Integrand size = 29, antiderivative size = 128 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {17 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {6 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))} \]
-17/2*a^3*arctanh(cos(d*x+c))/d-6*a^3*cot(d*x+c)/d-1/3*a^3*cot(d*x+c)^3/d- 3/2*a^3*cot(d*x+c)*csc(d*x+c)/d+2/3*a^3*cos(d*x+c)/d/(1-sin(d*x+c))^2+23/3 *a^3*cos(d*x+c)/d/(1-sin(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(287\) vs. \(2(128)=256\).
Time = 6.41 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.24 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (-\frac {17 \cot \left (\frac {1}{2} (c+d x)\right )}{6 d}-\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 d}-\frac {17 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {17 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {2}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {46 \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {17 \tan \left (\frac {1}{2} (c+d x)\right )}{6 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 d}\right ) \]
a^3*((-17*Cot[(c + d*x)/2])/(6*d) - (3*Csc[(c + d*x)/2]^2)/(8*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d) - (17*Log[Cos[(c + d*x)/2]])/(2*d) + (17*Log[Sin[(c + d*x)/2]])/(2*d) + (3*Sec[(c + d*x)/2]^2)/(8*d) + 2/(3*d* (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*Sin[(c + d*x)/2])/(3*d*(Cos[ (c + d*x)/2] - Sin[(c + d*x)/2])^3) + (46*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (17*Tan[(c + d*x)/2])/(6*d) + (Sec[(c + d* x)/2]^2*Tan[(c + d*x)/2])/(24*d))
Time = 0.41 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3351, 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.
| \(\displaystyle \int \csc ^4(c+d x) \sec ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{\sin (c+d x)^4 \cos (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle a^4 \int \left (\frac {\csc ^4(c+d x)}{a}+\frac {3 \csc ^3(c+d x)}{a}+\frac {5 \csc ^2(c+d x)}{a}+\frac {7 \csc (c+d x)}{a}+\frac {7}{a (1-\sin (c+d x))}+\frac {2}{a (1-\sin (c+d x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \left (-\frac {17 \text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}-\frac {6 \cot (c+d x)}{a d}+\frac {23 \cos (c+d x)}{3 a d (1-\sin (c+d x))}+\frac {2 \cos (c+d x)}{3 a d (1-\sin (c+d x))^2}-\frac {3 \cot (c+d x) \csc (c+d x)}{2 a d}\right )\) |
a^4*((-17*ArcTanh[Cos[c + d*x]])/(2*a*d) - (6*Cot[c + d*x])/(a*d) - Cot[c + d*x]^3/(3*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) + (2*Cos[c + d*x] )/(3*a*d*(1 - Sin[c + d*x])^2) + (23*Cos[c + d*x])/(3*a*d*(1 - Sin[c + d*x ])))
3.9.18.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Time = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.13
| method | result | size |
| parallelrisch | \(\frac {\left (204 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )+6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+6 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-846 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+1440 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-762\right ) a^{3}}{24 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(145\) |
| risch | \(\frac {-153 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+51 a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+459 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-289 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-511 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+501 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+189 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-327 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+80 a^{3}}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(196\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}-\frac {2}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{3}\right )}{d}\) | \(232\) |
| default | \(\frac {a^{3} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}-\frac {2}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{3}\right )}{d}\) | \(232\) |
1/24*(204*(tan(1/2*d*x+1/2*c)-1)^3*ln(tan(1/2*d*x+1/2*c))+tan(1/2*d*x+1/2* c)^6+6*tan(1/2*d*x+1/2*c)^5+45*tan(1/2*d*x+1/2*c)^4+cot(1/2*d*x+1/2*c)^3+6 *cot(1/2*d*x+1/2*c)^2-846*tan(1/2*d*x+1/2*c)^2+45*cot(1/2*d*x+1/2*c)+1440* tan(1/2*d*x+1/2*c)-762)*a^3/d/(tan(1/2*d*x+1/2*c)-1)^3
Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (114) = 228\).
