Integrand size = 12, antiderivative size = 88 \[ \int \frac {1}{(5+3 \tan (c+d x))^4} \, dx=-\frac {161 x}{334084}+\frac {60 \log (5 \cos (c+d x)+3 \sin (c+d x))}{83521 d}-\frac {1}{34 d (5+3 \tan (c+d x))^3}-\frac {15}{1156 d (5+3 \tan (c+d x))^2}-\frac {99}{19652 d (5+3 \tan (c+d x))} \]
-161/334084*x+60/83521*ln(5*cos(d*x+c)+3*sin(d*x+c))/d-1/34/d/(5+3*tan(d*x +c))^3-15/1156/d/(5+3*tan(d*x+c))^2-99/19652/d/(5+3*tan(d*x+c))
Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(5+3 \tan (c+d x))^4} \, dx=-\frac {(240-161 i) \log (i-\tan (c+d x))+(240+161 i) \log (i+\tan (c+d x))-480 \log (5+3 \tan (c+d x))+\frac {19652}{(5+3 \tan (c+d x))^3}+\frac {8670}{(5+3 \tan (c+d x))^2}+\frac {3366}{5+3 \tan (c+d x)}}{668168 d} \]
-1/668168*((240 - 161*I)*Log[I - Tan[c + d*x]] + (240 + 161*I)*Log[I + Tan [c + d*x]] - 480*Log[5 + 3*Tan[c + d*x]] + 19652/(5 + 3*Tan[c + d*x])^3 + 8670/(5 + 3*Tan[c + d*x])^2 + 3366/(5 + 3*Tan[c + d*x]))/d
Time = 0.62 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3964, 3042, 4012, 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.
| \(\displaystyle \int \frac {1}{(3 \tan (c+d x)+5)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(3 \tan (c+d x)+5)^4}dx\) |
| \(\Big \downarrow \) 3964 |
| \(\displaystyle \frac {1}{34} \int \frac {5-3 \tan (c+d x)}{(3 \tan (c+d x)+5)^3}dx-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{34} \int \frac {5-3 \tan (c+d x)}{(3 \tan (c+d x)+5)^3}dx-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{34} \int \frac {2 (8-15 \tan (c+d x))}{(3 \tan (c+d x)+5)^2}dx-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{17} \int \frac {8-15 \tan (c+d x)}{(3 \tan (c+d x)+5)^2}dx-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{17} \int \frac {8-15 \tan (c+d x)}{(3 \tan (c+d x)+5)^2}dx-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{17} \left (\frac {1}{34} \int -\frac {99 \tan (c+d x)+5}{3 \tan (c+d x)+5}dx-\frac {99}{34 d (3 \tan (c+d x)+5)}\right )-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{17} \left (-\frac {1}{34} \int \frac {99 \tan (c+d x)+5}{3 \tan (c+d x)+5}dx-\frac {99}{34 d (3 \tan (c+d x)+5)}\right )-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{17} \left (-\frac {1}{34} \int \frac {99 \tan (c+d x)+5}{3 \tan (c+d x)+5}dx-\frac {99}{34 d (3 \tan (c+d x)+5)}\right )-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 4014 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{17} \left (\frac {1}{34} \left (\frac {240}{17} \int \frac {3-5 \tan (c+d x)}{3 \tan (c+d x)+5}dx-\frac {161 x}{17}\right )-\frac {99}{34 d (3 \tan (c+d x)+5)}\right )-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{17} \left (\frac {1}{34} \left (\frac {240}{17} \int \frac {3-5 \tan (c+d x)}{3 \tan (c+d x)+5}dx-\frac {161 x}{17}\right )-\frac {99}{34 d (3 \tan (c+d x)+5)}\right )-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {1}{34} \left (\frac {1}{17} \left (\frac {1}{34} \left (\frac {240 \log (3 \sin (c+d x)+5 \cos (c+d x))}{17 d}-\frac {161 x}{17}\right )-\frac {99}{34 d (3 \tan (c+d x)+5)}\right )-\frac {15}{34 d (3 \tan (c+d x)+5)^2}\right )-\frac {1}{34 d (3 \tan (c+d x)+5)^3}\) |
-1/34*1/(d*(5 + 3*Tan[c + d*x])^3) + (-15/(34*d*(5 + 3*Tan[c + d*x])^2) + (((-161*x)/17 + (240*Log[5*Cos[c + d*x] + 3*Sin[c + d*x]])/(17*d))/34 - 99 /(34*d*(5 + 3*Tan[c + d*x])))/17)/34
3.6.2.3.1 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[(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 = 0.32 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{34 \left (5+3 \tan \left (d x +c \right )\right )^{3}}-\frac {15}{1156 \left (5+3 \tan \left (d x +c \right )\right )^{2}}-\frac {99}{19652 \left (5+3 \tan \left (d x +c \right )\right )}+\frac {60 \ln \left (5+3 \tan \left (d x +c \right )\right )}{83521}-\frac {30 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{83521}-\frac {161 \arctan \left (\tan \left (d x +c \right )\right )}{334084}}{d}\) | \(83\) |
