Integrand size = 25, antiderivative size = 147 \[ \int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=-\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {8 d^2 \cos (a+b x)}{b^3}-\frac {4 (c+d x)^2 \cos (a+b x)}{b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^2 \sec (a+b x)}{b}+\frac {8 d (c+d x) \sin (a+b x)}{b^2} \] Output:
-4*I*d*(d*x+c)*arctan(exp(I*(b*x+a)))/b^2+8*d^2*cos(b*x+a)/b^3-4*(d*x+c)^2 *cos(b*x+a)/b+2*I*d^2*polylog(2,-I*exp(I*(b*x+a)))/b^3-2*I*d^2*polylog(2,I *exp(I*(b*x+a)))/b^3-(d*x+c)^2*sec(b*x+a)/b+8*d*(d*x+c)*sin(b*x+a)/b^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(364\) vs. \(2(147)=294\).
Time = 2.55 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.48 \[ \int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\frac {4 b c d \text {arctanh}\left (\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )\right )+2 d^2 \left (2 \arctan (\cot (a)) \text {arctanh}\left (\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )\right )-\frac {\csc (a) \left ((b x-\arctan (\cot (a))) \left (\log \left (1-e^{i (b x-\arctan (\cot (a)))}\right )-\log \left (1+e^{i (b x-\arctan (\cot (a)))}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i (b x-\arctan (\cot (a)))}\right )-i \operatorname {PolyLog}\left (2,e^{i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {\csc ^2(a)}}\right )-b^2 (c+d x)^2 \sec (a)-4 \cos (b x) \left (\left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a)-2 b d (c+d x) \sin (a)\right )+4 \left (2 b d (c+d x) \cos (a)+\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a)\right ) \sin (b x)-\frac {b^2 (c+d x)^2 \sin \left (\frac {b x}{2}\right )}{\left (\cos \left (\frac {a}{2}\right )-\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )}+\frac {b^2 (c+d x)^2 \sin \left (\frac {b x}{2}\right )}{\left (\cos \left (\frac {a}{2}\right )+\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )}}{b^3} \] Input:
Integrate[(c + d*x)^2*Sec[a + b*x]^2*Sin[3*a + 3*b*x],x]
Output:
(4*b*c*d*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] + 2*d^2*(2*ArcTan[Cot[a]]*A rcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] - (Csc[a]*((b*x - ArcTan[Cot[a]])*(Lo g[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*(b*x - ArcTan[Cot[a]])) ]) + I*PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - I*PolyLog[2, E^(I*(b*x - ArcTan[Cot[a]]))]))/Sqrt[Csc[a]^2]) - b^2*(c + d*x)^2*Sec[a] - 4*Cos[b*x ]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] - 2*b*d*(c + d*x)*Sin[a]) + 4*(2*b*d* (c + d*x)*Cos[a] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x] - (b^2*(c + d*x)^2*Sin[(b*x)/2])/((Cos[a/2] - Sin[a/2])*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) + (b^2*(c + d*x)^2*Sin[(b*x)/2])/((Cos[a/2] + Sin[a/2])*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])))/b^3
Time = 0.42 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {4931, 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.
| \(\displaystyle \int (c+d x)^2 \sin (3 a+3 b x) \sec ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4931 |
| \(\displaystyle \int \left (3 (c+d x)^2 \sin (a+b x)-(c+d x)^2 \sin (a+b x) \tan ^2(a+b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {8 d^2 \cos (a+b x)}{b^3}+\frac {8 d (c+d x) \sin (a+b x)}{b^2}-\frac {4 (c+d x)^2 \cos (a+b x)}{b}-\frac {(c+d x)^2 \sec (a+b x)}{b}\) |
Input:
Int[(c + d*x)^2*Sec[a + b*x]^2*Sin[3*a + 3*b*x],x]
Output:
((-4*I)*d*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^2 + (8*d^2*Cos[a + b*x])/b^ 3 - (4*(c + d*x)^2*Cos[a + b*x])/b + ((2*I)*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 - ((2*I)*d^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - ((c + d*x)^2 *Sec[a + b*x])/b + (8*d*(c + d*x)*Sin[a + b*x])/b^2
Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int[ExpandTrigExpand[(e + f*x)^m*G[c + d*x] ^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && Member Q[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && E qQ[b*c - a*d, 0] && IGtQ[b/d, 1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (136 ) = 272\).
