Integrand size = 29, antiderivative size = 220 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {325 \log (1-\sin (c+d x))}{256 a d}+\frac {955 \log (1+\sin (c+d x))}{256 a d}-\frac {\sin (c+d x)}{a d}+\frac {\sin ^2(c+d x)}{2 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {15 a}{128 d (a-a \sin (c+d x))^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}+\frac {a^2}{6 d (a+a \sin (c+d x))^3}-\frac {55 a}{64 d (a+a \sin (c+d x))^2}+\frac {105}{32 d (a+a \sin (c+d x))} \]
325/256*ln(1-sin(d*x+c))/a/d+955/256*ln(1+sin(d*x+c))/a/d-sin(d*x+c)/a/d+1 /2*sin(d*x+c)^2/a/d+1/96*a^2/d/(a-a*sin(d*x+c))^3-15/128*a/d/(a-a*sin(d*x+ c))^2+95/128/d/(a-a*sin(d*x+c))-1/64*a^3/d/(a+a*sin(d*x+c))^4+1/6*a^2/d/(a +a*sin(d*x+c))^3-55/64*a/d/(a+a*sin(d*x+c))^2+105/32/d/(a+a*sin(d*x+c))
Time = 6.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {975 \log (1-\sin (c+d x))+2865 \log (1+\sin (c+d x))+\frac {8}{(1-\sin (c+d x))^3}-\frac {90}{(1-\sin (c+d x))^2}+\frac {570}{1-\sin (c+d x)}-768 \sin (c+d x)+384 \sin ^2(c+d x)-\frac {12}{(1+\sin (c+d x))^4}+\frac {128}{(1+\sin (c+d x))^3}-\frac {660}{(1+\sin (c+d x))^2}+\frac {2520}{1+\sin (c+d x)}}{768 a d} \]
(975*Log[1 - Sin[c + d*x]] + 2865*Log[1 + Sin[c + d*x]] + 8/(1 - Sin[c + d *x])^3 - 90/(1 - Sin[c + d*x])^2 + 570/(1 - Sin[c + d*x]) - 768*Sin[c + d* x] + 384*Sin[c + d*x]^2 - 12/(1 + Sin[c + d*x])^4 + 128/(1 + Sin[c + d*x]) ^3 - 660/(1 + Sin[c + d*x])^2 + 2520/(1 + Sin[c + d*x]))/(768*a*d)
Time = 0.41 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.94, 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.
\(\displaystyle \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)^{10}}{\cos (c+d x)^7 (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {a^7 \int \frac {\sin ^{10}(c+d x)}{(a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {a^{10} \sin ^{10}(c+d x)}{(a-a \sin (c+d x))^4 (\sin (c+d x) a+a)^5}d(a \sin (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (\frac {a^6}{16 (\sin (c+d x) a+a)^5}+\frac {a^5}{32 (a-a \sin (c+d x))^4}-\frac {a^5}{2 (\sin (c+d x) a+a)^4}-\frac {15 a^4}{64 (a-a \sin (c+d x))^3}+\frac {55 a^4}{32 (\sin (c+d x) a+a)^3}+\frac {95 a^3}{128 (a-a \sin (c+d x))^2}-\frac {105 a^3}{32 (\sin (c+d x) a+a)^2}-\frac {325 a^2}{256 (a-a \sin (c+d x))}+\frac {955 a^2}{256 (\sin (c+d x) a+a)}+\sin (c+d x) a-a\right )d(a \sin (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {a^6}{64 (a \sin (c+d x)+a)^4}+\frac {a^5}{96 (a-a \sin (c+d x))^3}+\frac {a^5}{6 (a \sin (c+d x)+a)^3}-\frac {15 a^4}{128 (a-a \sin (c+d x))^2}-\frac {55 a^4}{64 (a \sin (c+d x)+a)^2}+\frac {95 a^3}{128 (a-a \sin (c+d x))}+\frac {105 a^3}{32 (a \sin (c+d x)+a)}+\frac {1}{2} a^2 \sin ^2(c+d x)-a^2 \sin (c+d x)+\frac {325}{256} a^2 \log (a-a \sin (c+d x))+\frac {955}{256} a^2 \log (a \sin (c+d x)+a)}{a^3 d}\) |
((325*a^2*Log[a - a*Sin[c + d*x]])/256 + (955*a^2*Log[a + a*Sin[c + d*x]]) /256 - a^2*Sin[c + d*x] + (a^2*Sin[c + d*x]^2)/2 + a^5/(96*(a - a*Sin[c + d*x])^3) - (15*a^4)/(128*(a - a*Sin[c + d*x])^2) + (95*a^3)/(128*(a - a*Si n[c + d*x])) - a^6/(64*(a + a*Sin[c + d*x])^4) + a^5/(6*(a + a*Sin[c + d*x ])^3) - (55*a^4)/(64*(a + a*Sin[c + d*x])^2) + (105*a^3)/(32*(a + a*Sin[c + d*x])))/(a^3*d)
3.9.79.3.1 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 = 2.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.60
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {15}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {95}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{6 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {55}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {105}{32 \left (1+\sin \left (d x +c \right )\right )}+\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(133\) |
default | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\sin \left (d x +c \right )-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {15}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {95}{128 \left (\sin \left (d x +c \right )-1\right )}+\frac {325 \ln \left (\sin \left (d x +c \right )-1\right )}{256}-\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{6 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {55}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {105}{32 \left (1+\sin \left (d x +c \right )\right )}+\frac {955 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(133\) |
