Integrand size = 21, antiderivative size = 129 \[ \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx=-\frac {25 a^4 \log (1-\sin (c+d x))}{d}-\frac {16 a^4 \sin (c+d x)}{d}-\frac {9 a^4 \sin ^2(c+d x)}{2 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {a^6}{d (a-a \sin (c+d x))^2}-\frac {11 a^5}{d (a-a \sin (c+d x))} \]
-25*a^4*ln(1-sin(d*x+c))/d-16*a^4*sin(d*x+c)/d-9/2*a^4*sin(d*x+c)^2/d-4/3* a^4*sin(d*x+c)^3/d-1/4*a^4*sin(d*x+c)^4/d+a^6/d/(a-a*sin(d*x+c))^2-11*a^5/ d/(a-a*sin(d*x+c))
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.64 \[ \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx=-\frac {a^4 \left (300 \log (1-\sin (c+d x))+\frac {120-132 \sin (c+d x)}{(-1+\sin (c+d x))^2}+192 \sin (c+d x)+54 \sin ^2(c+d x)+16 \sin ^3(c+d x)+3 \sin ^4(c+d x)\right )}{12 d} \]
-1/12*(a^4*(300*Log[1 - Sin[c + d*x]] + (120 - 132*Sin[c + d*x])/(-1 + Sin [c + d*x])^2 + 192*Sin[c + d*x] + 54*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4))/d
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3186, 86, 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.
| \(\displaystyle \int \tan ^5(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^5 (a \sin (c+d x)+a)^4dx\) |
| \(\Big \downarrow \) 3186 |
| \(\displaystyle \frac {\int \frac {a^5 \sin ^5(c+d x) (\sin (c+d x) a+a)}{(a-a \sin (c+d x))^3}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\int \left (\frac {2 a^6}{(a-a \sin (c+d x))^3}-\frac {11 a^5}{(a-a \sin (c+d x))^2}+\frac {25 a^4}{a-a \sin (c+d x)}-\sin ^3(c+d x) a^3-4 \sin ^2(c+d x) a^3-9 \sin (c+d x) a^3-16 a^3\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^6}{(a-a \sin (c+d x))^2}-\frac {11 a^5}{a-a \sin (c+d x)}-\frac {1}{4} a^4 \sin ^4(c+d x)-\frac {4}{3} a^4 \sin ^3(c+d x)-\frac {9}{2} a^4 \sin ^2(c+d x)-16 a^4 \sin (c+d x)-25 a^4 \log (a-a \sin (c+d x))}{d}\) |
(-25*a^4*Log[a - a*Sin[c + d*x]] - 16*a^4*Sin[c + d*x] - (9*a^4*Sin[c + d* x]^2)/2 - (4*a^4*Sin[c + d*x]^3)/3 - (a^4*Sin[c + d*x]^4)/4 + a^6/(a - a*S in[c + d*x])^2 - (11*a^5)/(a - a*Sin[c + d*x]))/d
3.1.34.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x) ^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
Result contains complex when optimal does not.
Time = 26.93 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.76
| method | result | size |
| risch | \(25 i a^{4} x -\frac {i a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{6 d}+\frac {19 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {17 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {17 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {19 a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}+\frac {i a^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{6 d}+\frac {50 i a^{4} c}{d}+\frac {2 i \left (-11 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}-20 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+11 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}\right )}{\left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{4} d}-\frac {50 a^{4} \ln \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d}-\frac {a^{4} \cos \left (4 d x +4 c \right )}{32 d}\) | \(227\) |
| derivativedivides | \(\frac {a^{4} \left (\frac {\sin ^{10}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{4}-\left (\sin ^{6}\left (d x +c \right )\right )-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{2}-3 \left (\sin ^{2}\left (d x +c \right )\right )-6 \ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+6 a^{4} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(393\) |
| default | \(\frac {a^{4} \left (\frac {\sin ^{10}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{4}-\left (\sin ^{6}\left (d x +c \right )\right )-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{2}-3 \left (\sin ^{2}\left (d x +c \right )\right )-6 \ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+6 a^{4} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(393\) |
| parts | \(\frac {a^{4} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {a^{4} \left (\frac {\sin ^{10}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{10}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{8}\left (d x +c \right )\right )}{4}-\left (\sin ^{6}\left (d x +c \right )\right )-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{2}-3 \left (\sin ^{2}\left (d x +c \right )\right )-6 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {4 a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {6 a^{4} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {4 a^{4} \left (\frac {\sin ^{9}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {5 \left (\sin ^{9}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {5 \left (\sin ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}-\frac {35 \sin \left (d x +c \right )}{8}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(408\) |
