Integrand size = 21, antiderivative size = 157 \[ \int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx=-a^2 x-\frac {5 a b \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 \tan (c+d x)}{d}+\frac {5 a b \sec (c+d x) \tan (c+d x)}{8 d}-\frac {a^2 \tan ^3(c+d x)}{3 d}-\frac {5 a b \sec (c+d x) \tan ^3(c+d x)}{12 d}+\frac {a^2 \tan ^5(c+d x)}{5 d}+\frac {a b \sec (c+d x) \tan ^5(c+d x)}{3 d}+\frac {b^2 \tan ^7(c+d x)}{7 d} \] Output:
-a^2*x-5/8*a*b*arctanh(sin(d*x+c))/d+a^2*tan(d*x+c)/d+5/8*a*b*sec(d*x+c)*t an(d*x+c)/d-1/3*a^2*tan(d*x+c)^3/d-5/12*a*b*sec(d*x+c)*tan(d*x+c)^3/d+1/5* a^2*tan(d*x+c)^5/d+1/3*a*b*sec(d*x+c)*tan(d*x+c)^5/d+1/7*b^2*tan(d*x+c)^7/ d
Time = 1.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx=\frac {-840 a^2 \arctan (\tan (c+d x))-525 a b \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (175 a b (-1+7 \cos (2 (c+d x))) \sec ^5(c+d x)+105 a b \sec (c+d x) \left (-5+16 \tan ^4(c+d x)\right )+8 \left (105 a^2-35 a^2 \tan ^2(c+d x)+21 a^2 \tan ^4(c+d x)+15 b^2 \tan ^6(c+d x)\right )\right )}{840 d} \] Input:
Integrate[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^6,x]
Output:
(-840*a^2*ArcTan[Tan[c + d*x]] - 525*a*b*ArcTanh[Sin[c + d*x]] + Tan[c + d *x]*(175*a*b*(-1 + 7*Cos[2*(c + d*x)])*Sec[c + d*x]^5 + 105*a*b*Sec[c + d* x]*(-5 + 16*Tan[c + d*x]^4) + 8*(105*a^2 - 35*a^2*Tan[c + d*x]^2 + 21*a^2* Tan[c + d*x]^4 + 15*b^2*Tan[c + d*x]^6)))/(840*d)
Time = 0.41 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4374, 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.
| \(\displaystyle \int \tan ^6(c+d x) (a+b \sec (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot \left (c+d x+\frac {\pi }{2}\right )^6 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \int \left (a^2 \tan ^6(c+d x)+2 a b \tan ^6(c+d x) \sec (c+d x)+b^2 \tan ^6(c+d x) \sec ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 \tan ^5(c+d x)}{5 d}-\frac {a^2 \tan ^3(c+d x)}{3 d}+\frac {a^2 \tan (c+d x)}{d}-a^2 x-\frac {5 a b \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a b \tan ^5(c+d x) \sec (c+d x)}{3 d}-\frac {5 a b \tan ^3(c+d x) \sec (c+d x)}{12 d}+\frac {5 a b \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b^2 \tan ^7(c+d x)}{7 d}\) |
Input:
Int[(a + b*Sec[c + d*x])^2*Tan[c + d*x]^6,x]
Output:
-(a^2*x) - (5*a*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*Tan[c + d*x])/d + (5 *a*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (a^2*Tan[c + d*x]^3)/(3*d) - (5*a* b*Sec[c + d*x]*Tan[c + d*x]^3)/(12*d) + (a^2*Tan[c + d*x]^5)/(5*d) + (a*b* Sec[c + d*x]*Tan[c + d*x]^5)/(3*d) + (b^2*Tan[c + d*x]^7)/(7*d)
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Time = 1.98 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.06
| method | result | size |
| parts | \(\frac {a^{2} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {b^{2} \tan \left (d x +c \right )^{7}}{7 d}+\frac {2 a b \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(167\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-d x -c \right )+2 a b \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {b^{2} \sin \left (d x +c \right )^{7}}{7 \cos \left (d x +c \right )^{7}}}{d}\) | \(168\) |
| default | \(\frac {a^{2} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-d x -c \right )+2 a b \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {b^{2} \sin \left (d x +c \right )^{7}}{7 \cos \left (d x +c \right )^{7}}}{d}\) | \(168\) |
| risch | \(-a^{2} x -\frac {i \left (1155 a b \,{\mathrm e}^{13 i \left (d x +c \right )}-2520 a^{2} {\mathrm e}^{12 i \left (d x +c \right )}+840 b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+980 a b \,{\mathrm e}^{11 i \left (d x +c \right )}-10080 a^{2} {\mathrm e}^{10 i \left (d x +c \right )}+2975 a b \,{\mathrm e}^{9 i \left (d x +c \right )}-20440 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+4200 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-24640 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}-2975 a b \,{\mathrm e}^{5 i \left (d x +c \right )}-16968 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+2520 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-980 a b \,{\mathrm e}^{3 i \left (d x +c \right )}-6496 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-1155 a b \,{\mathrm e}^{i \left (d x +c \right )}-1288 a^{2}+120 b^{2}\right )}{420 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {5 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {5 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(282\) |
