Integrand size = 29, antiderivative size = 236 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {1795 \log (1+\sin (c+d x))}{256 a d}+\frac {5 \sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin ^3(c+d x)}{3 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {17 a}{128 d (a-a \sin (c+d x))^2}+\frac {125}{128 d (a-a \sin (c+d x))}+\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {3 a^2}{16 d (a+a \sin (c+d x))^3}+\frac {71 a}{64 d (a+a \sin (c+d x))^2}-\frac {5}{d (a+a \sin (c+d x))} \]
515/256*ln(1-sin(d*x+c))/a/d-1795/256*ln(1+sin(d*x+c))/a/d+5*sin(d*x+c)/a/ d-1/2*sin(d*x+c)^2/a/d+1/3*sin(d*x+c)^3/a/d+1/96*a^2/d/(a-a*sin(d*x+c))^3- 17/128*a/d/(a-a*sin(d*x+c))^2+125/128/d/(a-a*sin(d*x+c))+1/64*a^3/d/(a+a*s in(d*x+c))^4-3/16*a^2/d/(a+a*sin(d*x+c))^3+71/64*a/d/(a+a*sin(d*x+c))^2-5/ d/(a+a*sin(d*x+c))
Time = 6.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1545 \log (1-\sin (c+d x))-5385 \log (1+\sin (c+d x))+\frac {8}{(1-\sin (c+d x))^3}-\frac {102}{(1-\sin (c+d x))^2}+\frac {750}{1-\sin (c+d x)}+3840 \sin (c+d x)-384 \sin ^2(c+d x)+256 \sin ^3(c+d x)+\frac {12}{(1+\sin (c+d x))^4}-\frac {144}{(1+\sin (c+d x))^3}+\frac {852}{(1+\sin (c+d x))^2}-\frac {3840}{1+\sin (c+d x)}}{768 a d} \]
(1545*Log[1 - Sin[c + d*x]] - 5385*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d*x])^3 - 102/(1 - Sin[c + d*x])^2 + 750/(1 - Sin[c + d*x]) + 3840*Sin[c + d*x] - 384*Sin[c + d*x]^2 + 256*Sin[c + d*x]^3 + 12/(1 + Sin[c + d*x])^4 - 144/(1 + Sin[c + d*x])^3 + 852/(1 + Sin[c + d*x])^2 - 3840/(1 + Sin[c + d*x]))/(768*a*d)
Time = 0.43 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 27, 99, 2009}
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.
| \(\displaystyle \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^{11}}{\cos (c+d x)^7 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^7 \int \frac {\sin ^{11}(c+d x)}{(a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a^{11} \sin ^{11}(c+d x)}{(a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{a^4 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (-\frac {a^7}{16 (\sin (c+d x) a+a)^5}+\frac {a^6}{32 (a-a \sin (c+d x))^4}+\frac {9 a^6}{16 (\sin (c+d x) a+a)^4}-\frac {17 a^5}{64 (a-a \sin (c+d x))^3}-\frac {71 a^5}{32 (\sin (c+d x) a+a)^3}+\frac {125 a^4}{128 (a-a \sin (c+d x))^2}+\frac {5 a^4}{(\sin (c+d x) a+a)^2}-\frac {515 a^3}{256 (a-a \sin (c+d x))}-\frac {1795 a^3}{256 (\sin (c+d x) a+a)}+\sin ^2(c+d x) a^2-\sin (c+d x) a^2+5 a^2\right )d(a \sin (c+d x))}{a^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^7}{64 (a \sin (c+d x)+a)^4}+\frac {a^6}{96 (a-a \sin (c+d x))^3}-\frac {3 a^6}{16 (a \sin (c+d x)+a)^3}-\frac {17 a^5}{128 (a-a \sin (c+d x))^2}+\frac {71 a^5}{64 (a \sin (c+d x)+a)^2}+\frac {125 a^4}{128 (a-a \sin (c+d x))}-\frac {5 a^4}{a \sin (c+d x)+a}+\frac {1}{3} a^3 \sin ^3(c+d x)-\frac {1}{2} a^3 \sin ^2(c+d x)+5 a^3 \sin (c+d x)+\frac {515}{256} a^3 \log (a-a \sin (c+d x))-\frac {1795}{256} a^3 \log (a \sin (c+d x)+a)}{a^4 d}\) |
((515*a^3*Log[a - a*Sin[c + d*x]])/256 - (1795*a^3*Log[a + a*Sin[c + d*x]] )/256 + 5*a^3*Sin[c + d*x] - (a^3*Sin[c + d*x]^2)/2 + (a^3*Sin[c + d*x]^3) /3 + a^6/(96*(a - a*Sin[c + d*x])^3) - (17*a^5)/(128*(a - a*Sin[c + d*x])^ 2) + (125*a^4)/(128*(a - a*Sin[c + d*x])) + a^7/(64*(a + a*Sin[c + d*x])^4 ) - (3*a^6)/(16*(a + a*Sin[c + d*x])^3) + (71*a^5)/(64*(a + a*Sin[c + d*x] )^2) - (5*a^4)/(a + a*Sin[c + d*x]))/(a^4*d)