25*I*a^4*x-1/6*I/d*a^4*exp(3*I*(d*x+c))+19/16*a^4/d*exp(2*I*(d*x+c))+17/2* I/d*a^4*exp(I*(d*x+c))-17/2*I/d*a^4*exp(-I*(d*x+c))+19/16*a^4/d*exp(-2*I*( d*x+c))+1/6*I/d*a^4*exp(-3*I*(d*x+c))+50*I/d*a^4*c+2*I*(-11*exp(I*(d*x+c)) *a^4-20*I*a^4*exp(2*I*(d*x+c))+11*a^4*exp(3*I*(d*x+c)))/(-I+exp(I*(d*x+c)) )^4/d-50/d*a^4*ln(-I+exp(I*(d*x+c)))-1/32/d*a^4*cos(4*d*x+4*c)
Time = 0.34 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx=-\frac {24 \, a^{4} \cos \left (d x + c\right )^{6} - 272 \, a^{4} \cos \left (d x + c\right )^{4} - 2393 \, a^{4} \cos \left (d x + c\right )^{2} + 1906 \, a^{4} + 2400 \, {\left (a^{4} \cos \left (d x + c\right )^{2} + 2 \, a^{4} \sin \left (d x + c\right ) - 2 \, a^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \, {\left (8 \, a^{4} \cos \left (d x + c\right )^{4} - 96 \, a^{4} \cos \left (d x + c\right )^{2} + 181 \, a^{4}\right )} \sin \left (d x + c\right )}{96 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
-1/96*(24*a^4*cos(d*x + c)^6 - 272*a^4*cos(d*x + c)^4 - 2393*a^4*cos(d*x + c)^2 + 1906*a^4 + 2400*(a^4*cos(d*x + c)^2 + 2*a^4*sin(d*x + c) - 2*a^4)* log(-sin(d*x + c) + 1) - 10*(8*a^4*cos(d*x + c)^4 - 96*a^4*cos(d*x + c)^2 + 181*a^4)*sin(d*x + c))/(d*cos(d*x + c)^2 + 2*d*sin(d*x + c) - 2*d)
\[ \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx=a^{4} \left (\int 4 \sin {\left (c + d x \right )} \tan ^{5}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \tan ^{5}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \tan ^{5}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \tan ^{5}{\left (c + d x \right )}\, dx + \int \tan ^{5}{\left (c + d x \right )}\, dx\right ) \]
a**4*(Integral(4*sin(c + d*x)*tan(c + d*x)**5, x) + Integral(6*sin(c + d*x )**2*tan(c + d*x)**5, x) + Integral(4*sin(c + d*x)**3*tan(c + d*x)**5, x) + Integral(sin(c + d*x)**4*tan(c + d*x)**5, x) + Integral(tan(c + d*x)**5, x))
Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.84 \[ \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx=-\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 54 \, a^{4} \sin \left (d x + c\right )^{2} + 300 \, a^{4} \log \left (\sin \left (d x + c\right ) - 1\right ) + 192 \, a^{4} \sin \left (d x + c\right ) - \frac {12 \, {\left (11 \, a^{4} \sin \left (d x + c\right ) - 10 \, a^{4}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{12 \, d} \]
-1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 54*a^4*sin(d*x + c)^ 2 + 300*a^4*log(sin(d*x + c) - 1) + 192*a^4*sin(d*x + c) - 12*(11*a^4*sin( d*x + c) - 10*a^4)/(sin(d*x + c)^2 - 2*sin(d*x + c) + 1))/d
Timed out. \[ \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx=\text {Timed out} \]
Time = 7.45 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.94 \[ \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx=\frac {25\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {50\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d}-\frac {50\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-150\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {950\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}-\frac {1700\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {2180\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {2452\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {2180\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {1700\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {950\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-150\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+50\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
(25*a^4*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (50*a^4*log(tan(c/2 + (d*x)/2) - 1))/d - ((950*a^4*tan(c/2 + (d*x)/2)^3)/3 - 150*a^4*tan(c/2 + (d*x)/2)^2 - (1700*a^4*tan(c/2 + (d*x)/2)^4)/3 + (2180*a^4*tan(c/2 + (d*x)/2)^5)/3 - (2452*a^4*tan(c/2 + (d*x)/2)^6)/3 + (2180*a^4*tan(c/2 + (d*x)/2)^7)/3 - ( 1700*a^4*tan(c/2 + (d*x)/2)^8)/3 + (950*a^4*tan(c/2 + (d*x)/2)^9)/3 - 150* a^4*tan(c/2 + (d*x)/2)^10 + 50*a^4*tan(c/2 + (d*x)/2)^11 + 50*a^4*tan(c/2 + (d*x)/2))/(d*(10*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2) - 20*tan(c/ 2 + (d*x)/2)^3 + 31*tan(c/2 + (d*x)/2)^4 - 40*tan(c/2 + (d*x)/2)^5 + 44*ta n(c/2 + (d*x)/2)^6 - 40*tan(c/2 + (d*x)/2)^7 + 31*tan(c/2 + (d*x)/2)^8 - 2 0*tan(c/2 + (d*x)/2)^9 + 10*tan(c/2 + (d*x)/2)^10 - 4*tan(c/2 + (d*x)/2)^1 1 + tan(c/2 + (d*x)/2)^12 + 1))