| default | \(\frac {-\frac {1}{34 \left (5+3 \tan \left (d x +c \right )\right )^{3}}-\frac {15}{1156 \left (5+3 \tan \left (d x +c \right )\right )^{2}}-\frac {99}{19652 \left (5+3 \tan \left (d x +c \right )\right )}+\frac {60 \ln \left (5+3 \tan \left (d x +c \right )\right )}{83521}-\frac {30 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{83521}-\frac {161 \arctan \left (\tan \left (d x +c \right )\right )}{334084}}{d}\) | \(83\) |
| risch | \(-\frac {161 x}{334084}-\frac {60 i x}{83521}-\frac {120 i c}{83521 d}+\frac {\left (-\frac {5535}{48776264}+\frac {351 i}{48776264}\right ) \left (84388 \,{\mathrm e}^{4 i \left (d x +c \right )}+127585 i {\mathrm e}^{2 i \left (d x +c \right )}+108987 \,{\mathrm e}^{2 i \left (d x +c \right )}-13133+79235 i\right )}{d \left (17 \,{\mathrm e}^{2 i \left (d x +c \right )}+8+15 i\right )^{3}}+\frac {60 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {8}{17}+\frac {15 i}{17}\right )}{83521 d}\) | \(97\) |
| norman | \(\frac {-\frac {20125 x}{334084}-\frac {36225 x \tan \left (d x +c \right )}{334084}-\frac {21735 x \left (\tan ^{2}\left (d x +c \right )\right )}{334084}-\frac {4347 x \left (\tan ^{3}\left (d x +c \right )\right )}{334084}-\frac {1082}{4913 d}-\frac {3735 \tan \left (d x +c \right )}{19652 d}-\frac {891 \left (\tan ^{2}\left (d x +c \right )\right )}{19652 d}}{\left (5+3 \tan \left (d x +c \right )\right )^{3}}+\frac {60 \ln \left (5+3 \tan \left (d x +c \right )\right )}{83521 d}-\frac {30 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{83521 d}\) | \(111\) |
| parallelrisch | \(\frac {-117369 \left (\tan ^{3}\left (d x +c \right )\right ) x d -1986552+174960 \ln \left (\frac {5}{3}+\tan \left (d x +c \right )\right ) \left (\tan ^{3}\left (d x +c \right )\right )-87480 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \left (\tan ^{3}\left (d x +c \right )\right )-586845 \left (\tan ^{2}\left (d x +c \right )\right ) x d +874800 \ln \left (\frac {5}{3}+\tan \left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right )-437400 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \left (\tan ^{2}\left (d x +c \right )\right )-978075 \tan \left (d x +c \right ) x d +1458000 \ln \left (\frac {5}{3}+\tan \left (d x +c \right )\right ) \tan \left (d x +c \right )-729000 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )-543375 d x -408969 \left (\tan ^{2}\left (d x +c \right )\right )+810000 \ln \left (\frac {5}{3}+\tan \left (d x +c \right )\right )-405000 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-1714365 \tan \left (d x +c \right )}{9020268 d \left (5+3 \tan \left (d x +c \right )\right )^{3}}\) | \(216\) |
1/d*(-1/34/(5+3*tan(d*x+c))^3-15/1156/(5+3*tan(d*x+c))^2-99/19652/(5+3*tan (d*x+c))+60/83521*ln(5+3*tan(d*x+c))-30/83521*ln(1+tan(d*x+c)^2)-161/33408 4*arctan(tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (78) = 156\).
Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(5+3 \tan (c+d x))^4} \, dx=-\frac {27 \, {\left (161 \, d x - 305\right )} \tan \left (d x + c\right )^{3} + 27 \, {\left (805 \, d x - 964\right )} \tan \left (d x + c\right )^{2} + 20125 \, d x - 120 \, {\left (27 \, \tan \left (d x + c\right )^{3} + 135 \, \tan \left (d x + c\right )^{2} + 225 \, \tan \left (d x + c\right ) + 125\right )} \log \left (\frac {9 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 25}{\tan \left (d x + c\right )^{2} + 1}\right ) + 45 \, {\left (805 \, d x - 114\right )} \tan \left (d x + c\right ) + 35451}{334084 \, {\left (27 \, d \tan \left (d x + c\right )^{3} + 135 \, d \tan \left (d x + c\right )^{2} + 225 \, d \tan \left (d x + c\right ) + 125 \, d\right )}} \]
-1/334084*(27*(161*d*x - 305)*tan(d*x + c)^3 + 27*(805*d*x - 964)*tan(d*x + c)^2 + 20125*d*x - 120*(27*tan(d*x + c)^3 + 135*tan(d*x + c)^2 + 225*tan (d*x + c) + 125)*log((9*tan(d*x + c)^2 + 30*tan(d*x + c) + 25)/(tan(d*x + c)^2 + 1)) + 45*(805*d*x - 114)*tan(d*x + c) + 35451)/(27*d*tan(d*x + c)^3 + 135*d*tan(d*x + c)^2 + 225*d*tan(d*x + c) + 125*d)
Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (76) = 152\).