3.9.78.3.1 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_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 2.93 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.61
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+5 \sin \left (d x +c \right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {17}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {125}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {515 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {3}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {71}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5}{1+\sin \left (d x +c \right )}-\frac {1795 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(143\) |
| default | \(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+5 \sin \left (d x +c \right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {17}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {125}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {515 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {3}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {71}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5}{1+\sin \left (d x +c \right )}-\frac {1795 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(143\) |
| risch | \(\frac {5 i x}{a}+\frac {i {\mathrm e}^{3 i \left (d x +c \right )}}{24 d a}+\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 a d}-\frac {21 i {\mathrm e}^{i \left (d x +c \right )}}{8 a d}+\frac {21 i {\mathrm e}^{-i \left (d x +c \right )}}{8 a d}+\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a d}-\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d a}+\frac {10 i c}{d a}-\frac {i \left (18313 \,{\mathrm e}^{5 i \left (d x +c \right )}+750 i {\mathrm e}^{12 i \left (d x +c \right )}+1690 i {\mathrm e}^{10 i \left (d x +c \right )}+1196 i {\mathrm e}^{8 i \left (d x +c \right )}-1196 i {\mathrm e}^{6 i \left (d x +c \right )}-1690 i {\mathrm e}^{4 i \left (d x +c \right )}-750 i {\mathrm e}^{2 i \left (d x +c \right )}+2295 \,{\mathrm e}^{i \left (d x +c \right )}+2295 \,{\mathrm e}^{13 i \left (d x +c \right )}+18313 \,{\mathrm e}^{9 i \left (d x +c \right )}+8150 \,{\mathrm e}^{11 i \left (d x +c \right )}+8150 \,{\mathrm e}^{3 i \left (d x +c \right )}+21332 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}+\frac {515 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}-\frac {1795 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}\) | \(354\) |
| parallelrisch | \(\frac {7724+1920 \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1545 \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+5385 \left (-20-\sin \left (7 d x +7 c \right )-5 \sin \left (5 d x +5 c \right )-9 \sin \left (3 d x +3 c \right )-5 \sin \left (d x +c \right )-2 \cos \left (6 d x +6 c \right )-12 \cos \left (4 d x +4 c \right )-30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+31620 \sin \left (3 d x +3 c \right )+17260 \sin \left (5 d x +5 c \right )+3200 \sin \left (7 d x +7 c \right )+16 \sin \left (9 d x +9 c \right )+16 \cos \left (10 d x +10 c \right )+3822 \cos \left (2 d x +2 c \right )-5100 \cos \left (4 d x +4 c \right )-5630 \cos \left (6 d x +6 c \right )-832 \cos \left (8 d x +8 c \right )+18056 \sin \left (d x +c \right )}{384 a d \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right )}\) | \(460\) |