Time = 0.29 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.12 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {160 \, a^{3} \cos \left (d x + c\right )^{5} - 58 \, a^{3} \cos \left (d x + c\right )^{4} - 356 \, a^{3} \cos \left (d x + c\right )^{3} + 70 \, a^{3} \cos \left (d x + c\right )^{2} + 200 \, a^{3} \cos \left (d x + c\right ) - 8 \, a^{3} - 51 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{4} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 51 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{4} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (80 \, a^{3} \cos \left (d x + c\right )^{4} + 109 \, a^{3} \cos \left (d x + c\right )^{3} - 69 \, a^{3} \cos \left (d x + c\right )^{2} - 104 \, a^{3} \cos \left (d x + c\right ) - 4 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 4 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \]
1/12*(160*a^3*cos(d*x + c)^5 - 58*a^3*cos(d*x + c)^4 - 356*a^3*cos(d*x + c )^3 + 70*a^3*cos(d*x + c)^2 + 200*a^3*cos(d*x + c) - 8*a^3 - 51*(a^3*cos(d *x + c)^5 + 2*a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^3 - 4*a^3*cos(d*x + c)^2 + a^3*cos(d*x + c) + 2*a^3 - (a^3*cos(d*x + c)^4 - a^3*cos(d*x + c)^3 - 3*a^3*cos(d*x + c)^2 + a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))*log(1/2* cos(d*x + c) + 1/2) + 51*(a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 2*a^ 3*cos(d*x + c)^3 - 4*a^3*cos(d*x + c)^2 + a^3*cos(d*x + c) + 2*a^3 - (a^3* cos(d*x + c)^4 - a^3*cos(d*x + c)^3 - 3*a^3*cos(d*x + c)^2 + a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*(80*a^3*cos(d* x + c)^4 + 109*a^3*cos(d*x + c)^3 - 69*a^3*cos(d*x + c)^2 - 104*a^3*cos(d* x + c) - 4*a^3)*sin(d*x + c))/(d*cos(d*x + c)^5 + 2*d*cos(d*x + c)^4 - 2*d *cos(d*x + c)^3 - 4*d*cos(d*x + c)^2 + d*cos(d*x + c) - (d*cos(d*x + c)^4 - d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 + d*cos(d*x + c) + 2*d)*sin(d*x + c) + 2*d)
Timed out. \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.60 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {12 \, {\left (\tan \left (d x + c\right )^{3} - \frac {3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} - \frac {9 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} + 9 \, \tan \left (d x + c\right )\right )} a^{3} + 3 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
1/12*(12*(tan(d*x + c)^3 - 3/tan(d*x + c) + 6*tan(d*x + c))*a^3 + 4*(tan(d *x + c)^3 - (9*tan(d*x + c)^2 + 1)/tan(d*x + c)^3 + 9*tan(d*x + c))*a^3 + 3*a^3*(2*(15*cos(d*x + c)^4 - 10*cos(d*x + c)^2 - 2)/(cos(d*x + c)^5 - cos (d*x + c)^3) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 2*a^ 3*(2*(3*cos(d*x + c)^2 + 1)/cos(d*x + c)^3 - 3*log(cos(d*x + c) + 1) + 3*l og(cos(d*x + c) - 1)))/d
Time = 0.36 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.52 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 204 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 69 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {187 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 405 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 394 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3}}}{24 \, d} \]
1/24*(a^3*tan(1/2*d*x + 1/2*c)^3 + 9*a^3*tan(1/2*d*x + 1/2*c)^2 + 204*a^3* log(abs(tan(1/2*d*x + 1/2*c))) + 69*a^3*tan(1/2*d*x + 1/2*c) - (187*a^3*ta n(1/2*d*x + 1/2*c)^6 - 60*a^3*tan(1/2*d*x + 1/2*c)^5 - 405*a^3*tan(1/2*d*x + 1/2*c)^4 + 394*a^3*tan(1/2*d*x + 1/2*c)^3 - 45*a^3*tan(1/2*d*x + 1/2*c) ^2 - 6*a^3*tan(1/2*d*x + 1/2*c) - a^3)/(tan(1/2*d*x + 1/2*c)^2 - tan(1/2*d *x + 1/2*c))^3)/d
Time = 14.33 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.87 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\left (6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+45\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-581\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+897\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-303\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-181\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+45\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-204\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+612\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-612\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+204\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1\right )}{24\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3} \]
(a^3*(6*tan(c/2 + (d*x)/2) + 45*tan(c/2 + (d*x)/2)^2 - 581*tan(c/2 + (d*x) /2)^3 + 897*tan(c/2 + (d*x)/2)^4 - 303*tan(c/2 + (d*x)/2)^5 - 181*tan(c/2 + (d*x)/2)^6 + 45*tan(c/2 + (d*x)/2)^7 + 6*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^9 - 204*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^3 + 612*log( tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^4 - 612*log(tan(c/2 + (d*x)/2))*tan (c/2 + (d*x)/2)^5 + 204*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^6 + 1)) /(24*d*tan(c/2 + (d*x)/2)^3*(tan(c/2 + (d*x)/2) - 1)^3)