Time = 0.45 (sec) , antiderivative size = 790, normalized size of antiderivative = 8.98 \[ \int \frac {1}{(5+3 \tan (c+d x))^4} \, dx=\text {Too large to display} \]
Piecewise((-4347*d*x*tan(c + d*x)**3/(9020268*d*tan(c + d*x)**3 + 45101340 *d*tan(c + d*x)**2 + 75168900*d*tan(c + d*x) + 41760500*d) - 21735*d*x*tan (c + d*x)**2/(9020268*d*tan(c + d*x)**3 + 45101340*d*tan(c + d*x)**2 + 751 68900*d*tan(c + d*x) + 41760500*d) - 36225*d*x*tan(c + d*x)/(9020268*d*tan (c + d*x)**3 + 45101340*d*tan(c + d*x)**2 + 75168900*d*tan(c + d*x) + 4176 0500*d) - 20125*d*x/(9020268*d*tan(c + d*x)**3 + 45101340*d*tan(c + d*x)** 2 + 75168900*d*tan(c + d*x) + 41760500*d) + 6480*log(3*tan(c + d*x) + 5)*t an(c + d*x)**3/(9020268*d*tan(c + d*x)**3 + 45101340*d*tan(c + d*x)**2 + 7 5168900*d*tan(c + d*x) + 41760500*d) + 32400*log(3*tan(c + d*x) + 5)*tan(c + d*x)**2/(9020268*d*tan(c + d*x)**3 + 45101340*d*tan(c + d*x)**2 + 75168 900*d*tan(c + d*x) + 41760500*d) + 54000*log(3*tan(c + d*x) + 5)*tan(c + d *x)/(9020268*d*tan(c + d*x)**3 + 45101340*d*tan(c + d*x)**2 + 75168900*d*t an(c + d*x) + 41760500*d) + 30000*log(3*tan(c + d*x) + 5)/(9020268*d*tan(c + d*x)**3 + 45101340*d*tan(c + d*x)**2 + 75168900*d*tan(c + d*x) + 417605 00*d) - 3240*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(9020268*d*tan(c + d *x)**3 + 45101340*d*tan(c + d*x)**2 + 75168900*d*tan(c + d*x) + 41760500*d ) - 16200*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(9020268*d*tan(c + d*x) **3 + 45101340*d*tan(c + d*x)**2 + 75168900*d*tan(c + d*x) + 41760500*d) - 27000*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(9020268*d*tan(c + d*x)**3 + 45101340*d*tan(c + d*x)**2 + 75168900*d*tan(c + d*x) + 41760500*d) - 15...
Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(5+3 \tan (c+d x))^4} \, dx=-\frac {161 \, d x + 161 \, c + \frac {17 \, {\left (891 \, \tan \left (d x + c\right )^{2} + 3735 \, \tan \left (d x + c\right ) + 4328\right )}}{27 \, \tan \left (d x + c\right )^{3} + 135 \, \tan \left (d x + c\right )^{2} + 225 \, \tan \left (d x + c\right ) + 125} + 120 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 240 \, \log \left (3 \, \tan \left (d x + c\right ) + 5\right )}{334084 \, d} \]
-1/334084*(161*d*x + 161*c + 17*(891*tan(d*x + c)^2 + 3735*tan(d*x + c) + 4328)/(27*tan(d*x + c)^3 + 135*tan(d*x + c)^2 + 225*tan(d*x + c) + 125) + 120*log(tan(d*x + c)^2 + 1) - 240*log(3*tan(d*x + c) + 5))/d
Time = 0.39 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(5+3 \tan (c+d x))^4} \, dx=-\frac {161 \, d x + 161 \, c + \frac {11880 \, \tan \left (d x + c\right )^{3} + 74547 \, \tan \left (d x + c\right )^{2} + 162495 \, \tan \left (d x + c\right ) + 128576}{{\left (3 \, \tan \left (d x + c\right ) + 5\right )}^{3}} + 120 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 240 \, \log \left ({\left | 3 \, \tan \left (d x + c\right ) + 5 \right |}\right )}{334084 \, d} \]
-1/334084*(161*d*x + 161*c + (11880*tan(d*x + c)^3 + 74547*tan(d*x + c)^2 + 162495*tan(d*x + c) + 128576)/(3*tan(d*x + c) + 5)^3 + 120*log(tan(d*x + c)^2 + 1) - 240*log(abs(3*tan(d*x + c) + 5)))/d
Time = 4.51 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(5+3 \tan (c+d x))^4} \, dx=\frac {60\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+\frac {5}{3}\right )}{83521\,d}-\frac {\frac {33\,{\mathrm {tan}\left (c+d\,x\right )}^2}{19652}+\frac {415\,\mathrm {tan}\left (c+d\,x\right )}{58956}+\frac {1082}{132651}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^3+5\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {25\,\mathrm {tan}\left (c+d\,x\right )}{3}+\frac {125}{27}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {30}{83521}+\frac {161}{668168}{}\mathrm {i}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-\frac {30}{83521}-\frac {161}{668168}{}\mathrm {i}\right )}{d} \]