risch | \(-\frac {5 i x}{a}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 a d}+\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 a d}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a d}-\frac {10 i c}{d a}+\frac {i \left (-1890 i {\mathrm e}^{12 i \left (d x +c \right )}+975 \,{\mathrm e}^{13 i \left (d x +c \right )}-3030 i {\mathrm e}^{10 i \left (d x +c \right )}+7110 \,{\mathrm e}^{11 i \left (d x +c \right )}-2932 i {\mathrm e}^{8 i \left (d x +c \right )}+18609 \,{\mathrm e}^{9 i \left (d x +c \right )}+2932 i {\mathrm e}^{6 i \left (d x +c \right )}+25460 \,{\mathrm e}^{7 i \left (d x +c \right )}+3030 i {\mathrm e}^{4 i \left (d x +c \right )}+18609 \,{\mathrm e}^{5 i \left (d x +c \right )}+1890 i {\mathrm e}^{2 i \left (d x +c \right )}+7110 \,{\mathrm e}^{3 i \left (d x +c \right )}+975 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}+\frac {325 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}+\frac {955 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}\) | \(318\) |
parallelrisch | \(\frac {340+1920 \left (-20-\sin \left (7 d x +7 c \right )-5 \sin \left (5 d x +5 c \right )-9 \sin \left (3 d x +3 c \right )-5 \sin \left (d x +c \right )-2 \cos \left (6 d x +6 c \right )-12 \cos \left (4 d x +4 c \right )-30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+975 \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2865 \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-8100 \sin \left (3 d x +3 c \right )-4300 \sin \left (5 d x +5 c \right )-1760 \sin \left (7 d x +7 c \right )-48 \sin \left (9 d x +9 c \right )-126 \cos \left (2 d x +2 c \right )-180 \cos \left (4 d x +4 c \right )-130 \cos \left (6 d x +6 c \right )+96 \cos \left (8 d x +8 c \right )-1928 \sin \left (d x +c \right )}{384 a d \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right )}\) | \(449\) |
1/d/a*(1/2*sin(d*x+c)^2-sin(d*x+c)-1/96/(sin(d*x+c)-1)^3-15/128/(sin(d*x+c )-1)^2-95/128/(sin(d*x+c)-1)+325/256*ln(sin(d*x+c)-1)-1/64/(1+sin(d*x+c))^ 4+1/6/(1+sin(d*x+c))^3-55/64/(1+sin(d*x+c))^2+105/32/(1+sin(d*x+c))+955/25 6*ln(1+sin(d*x+c)))
Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {384 \, \cos \left (d x + c\right )^{8} + 1374 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{4} - 132 \, \cos \left (d x + c\right )^{2} + 2865 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 975 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (192 \, \cos \left (d x + c\right )^{8} + 288 \, \cos \left (d x + c\right )^{6} - 945 \, \cos \left (d x + c\right )^{4} + 330 \, \cos \left (d x + c\right )^{2} - 56\right )} \sin \left (d x + c\right ) + 16}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
1/768*(384*cos(d*x + c)^8 + 1374*cos(d*x + c)^6 + 630*cos(d*x + c)^4 - 132 *cos(d*x + c)^2 + 2865*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log( sin(d*x + c) + 1) + 975*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log (-sin(d*x + c) + 1) - 2*(192*cos(d*x + c)^8 + 288*cos(d*x + c)^6 - 945*cos (d*x + c)^4 + 330*cos(d*x + c)^2 - 56)*sin(d*x + c) + 16)/(a*d*cos(d*x + c )^6*sin(d*x + c) + a*d*cos(d*x + c)^6)
Timed out. \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (975 \, \sin \left (d x + c\right )^{6} - 945 \, \sin \left (d x + c\right )^{5} - 3240 \, \sin \left (d x + c\right )^{4} + 1560 \, \sin \left (d x + c\right )^{3} + 3489 \, \sin \left (d x + c\right )^{2} - 671 \, \sin \left (d x + c\right ) - 1232\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} + \frac {384 \, {\left (\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )\right )}}{a} + \frac {2865 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {975 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
