Integrand size = 29, antiderivative size = 110 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {11 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {17 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))} \] Output:
-11/2*a^3*arctanh(cos(d*x+c))/d-3*a^3*cot(d*x+c)/d-1/2*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+17/3*a^3*cos(d*x+c)/d/(1-si n(d*x+c))
Time = 2.46 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.73 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-36 \cot \left (\frac {1}{2} (c+d x)\right )-3 \csc ^2\left (\frac {1}{2} (c+d x)\right )-132 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+132 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\frac {16}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {32 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {272 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+36 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{24 d} \] Input:
Integrate[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(-36*Cot[(c + d*x)/2] - 3*Csc[(c + d*x)/2]^2 - 132*Log[Cos[(c + d*x)/ 2]] + 132*Log[Sin[(c + d*x)/2]] + 3*Sec[(c + d*x)/2]^2 + 16/(Cos[(c + d*x) /2] - Sin[(c + d*x)/2])^2 + (32*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[ (c + d*x)/2])^3 + (272*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x) /2]) + 36*Tan[(c + d*x)/2]))/(24*d)
Time = 0.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04, 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.
| \(\displaystyle \int \csc ^3(c+d x) \sec ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{\sin (c+d x)^3 \cos (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle a^4 \int \left (\frac {\csc ^3(c+d x)}{a}+\frac {3 \csc ^2(c+d x)}{a}+\frac {5 \csc (c+d x)}{a}+\frac {5}{a (1-\sin (c+d x))}+\frac {2}{a (1-\sin (c+d x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \left (-\frac {11 \text {arctanh}(\cos (c+d x))}{2 a d}-\frac {3 \cot (c+d x)}{a d}+\frac {17 \cos (c+d x)}{3 a d (1-\sin (c+d x))}+\frac {2 \cos (c+d x)}{3 a d (1-\sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}\right )\) |
Input:
Int[Csc[c + d*x]^3*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
a^4*((-11*ArcTanh[Cos[c + d*x]])/(2*a*d) - (3*Cot[c + d*x])/(a*d) - (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) + (17*Cos[c + d*x])/(3*a*d*(1 - Sin[c + d*x])))
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 = 1.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.08
| method | result | size |
| parallelrisch | \(\frac {\left (44 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-188 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+9 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+314 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {502}{3}\right ) a^{3}}{8 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(119\) |
| risch | \(\frac {a^{3} \left (-99 i {\mathrm e}^{5 i \left (d x +c \right )}+33 \,{\mathrm e}^{6 i \left (d x +c \right )}+210 i {\mathrm e}^{3 i \left (d x +c \right )}-154 \,{\mathrm e}^{4 i \left (d x +c \right )}-123 i {\mathrm e}^{i \left (d x +c \right )}+161 \,{\mathrm e}^{2 i \left (d x +c \right )}-52\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(148\) |
| derivativedivides | \(\frac {-a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(188\) |
| default | \(\frac {-a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{3} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(188\) |
Input:
int(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/8*(44*(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) ^5+9*tan(1/2*d*x+1/2*c)^4+cot(1/2*d*x+1/2*c)^2-188*tan(1/2*d*x+1/2*c)^2+9* cot(1/2*d*x+1/2*c)+314*tan(1/2*d*x+1/2*c)-502/3)*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. 428 vs. \(2 (98) = 196\).
Time = 0.11 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.89 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {104 \, a^{3} \cos \left (d x + c\right )^{4} + 142 \, a^{3} \cos \left (d x + c\right )^{3} - 90 \, a^{3} \cos \left (d x + c\right )^{2} - 136 \, a^{3} \cos \left (d x + c\right ) - 8 \, a^{3} + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (52 \, a^{3} \cos \left (d x + c\right )^{3} - 19 \, a^{3} \cos \left (d x + c\right )^{2} - 64 \, a^{3} \cos \left (d x + c\right ) + 4 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - 2 \, d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/12*(104*a^3*cos(d*x + c)^4 + 142*a^3*cos(d*x + c)^3 - 90*a^3*cos(d*x + c)^2 - 136*a^3*cos(d*x + c) - 8*a^3 + 33*(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 + (a^3*cos(d*x + c)^3 + 2*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) - 33*(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 + (a^3*cos(d*x + c)^3 + 2*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*(52*a^3*cos(d*x + c)^3 - 19*a^3*cos(d*x + c)^2 - 64*a^3 *cos(d*x + c) + 4*a^3)*sin(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) + (d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 - d*cos(d*x + c) - 2*d)*sin(d*x + c) + 2*d)
Timed out. \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(csc(d*x+c)**3*sec(d*x+c)**4*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.65 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {12 \, {\left (\tan \left (d x + c\right )^{3} - \frac {3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
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 + 3*tan(d*x + c))*a^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)) + 6*a^3*(2*(3*cos(d*x + c)^2 + 1)/cos(d*x + c)^3 - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)))/d
Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.36 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 132 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {3 \, {\left (66 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {16 \, {\left (21 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{24 \, d} \] Input:
integrate(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/24*(3*a^3*tan(1/2*d*x + 1/2*c)^2 + 132*a^3*log(abs(tan(1/2*d*x + 1/2*c)) ) + 36*a^3*tan(1/2*d*x + 1/2*c) - 3*(66*a^3*tan(1/2*d*x + 1/2*c)^2 + 12*a^ 3*tan(1/2*d*x + 1/2*c) + a^3)/tan(1/2*d*x + 1/2*c)^2 - 16*(21*a^3*tan(1/2* d*x + 1/2*c)^2 - 36*a^3*tan(1/2*d*x + 1/2*c) + 19*a^3)/(tan(1/2*d*x + 1/2* c) - 1)^3)/d
Time = 31.40 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.66 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {11\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {-62\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {227\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {403\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}+\frac {9\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a^3}{2}}{d\,\left (-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \] Input:
int((a + a*sin(c + d*x))^3/(cos(c + d*x)^4*sin(c + d*x)^3),x)
Output:
(a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (11*a^3*log(tan(c/2 + (d*x)/2)))/(2*d) - ((227*a^3*tan(c/2 + (d*x)/2)^3)/2 - (403*a^3*tan(c/2 + (d*x)/2)^2)/6 - 6 2*a^3*tan(c/2 + (d*x)/2)^4 + a^3/2 + (9*a^3*tan(c/2 + (d*x)/2))/2)/(d*(4*t an(c/2 + (d*x)/2)^2 - 12*tan(c/2 + (d*x)/2)^3 + 12*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^5)) + (3*a^3*tan(c/2 + (d*x)/2))/(2*d)
Time = 0.19 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.07 \[ \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (132 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-396 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+396 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-132 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-188 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+378 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-314 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \] Input:
int(csc(d*x+c)^3*sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(132*log(tan((c + d*x)/2))*tan((c + d*x)/2)**5 - 396*log(tan((c + d* x)/2))*tan((c + d*x)/2)**4 + 396*log(tan((c + d*x)/2))*tan((c + d*x)/2)**3 - 132*log(tan((c + d*x)/2))*tan((c + d*x)/2)**2 + 3*tan((c + d*x)/2)**7 + 27*tan((c + d*x)/2)**6 - 188*tan((c + d*x)/2)**5 + 378*tan((c + d*x)/2)** 3 - 314*tan((c + d*x)/2)**2 + 27*tan((c + d*x)/2) + 3))/(24*tan((c + d*x)/ 2)**2*d*(tan((c + d*x)/2)**3 - 3*tan((c + d*x)/2)**2 + 3*tan((c + d*x)/2) - 1))