Input:
int((a+b*sec(d*x+c))^2*tan(d*x+c)^6,x,method=_RETURNVERBOSE)
Output:
a^2/d*(1/5*tan(d*x+c)^5-1/3*tan(d*x+c)^3+tan(d*x+c)-arctan(tan(d*x+c)))+1/ 7*b^2*tan(d*x+c)^7/d+2*a*b/d*(1/6*sin(d*x+c)^7/cos(d*x+c)^6-1/24*sin(d*x+c )^7/cos(d*x+c)^4+1/16*sin(d*x+c)^7/cos(d*x+c)^2+1/16*sin(d*x+c)^5+5/48*sin (d*x+c)^3+5/16*sin(d*x+c)-5/16*ln(sec(d*x+c)+tan(d*x+c)))
Time = 0.13 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx=-\frac {1680 \, a^{2} d x \cos \left (d x + c\right )^{7} + 525 \, a b \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 525 \, a b \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (1155 \, a b \cos \left (d x + c\right )^{5} + 8 \, {\left (161 \, a^{2} - 15 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 910 \, a b \cos \left (d x + c\right )^{3} - 8 \, {\left (77 \, a^{2} - 45 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 280 \, a b \cos \left (d x + c\right ) + 24 \, {\left (7 \, a^{2} - 15 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{1680 \, d \cos \left (d x + c\right )^{7}} \] Input:
integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^6,x, algorithm="fricas")
Output:
-1/1680*(1680*a^2*d*x*cos(d*x + c)^7 + 525*a*b*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 525*a*b*cos(d*x + c)^7*log(-sin(d*x + c) + 1) - 2*(1155*a*b*co s(d*x + c)^5 + 8*(161*a^2 - 15*b^2)*cos(d*x + c)^6 - 910*a*b*cos(d*x + c)^ 3 - 8*(77*a^2 - 45*b^2)*cos(d*x + c)^4 + 280*a*b*cos(d*x + c) + 24*(7*a^2 - 15*b^2)*cos(d*x + c)^2 + 120*b^2)*sin(d*x + c))/(d*cos(d*x + c)^7)
\[ \int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \tan ^{6}{\left (c + d x \right )}\, dx \] Input:
integrate((a+b*sec(d*x+c))**2*tan(d*x+c)**6,x)
Output:
Integral((a + b*sec(c + d*x))**2*tan(c + d*x)**6, x)
Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.96 \[ \int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx=\frac {240 \, b^{2} \tan \left (d x + c\right )^{7} + 112 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{2} - 35 \, a b {\left (\frac {2 \, {\left (33 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{1680 \, d} \] Input:
integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^6,x, algorithm="maxima")
Output:
1/1680*(240*b^2*tan(d*x + c)^7 + 112*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c))*a^2 - 35*a*b*(2*(33*sin(d*x + c)^5 - 40 *sin(d*x + c)^3 + 15*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3* sin(d*x + c)^2 - 1) + 15*log(sin(d*x + c) + 1) - 15*log(sin(d*x + c) - 1)) )/d
Time = 0.54 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.80 \[ \int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx=-\frac {840 \, {\left (d x + c\right )} a^{2} + 525 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 525 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (840 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 525 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 6160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3500 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 19768 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 9905 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 28896 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7680 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 19768 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9905 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6160 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3500 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 840 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 525 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{840 \, d} \] Input:
integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^6,x, algorithm="giac")
Output:
