Integrand size = 21, antiderivative size = 78 \[ \int \frac {\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {x}{a}+\frac {3 \text {arctanh}(\sin (c+d x))}{8 a d}+\frac {(8-3 \sec (c+d x)) \tan (c+d x)}{8 a d}-\frac {(4-3 \sec (c+d x)) \tan ^3(c+d x)}{12 a d} \] Output:
-x/a+3/8*arctanh(sin(d*x+c))/a/d+1/8*(8-3*sec(d*x+c))*tan(d*x+c)/a/d-1/12* (4-3*sec(d*x+c))*tan(d*x+c)^3/a/d
Leaf count is larger than twice the leaf count of optimal. \(893\) vs. \(2(78)=156\).
Time = 7.01 (sec) , antiderivative size = 893, normalized size of antiderivative = 11.45 \[ \int \frac {\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[Tan[c + d*x]^6/(a + a*Sec[c + d*x]),x]
Output:
(-2*x*Cos[c/2 + (d*x)/2]^2*Sec[c + d*x])/(a + a*Sec[c + d*x]) - (3*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x])/( 4*d*(a + a*Sec[c + d*x])) + (3*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x])/(4*d*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x])/(8*d*(a + a*Sec[c + d*x])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) - (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*Sin[(d*x)/2] )/(3*d*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Si n[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*(-19*Cos[c/2] + 11*Sin[c/2]))/(24*d*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (8*Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*S in[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d *x)/2] - Sin[c/2 + (d*x)/2])) - (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x])/(8*d*( a + a*Sec[c + d*x])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) - (Cos[c/ 2 + (d*x)/2]^2*Sec[c + d*x]*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d*x])*(Cos[c /2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*(19*Cos[c/2] + 11*Sin[c/2]))/(24*d*(a + a*Sec[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (8*Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*Sin[(d*x)/2])/(3*d*(a + a*Sec[c + d* x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))
Time = 0.45 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4376, 25, 3042, 4369, 3042, 4369, 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.
| \(\displaystyle \int \frac {\tan ^6(c+d x)}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot \left (c+d x+\frac {\pi }{2}\right )^6}{a \csc \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int -\left ((a-a \sec (c+d x)) \tan ^4(c+d x)\right )dx}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (a-a \sec (c+d x)) \tan ^4(c+d x)dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \cot \left (c+d x+\frac {\pi }{2}\right )^4 \left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}\) |
| \(\Big \downarrow \) 4369 |
| \(\displaystyle -\frac {\frac {\tan ^3(c+d x) (4 a-3 a \sec (c+d x))}{12 d}-\frac {1}{4} \int (4 a-3 a \sec (c+d x)) \tan ^2(c+d x)dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\tan ^3(c+d x) (4 a-3 a \sec (c+d x))}{12 d}-\frac {1}{4} \int \cot \left (c+d x+\frac {\pi }{2}\right )^2 \left (4 a-3 a \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}\) |
| \(\Big \downarrow \) 4369 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \int (8 a-3 a \sec (c+d x))dx-\frac {\tan (c+d x) (8 a-3 a \sec (c+d x))}{2 d}\right )+\frac {\tan ^3(c+d x) (4 a-3 a \sec (c+d x))}{12 d}}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (8 a x-\frac {3 a \text {arctanh}(\sin (c+d x))}{d}\right )-\frac {\tan (c+d x) (8 a-3 a \sec (c+d x))}{2 d}\right )+\frac {\tan ^3(c+d x) (4 a-3 a \sec (c+d x))}{12 d}}{a^2}\) |
Input:
Int[Tan[c + d*x]^6/(a + a*Sec[c + d*x]),x]
Output:
-((((4*a - 3*a*Sec[c + d*x])*Tan[c + d*x]^3)/(12*d) + ((8*a*x - (3*a*ArcTa nh[Sin[c + d*x]])/d)/2 - ((8*a - 3*a*Sec[c + d*x])*Tan[c + d*x])/(2*d))/4) /a^2)
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-e)*(e*Cot[c + d*x])^(m - 1)*((a*m + b*(m - 1)*Csc [c + d*x])/(d*m*(m - 1))), x] - Simp[e^2/m Int[(e*Cot[c + d*x])^(m - 2)*( a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m , 1]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.94
| method | result | size |
| risch | \(-\frac {x}{a}+\frac {i \left (15 \,{\mathrm e}^{7 i \left (d x +c \right )}+48 \,{\mathrm e}^{6 i \left (d x +c \right )}-9 \,{\mathrm e}^{5 i \left (d x +c \right )}+96 \,{\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{3 i \left (d x +c \right )}+80 \,{\mathrm e}^{2 i \left (d x +c \right )}-15 \,{\mathrm e}^{i \left (d x +c \right )}+32\right )}{12 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 a d}\) | \(151\) |
| derivativedivides | \(\frac {-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {5}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {5}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}}{a d}\) | \(170\) |
| default | \(\frac {-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {5}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {5}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {11}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8}}{a d}\) | \(170\) |
Input:
int(tan(d*x+c)^6/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-x/a+1/12*I*(15*exp(7*I*(d*x+c))+48*exp(6*I*(d*x+c))-9*exp(5*I*(d*x+c))+96 *exp(4*I*(d*x+c))+9*exp(3*I*(d*x+c))+80*exp(2*I*(d*x+c))-15*exp(I*(d*x+c)) +32)/d/a/(exp(2*I*(d*x+c))+1)^4-3/8/a/d*ln(exp(I*(d*x+c))-I)+3/8/a/d*ln(ex p(I*(d*x+c))+I)
Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.37 \[ \int \frac {\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {48 \, d x \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) + 9 \, \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (32 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )^{2} - 8 \, \cos \left (d x + c\right ) + 6\right )} \sin \left (d x + c\right )}{48 \, a d \cos \left (d x + c\right )^{4}} \] Input:
integrate(tan(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="fricas")
Output:
-1/48*(48*d*x*cos(d*x + c)^4 - 9*cos(d*x + c)^4*log(sin(d*x + c) + 1) + 9* cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*(32*cos(d*x + c)^3 - 15*cos(d*x + c)^2 - 8*cos(d*x + c) + 6)*sin(d*x + c))/(a*d*cos(d*x + c)^4)
\[ \int \frac {\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\tan ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(tan(d*x+c)**6/(a+a*sec(d*x+c)),x)
Output:
Integral(tan(c + d*x)**6/(sec(c + d*x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (72) = 144\).