1/d/a*(1/3*sin(d*x+c)^3-1/2*sin(d*x+c)^2+5*sin(d*x+c)-1/96/(sin(d*x+c)-1)^ 3-17/128/(sin(d*x+c)-1)^2-125/128/(sin(d*x+c)-1)+515/256*ln(sin(d*x+c)-1)+ 1/64/(1+sin(d*x+c))^4-3/16/(1+sin(d*x+c))^3+71/64/(1+sin(d*x+c))^2-5/(1+si n(d*x+c))-1795/256*ln(1+sin(d*x+c)))
Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.88 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {256 \, \cos \left (d x + c\right )^{10} - 3968 \, \cos \left (d x + c\right )^{8} - 686 \, \cos \left (d x + c\right )^{6} + 2810 \, \cos \left (d x + c\right )^{4} - 796 \, \cos \left (d x + c\right )^{2} - 5385 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 1545 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (64 \, \cos \left (d x + c\right )^{8} + 1952 \, \cos \left (d x + c\right )^{6} + 375 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
1/768*(256*cos(d*x + c)^10 - 3968*cos(d*x + c)^8 - 686*cos(d*x + c)^6 + 28 10*cos(d*x + c)^4 - 796*cos(d*x + c)^2 - 5385*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) + 1545*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(-sin(d*x + c) + 1) + 2*(64*cos(d*x + c)^8 + 1952* cos(d*x + c)^6 + 375*cos(d*x + c)^4 - 70*cos(d*x + c)^2 + 8)*sin(d*x + c) + 112)/(a*d*cos(d*x + c)^6*sin(d*x + c) + a*d*cos(d*x + c)^6)
Timed out. \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (2295 \, \sin \left (d x + c\right )^{6} + 375 \, \sin \left (d x + c\right )^{5} - 5480 \, \sin \left (d x + c\right )^{4} - 680 \, \sin \left (d x + c\right )^{3} + 4473 \, \sin \left (d x + c\right )^{2} + 313 \, \sin \left (d x + c\right ) - 1232\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {128 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 30 \, \sin \left (d x + c\right )\right )}}{a} + \frac {5385 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {1545 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
-1/768*(2*(2295*sin(d*x + c)^6 + 375*sin(d*x + c)^5 - 5480*sin(d*x + c)^4 - 680*sin(d*x + c)^3 + 4473*sin(d*x + c)^2 + 313*sin(d*x + c) - 1232)/(a*s in(d*x + c)^7 + a*sin(d*x + c)^6 - 3*a*sin(d*x + c)^5 - 3*a*sin(d*x + c)^4 + 3*a*sin(d*x + c)^3 + 3*a*sin(d*x + c)^2 - a*sin(d*x + c) - a) - 128*(2* sin(d*x + c)^3 - 3*sin(d*x + c)^2 + 30*sin(d*x + c))/a + 5385*log(sin(d*x + c) + 1)/a - 1545*log(sin(d*x + c) - 1)/a)/d
Time = 0.42 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.76 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {21540 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {6180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {512 \, {\left (2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} \sin \left (d x + c\right )^{2} + 30 \, a^{2} \sin \left (d x + c\right )\right )}}{a^{3}} + \frac {2 \, {\left (5665 \, \sin \left (d x + c\right )^{3} - 15495 \, \sin \left (d x + c\right )^{2} + 14199 \, \sin \left (d x + c\right ) - 4353\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {44875 \, \sin \left (d x + c\right )^{4} + 164140 \, \sin \left (d x + c\right )^{3} + 226578 \, \sin \left (d x + c\right )^{2} + 139660 \, \sin \left (d x + c\right ) + 32395}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