1/768*(2*(975*sin(d*x + c)^6 - 945*sin(d*x + c)^5 - 3240*sin(d*x + c)^4 + 1560*sin(d*x + c)^3 + 3489*sin(d*x + c)^2 - 671*sin(d*x + c) - 1232)/(a*si n(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) + 384*(sin (d*x + c)^2 - 2*sin(d*x + c))/a + 2865*log(sin(d*x + c) + 1)/a + 975*log(s in(d*x + c) - 1)/a)/d
Time = 0.45 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.73 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {11460 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {3900 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {1536 \, {\left (a \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )\right )}}{a^{2}} - \frac {2 \, {\left (3575 \, \sin \left (d x + c\right )^{3} - 9585 \, \sin \left (d x + c\right )^{2} + 8625 \, \sin \left (d x + c\right ) - 2599\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {23875 \, \sin \left (d x + c\right )^{4} + 85420 \, \sin \left (d x + c\right )^{3} + 115650 \, \sin \left (d x + c\right )^{2} + 70028 \, \sin \left (d x + c\right ) + 15971}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
1/3072*(11460*log(abs(sin(d*x + c) + 1))/a + 3900*log(abs(sin(d*x + c) - 1 ))/a + 1536*(a*sin(d*x + c)^2 - 2*a*sin(d*x + c))/a^2 - 2*(3575*sin(d*x + c)^3 - 9585*sin(d*x + c)^2 + 8625*sin(d*x + c) - 2599)/(a*(sin(d*x + c) - 1)^3) - (23875*sin(d*x + c)^4 + 85420*sin(d*x + c)^3 + 115650*sin(d*x + c) ^2 + 70028*sin(d*x + c) + 15971)/(a*(sin(d*x + c) + 1)^4))/d
Time = 11.07 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.33 \[ \int \frac {\sin ^3(c+d x) \tan ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {325\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{128\,a\,d}+\frac {955\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{128\,a\,d}+\frac {-\frac {315\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{32}+\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{8}+\frac {195\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{32}-\frac {1217\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}-\frac {2389\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{96}+\frac {1189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{24}+\frac {767\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{32}+\frac {6845\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{96}+\frac {767\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{32}+\frac {1189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{24}-\frac {2389\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{96}-\frac {1217\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {195\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}+\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {315\,\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 )}^{18}+2\,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 )}^{15}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,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+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-3\,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} \]
(325*log(tan(c/2 + (d*x)/2) - 1))/(128*a*d) + (955*log(tan(c/2 + (d*x)/2) + 1))/(128*a*d) + ((5*tan(c/2 + (d*x)/2)^2)/32 - (315*tan(c/2 + (d*x)/2))/ 64 + (265*tan(c/2 + (d*x)/2)^3)/8 + (195*tan(c/2 + (d*x)/2)^4)/32 - (1217* tan(c/2 + (d*x)/2)^5)/16 - (2389*tan(c/2 + (d*x)/2)^6)/96 + (1189*tan(c/2 + (d*x)/2)^7)/24 + (767*tan(c/2 + (d*x)/2)^8)/32 + (6845*tan(c/2 + (d*x)/2 )^9)/96 + (767*tan(c/2 + (d*x)/2)^10)/32 + (1189*tan(c/2 + (d*x)/2)^11)/24 - (2389*tan(c/2 + (d*x)/2)^12)/96 - (1217*tan(c/2 + (d*x)/2)^13)/16 + (19 5*tan(c/2 + (d*x)/2)^14)/32 + (265*tan(c/2 + (d*x)/2)^15)/8 + (5*tan(c/2 + (d*x)/2)^16)/32 - (315*tan(c/2 + (d*x)/2)^17)/64)/(d*(a + 2*a*tan(c/2 + ( d*x)/2) - 3*a*tan(c/2 + (d*x)/2)^2 - 8*a*tan(c/2 + (d*x)/2)^3 + 8*a*tan(c/ 2 + (d*x)/2)^5 + 8*a*tan(c/2 + (d*x)/2)^6 + 8*a*tan(c/2 + (d*x)/2)^7 - 6*a *tan(c/2 + (d*x)/2)^8 - 20*a*tan(c/2 + (d*x)/2)^9 - 6*a*tan(c/2 + (d*x)/2) ^10 + 8*a*tan(c/2 + (d*x)/2)^11 + 8*a*tan(c/2 + (d*x)/2)^12 + 8*a*tan(c/2 + (d*x)/2)^13 - 8*a*tan(c/2 + (d*x)/2)^15 - 3*a*tan(c/2 + (d*x)/2)^16 + 2* a*tan(c/2 + (d*x)/2)^17 + a*tan(c/2 + (d*x)/2)^18)) - (5*log(tan(c/2 + (d* x)/2)^2 + 1))/(a*d)