-1/840*(840*(d*x + c)*a^2 + 525*a*b*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 5 25*a*b*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(840*a^2*tan(1/2*d*x + 1/2*c )^13 - 525*a*b*tan(1/2*d*x + 1/2*c)^13 - 6160*a^2*tan(1/2*d*x + 1/2*c)^11 + 3500*a*b*tan(1/2*d*x + 1/2*c)^11 + 19768*a^2*tan(1/2*d*x + 1/2*c)^9 - 99 05*a*b*tan(1/2*d*x + 1/2*c)^9 - 28896*a^2*tan(1/2*d*x + 1/2*c)^7 + 7680*b^ 2*tan(1/2*d*x + 1/2*c)^7 + 19768*a^2*tan(1/2*d*x + 1/2*c)^5 + 9905*a*b*tan (1/2*d*x + 1/2*c)^5 - 6160*a^2*tan(1/2*d*x + 1/2*c)^3 - 3500*a*b*tan(1/2*d *x + 1/2*c)^3 + 840*a^2*tan(1/2*d*x + 1/2*c) + 525*a*b*tan(1/2*d*x + 1/2*c ))/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d
Time = 11.34 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.57 \[ \int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx=\frac {\left (\frac {5\,a\,b}{4}-2\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {44\,a^2}{3}-\frac {25\,a\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {283\,a\,b}{12}-\frac {706\,a^2}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {344\,a^2}{5}-\frac {128\,b^2}{7}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-\frac {706\,a^2}{15}-\frac {283\,b\,a}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {44\,a^2}{3}+\frac {25\,b\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-2\,a^2-\frac {5\,b\,a}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {2\,a^2\,\mathrm {atan}\left (\frac {64\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,a^6+25\,a^4\,b^2}+\frac {25\,a^4\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,a^6+25\,a^4\,b^2}\right )}{d}-\frac {5\,a\,b\,\mathrm {atanh}\left (\frac {40\,a^5\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{40\,a^5\,b+\frac {125\,a^3\,b^3}{8}}+\frac {125\,a^3\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,\left (40\,a^5\,b+\frac {125\,a^3\,b^3}{8}\right )}\right )}{4\,d} \] Input:
int(tan(c + d*x)^6*(a + b/cos(c + d*x))^2,x)
Output:
(tan(c/2 + (d*x)/2)^7*((344*a^2)/5 - (128*b^2)/7) + tan(c/2 + (d*x)/2)^13* ((5*a*b)/4 - 2*a^2) + tan(c/2 + (d*x)/2)^3*((25*a*b)/3 + (44*a^2)/3) - tan (c/2 + (d*x)/2)^11*((25*a*b)/3 - (44*a^2)/3) - tan(c/2 + (d*x)/2)^5*((283* a*b)/12 + (706*a^2)/15) + tan(c/2 + (d*x)/2)^9*((283*a*b)/12 - (706*a^2)/1 5) - tan(c/2 + (d*x)/2)*((5*a*b)/4 + 2*a^2))/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^ 8 + 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2 )^14 - 1)) - (2*a^2*atan((64*a^6*tan(c/2 + (d*x)/2))/(64*a^6 + 25*a^4*b^2) + (25*a^4*b^2*tan(c/2 + (d*x)/2))/(64*a^6 + 25*a^4*b^2)))/d - (5*a*b*atan h((40*a^5*b*tan(c/2 + (d*x)/2))/(40*a^5*b + (125*a^3*b^3)/8) + (125*a^3*b^ 3*tan(c/2 + (d*x)/2))/(8*(40*a^5*b + (125*a^3*b^3)/8))))/(4*d)
Time = 0.17 (sec) , antiderivative size = 703, normalized size of antiderivative = 4.48 \[ \int (a+b \sec (c+d x))^2 \tan ^6(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+b*sec(d*x+c))^2*tan(d*x+c)^6,x)
Output:
(525*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a*b - 1575*cos (c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a*b + 1575*cos(c + d*x )*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a*b - 525*cos(c + d*x)*log(tan ((c + d*x)/2) - 1)*a*b - 525*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a*b + 1575*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)** 4*a*b - 1575*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a*b + 525*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a*b + 168*cos(c + d*x)*sin(c + d*x)**6*tan(c + d*x)**5*a**2 - 280*cos(c + d*x)*sin(c + d*x)**6*tan(c + d* x)**3*a**2 + 840*cos(c + d*x)*sin(c + d*x)**6*tan(c + d*x)*a**2 - 840*cos( c + d*x)*sin(c + d*x)**6*a**2*d*x - 1155*cos(c + d*x)*sin(c + d*x)**5*a*b - 504*cos(c + d*x)*sin(c + d*x)**4*tan(c + d*x)**5*a**2 + 840*cos(c + d*x) *sin(c + d*x)**4*tan(c + d*x)**3*a**2 - 2520*cos(c + d*x)*sin(c + d*x)**4* tan(c + d*x)*a**2 + 2520*cos(c + d*x)*sin(c + d*x)**4*a**2*d*x + 1400*cos( c + d*x)*sin(c + d*x)**3*a*b + 504*cos(c + d*x)*sin(c + d*x)**2*tan(c + d* x)**5*a**2 - 840*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)**3*a**2 + 2520* cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)*a**2 - 2520*cos(c + d*x)*sin(c + d*x)**2*a**2*d*x - 525*cos(c + d*x)*sin(c + d*x)*a*b - 168*cos(c + d*x)*t an(c + d*x)**5*a**2 + 280*cos(c + d*x)*tan(c + d*x)**3*a**2 - 840*cos(c + d*x)*tan(c + d*x)*a**2 + 840*cos(c + d*x)*a**2*d*x - 120*sin(c + d*x)**7*b **2)/(840*cos(c + d*x)*d*(sin(c + d*x)**6 - 3*sin(c + d*x)**4 + 3*sin(c...