-1/3072*(21540*log(abs(sin(d*x + c) + 1))/a - 6180*log(abs(sin(d*x + c) - 1))/a - 512*(2*a^2*sin(d*x + c)^3 - 3*a^2*sin(d*x + c)^2 + 30*a^2*sin(d*x + c))/a^3 + 2*(5665*sin(d*x + c)^3 - 15495*sin(d*x + c)^2 + 14199*sin(d*x + c) - 4353)/(a*(sin(d*x + c) - 1)^3) - (44875*sin(d*x + c)^4 + 164140*sin (d*x + c)^3 + 226578*sin(d*x + c)^2 + 139660*sin(d*x + c) + 32395)/(a*(sin (d*x + c) + 1)^4))/d
Time = 11.89 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.40 \[ \int \frac {\sin ^4(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {515\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{128\,a\,d}-\frac {1795\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{128\,a\,d}-\frac {-\frac {1155\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{64}-\frac {835\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{32}+\frac {3205\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {305\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{4}+\frac {41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{16}+\frac {53\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{24}-\frac {5521\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{48}-\frac {2387\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{12}+\frac {6697\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{96}+\frac {6901\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{48}+\frac {6697\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{96}-\frac {2387\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{12}-\frac {5521\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{48}+\frac {53\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{24}+\frac {41\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {305\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {3205\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}-\frac {835\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {1155\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}+\frac {5\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]
(515*log(tan(c/2 + (d*x)/2) - 1))/(128*a*d) - (1795*log(tan(c/2 + (d*x)/2) + 1))/(128*a*d) - ((3205*tan(c/2 + (d*x)/2)^3)/64 - (835*tan(c/2 + (d*x)/ 2)^2)/32 - (1155*tan(c/2 + (d*x)/2))/64 + (305*tan(c/2 + (d*x)/2)^4)/4 + ( 41*tan(c/2 + (d*x)/2)^5)/16 + (53*tan(c/2 + (d*x)/2)^6)/24 - (5521*tan(c/2 + (d*x)/2)^7)/48 - (2387*tan(c/2 + (d*x)/2)^8)/12 + (6697*tan(c/2 + (d*x) /2)^9)/96 + (6901*tan(c/2 + (d*x)/2)^10)/48 + (6697*tan(c/2 + (d*x)/2)^11) /96 - (2387*tan(c/2 + (d*x)/2)^12)/12 - (5521*tan(c/2 + (d*x)/2)^13)/48 + (53*tan(c/2 + (d*x)/2)^14)/24 + (41*tan(c/2 + (d*x)/2)^15)/16 + (305*tan(c /2 + (d*x)/2)^16)/4 + (3205*tan(c/2 + (d*x)/2)^17)/64 - (835*tan(c/2 + (d* x)/2)^18)/32 - (1155*tan(c/2 + (d*x)/2)^19)/64)/(d*(a + 2*a*tan(c/2 + (d*x )/2) - 2*a*tan(c/2 + (d*x)/2)^2 - 6*a*tan(c/2 + (d*x)/2)^3 - 3*a*tan(c/2 + (d*x)/2)^4 + 8*a*tan(c/2 + (d*x)/2)^6 + 16*a*tan(c/2 + (d*x)/2)^7 + 2*a*t an(c/2 + (d*x)/2)^8 - 12*a*tan(c/2 + (d*x)/2)^9 - 12*a*tan(c/2 + (d*x)/2)^ 10 - 12*a*tan(c/2 + (d*x)/2)^11 + 2*a*tan(c/2 + (d*x)/2)^12 + 16*a*tan(c/2 + (d*x)/2)^13 + 8*a*tan(c/2 + (d*x)/2)^14 - 3*a*tan(c/2 + (d*x)/2)^16 - 6 *a*tan(c/2 + (d*x)/2)^17 - 2*a*tan(c/2 + (d*x)/2)^18 + 2*a*tan(c/2 + (d*x) /2)^19 + a*tan(c/2 + (d*x)/2)^20)) + (5*log(tan(c/2 + (d*x)/2)^2 + 1))/(a* d)