\left (x \right )} + 1 \right )} \tan ^{2}{\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} + \frac {168 \log {\left (\tan ^{2}{\left (x \right )} + 1 \right )} \tan {\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} + \frac {28 \log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {435 \tan {\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {320}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} \]
-603*x*tan(x)**2/(2250*tan(x)**2 + 1500*tan(x) + 250) - 402*x*tan(x)/(2250 *tan(x)**2 + 1500*tan(x) + 250) - 67*x/(2250*tan(x)**2 + 1500*tan(x) + 250 ) - 504*log(3*tan(x) + 1)*tan(x)**2/(2250*tan(x)**2 + 1500*tan(x) + 250) - 336*log(3*tan(x) + 1)*tan(x)/(2250*tan(x)**2 + 1500*tan(x) + 250) - 56*lo g(3*tan(x) + 1)/(2250*tan(x)**2 + 1500*tan(x) + 250) + 252*log(tan(x)**2 + 1)*tan(x)**2/(2250*tan(x)**2 + 1500*tan(x) + 250) + 168*log(tan(x)**2 + 1 )*tan(x)/(2250*tan(x)**2 + 1500*tan(x) + 250) + 28*log(tan(x)**2 + 1)/(225 0*tan(x)**2 + 1500*tan(x) + 250) - 435*tan(x)/(2250*tan(x)**2 + 1500*tan(x ) + 250) - 320/(2250*tan(x)**2 + 1500*tan(x) + 250)
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05 \[ \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx=-\frac {67}{250} \, x - \frac {87 \, \tan \left (x\right ) + 64}{50 \, {\left (9 \, \tan \left (x\right )^{2} + 6 \, \tan \left (x\right ) + 1\right )}} + \frac {14}{125} \, \log \left (\tan \left (x\right )^{2} + 1\right ) - \frac {28}{125} \, \log \left (3 \, \tan \left (x\right ) + 1\right ) \]
-67/250*x - 1/50*(87*tan(x) + 64)/(9*tan(x)^2 + 6*tan(x) + 1) + 14/125*log (tan(x)^2 + 1) - 28/125*log(3*tan(x) + 1)
Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx=-\frac {67}{250} \, x - \frac {87 \, \tan \left (x\right ) + 64}{50 \, {\left (3 \, \tan \left (x\right ) + 1\right )}^{2}} + \frac {14}{125} \, \log \left (\tan \left (x\right )^{2} + 1\right ) - \frac {28}{125} \, \log \left ({\left | 3 \, \tan \left (x\right ) + 1 \right |}\right ) \]
-67/250*x - 1/50*(87*tan(x) + 64)/(3*tan(x) + 1)^2 + 14/125*log(tan(x)^2 + 1) - 28/125*log(abs(3*tan(x) + 1))
Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx=-\frac {28\,\ln \left (\mathrm {tan}\left (x\right )+\frac {1}{3}\right )}{125}-\frac {\frac {29\,\mathrm {tan}\left (x\right )}{150}+\frac {32}{225}}{{\mathrm {tan}\left (x\right )}^2+\frac {2\,\mathrm {tan}\left (x\right )}{3}+\frac {1}{9}}+\ln \left (\mathrm {tan}\left (x\right )-\mathrm {i}\right )\,\left (\frac {14}{125}+\frac {67}{500}{}\mathrm {i}\right )+\ln \left (\mathrm {tan}\left (x\right )+1{}\mathrm {i}\right )\,\left (\frac {14}{125}-\frac {67}{500}{}\mathrm {i}\right ) \]
log(tan(x) - 1i)*(14/125 + 67i/500) - (28*log(tan(x) + 1/3))/125 + log(tan (x) + 1i)*(14/125 - 67i/500) - ((29*tan(x))/150 + 32/225)/((2*tan(x))/3 + tan(x)^2 + 1/9)
Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.48 \[ \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx=\frac {504 \,\mathrm {log}\left (\tan \left (x \right )^{2}+1\right ) \tan \left (x \right )^{2}+336 \,\mathrm {log}\left (\tan \left (x \right )^{2}+1\right ) \tan \left (x \right )+56 \,\mathrm {log}\left (\tan \left (x \right )^{2}+1\right )-1008 \,\mathrm {log}\left (3 \tan \left (x \right )+1\right ) \tan \left (x \right )^{2}-672 \,\mathrm {log}\left (3 \tan \left (x \right )+1\right ) \tan \left (x \right )-112 \,\mathrm {log}\left (3 \tan \left (x \right )+1\right )-1206 \tan \left (x \right )^{2} x +1305 \tan \left (x \right )^{2}-804 \tan \left (x \right ) x -134 x -495}{4500 \tan \left (x \right )^{2}+3000 \tan \left (x \right )+500} \]