Time = 0.12 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.17 \[ \int \frac {\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {2 \, {\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {71 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {137 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {33 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a - \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {48 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a}}{24 \, d} \] Input:
integrate(tan(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="maxima")
Output:
1/24*(2*(15*sin(d*x + c)/(cos(d*x + c) + 1) - 71*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 137*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 33*sin(d*x + c)^7/( cos(d*x + c) + 1)^7)/(a - 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a*si n(d*x + c)^4/(cos(d*x + c) + 1)^4 - 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^ 6 + a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) - 48*arctan(sin(d*x + c)/(cos(d *x + c) + 1))/a + 9*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a - 9*log(sin (d*x + c)/(cos(d*x + c) + 1) - 1)/a)/d
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.58 \[ \int \frac {\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {24 \, {\left (d x + c\right )}}{a} - \frac {9 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} + \frac {9 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (33 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 137 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 71 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a}}{24 \, d} \] Input:
integrate(tan(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="giac")
Output:
-1/24*(24*(d*x + c)/a - 9*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a + 9*log(abs (tan(1/2*d*x + 1/2*c) - 1))/a + 2*(33*tan(1/2*d*x + 1/2*c)^7 - 137*tan(1/2 *d*x + 1/2*c)^5 + 71*tan(1/2*d*x + 1/2*c)^3 - 15*tan(1/2*d*x + 1/2*c))/((t an(1/2*d*x + 1/2*c)^2 - 1)^4*a))/d
Time = 13.12 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.78 \[ \int \frac {\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a\,d}-\frac {x}{a}+\frac {-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {137\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}-\frac {71\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \] Input:
int(tan(c + d*x)^6/(a + a/cos(c + d*x)),x)
Output:
(3*atanh(tan(c/2 + (d*x)/2)))/(4*a*d) - x/a + ((5*tan(c/2 + (d*x)/2))/4 - (71*tan(c/2 + (d*x)/2)^3)/12 + (137*tan(c/2 + (d*x)/2)^5)/12 - (11*tan(c/2 + (d*x)/2)^7)/4)/(d*(a - 4*a*tan(c/2 + (d*x)/2)^2 + 6*a*tan(c/2 + (d*x)/2 )^4 - 4*a*tan(c/2 + (d*x)/2)^6 + a*tan(c/2 + (d*x)/2)^8))
Time = 0.16 (sec) , antiderivative size = 463, normalized size of antiderivative = 5.94 \[ \int \frac {\tan ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {-45 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4}+90 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}-45 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+45 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4}-90 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}+45 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )^{5}-40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )^{3}+120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} \tan \left (d x +c \right )-120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} d x +75 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )^{5}+80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )^{3}-240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )+240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} d x -45 \cos \left (d x +c \right ) \sin \left (d x +c \right )+24 \cos \left (d x +c \right ) \tan \left (d x +c \right )^{5}-40 \cos \left (d x +c \right ) \tan \left (d x +c \right )^{3}+120 \cos \left (d x +c \right ) \tan \left (d x +c \right )-120 \cos \left (d x +c \right ) d x -24 \sin \left (d x +c \right )^{5}}{120 \cos \left (d x +c \right ) a d \left (\sin \left (d x +c \right )^{4}-2 \sin \left (d x +c \right )^{2}+1\right )} \] Input:
int(tan(d*x+c)^6/(a+a*sec(d*x+c)),x)
Output:
( - 45*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 + 90*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2 - 45*cos(c + d*x)*log(tan( (c + d*x)/2) - 1) + 45*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x) **4 - 90*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 + 45*cos(c + d*x)*log(tan((c + d*x)/2) + 1) + 24*cos(c + d*x)*sin(c + d*x)**4*tan(c + d*x)**5 - 40*cos(c + d*x)*sin(c + d*x)**4*tan(c + d*x)**3 + 120*cos(c + d*x)*sin(c + d*x)**4*tan(c + d*x) - 120*cos(c + d*x)*sin(c + d*x)**4*d*x + 75*cos(c + d*x)*sin(c + d*x)**3 - 48*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)**5 + 80*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)**3 - 240*cos(c + d *x)*sin(c + d*x)**2*tan(c + d*x) + 240*cos(c + d*x)*sin(c + d*x)**2*d*x - 45*cos(c + d*x)*sin(c + d*x) + 24*cos(c + d*x)*tan(c + d*x)**5 - 40*cos(c + d*x)*tan(c + d*x)**3 + 120*cos(c + d*x)*tan(c + d*x) - 120*cos(c + d*x)* d*x - 24*sin(c + d*x)**5)/(120*cos(c + d*x)*a*d*(sin(c + d*x)**4 - 2*sin(c + d*x)**2 + 1))