Time = 3.87 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.27
| method | result | size |
| default | \(-\frac {4 c^{2} \cos \left (b x +a \right )}{b}-\frac {c^{2}}{b \cos \left (b x +a \right )}+\frac {4 d^{2} \left (-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )-2 a \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-a^{2} \cos \left (b x +a \right )\right )}{b^{3}}-\frac {d^{2} \left (\frac {\left (b x +a \right )^{2}}{\cos \left (b x +a \right )}+2 \left (b x +a \right ) \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right )-2 \left (b x +a \right ) \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )-2 i \operatorname {dilog}\left (i {\mathrm e}^{i \left (b x +a \right )}+1\right )+2 i \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )-2 a \left (\frac {b x +a}{\cos \left (b x +a \right )}-\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )\right )+\frac {a^{2}}{\cos \left (b x +a \right )}\right )}{b^{3}}+\frac {8 c d \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )+a \cos \left (b x +a \right )\right )}{b^{2}}-\frac {2 c d x}{b \cos \left (b x +a \right )}+\frac {2 c d \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b^{2}}\) | \(334\) |
| risch | \(-\frac {2 \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +2 i b \,d^{2} x +b^{2} c^{2}+2 i b c d -2 d^{2}\right ) {\mathrm e}^{i \left (b x +a \right )}}{b^{3}}-\frac {2 \left (x^{2} d^{2} b^{2}+2 b^{2} c d x -2 i b \,d^{2} x +b^{2} c^{2}-2 i b c d -2 d^{2}\right ) {\mathrm e}^{-i \left (b x +a \right )}}{b^{3}}-\frac {2 \,{\mathrm e}^{i \left (b x +a \right )} \left (x^{2} d^{2}+2 c d x +c^{2}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}-\frac {4 i d c \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {2 d^{2} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}-\frac {2 d^{2} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{b^{3}}+\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}+\frac {2 i d^{2} \operatorname {dilog}\left (i {\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}-\frac {2 i d^{2} \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {4 i d^{2} a \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}\) | \(345\) |
Input:
int((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x,method=_RETURNVERBOSE)
Output:
-4*c^2/b*cos(b*x+a)-c^2/b/cos(b*x+a)+4*d^2/b^3*(-(b*x+a)^2*cos(b*x+a)+2*co s(b*x+a)+2*(b*x+a)*sin(b*x+a)-2*a*(sin(b*x+a)-(b*x+a)*cos(b*x+a))-a^2*cos( b*x+a))-d^2/b^3*((b*x+a)^2/cos(b*x+a)+2*(b*x+a)*ln(I*exp(I*(b*x+a))+1)-2*( b*x+a)*ln(1-I*exp(I*(b*x+a)))-2*I*dilog(I*exp(I*(b*x+a))+1)+2*I*dilog(1-I* exp(I*(b*x+a)))-2*a*((b*x+a)/cos(b*x+a)-ln(sec(b*x+a)+tan(b*x+a)))+a^2/cos (b*x+a))+8*c*d/b^2*(sin(b*x+a)-(b*x+a)*cos(b*x+a)+a*cos(b*x+a))-2*c*d/b/co s(b*x+a)*x+2*c*d/b^2*ln(sec(b*x+a)+tan(b*x+a))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (129) = 258\).
Time = 0.11 (sec) , antiderivative size = 513, normalized size of antiderivative = 3.49 \[ \int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=-\frac {b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + i \, d^{2} \cos \left (b x + a\right ) {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + i \, d^{2} \cos \left (b x + a\right ) {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - i \, d^{2} \cos \left (b x + a\right ) {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - i \, d^{2} \cos \left (b x + a\right ) {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + 4 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )^{2} - {\left (b c d - a d^{2}\right )} \cos \left (b x + a\right ) \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b c d - a d^{2}\right )} \cos \left (b x + a\right ) \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - {\left (b d^{2} x + a d^{2}\right )} \cos \left (b x + a\right ) \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b d^{2} x + a d^{2}\right )} \cos \left (b x + a\right ) \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b d^{2} x + a d^{2}\right )} \cos \left (b x + a\right ) \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b d^{2} x + a d^{2}\right )} \cos \left (b x + a\right ) \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b c d - a d^{2}\right )} \cos \left (b x + a\right ) \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b c d - a d^{2}\right )} \cos \left (b x + a\right ) \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - 8 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )}{b^{3} \cos \left (b x + a\right )} \] Input:
integrate((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="fricas")
Output:
-(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + I*d^2*cos(b*x + a)*dilog(I*cos(b*x + a) + sin(b*x + a)) + I*d^2*cos(b*x + a)*dilog(I*cos(b*x + a) - sin(b*x + a)) - I*d^2*cos(b*x + a)*dilog(-I*cos(b*x + a) + sin(b*x + a)) - I*d^2*c os(b*x + a)*dilog(-I*cos(b*x + a) - sin(b*x + a)) + 4*(b^2*d^2*x^2 + 2*b^2 *c*d*x + b^2*c^2 - 2*d^2)*cos(b*x + a)^2 - (b*c*d - a*d^2)*cos(b*x + a)*lo g(cos(b*x + a) + I*sin(b*x + a) + I) + (b*c*d - a*d^2)*cos(b*x + a)*log(co s(b*x + a) - I*sin(b*x + a) + I) - (b*d^2*x + a*d^2)*cos(b*x + a)*log(I*co s(b*x + a) + sin(b*x + a) + 1) + (b*d^2*x + a*d^2)*cos(b*x + a)*log(I*cos( b*x + a) - sin(b*x + a) + 1) - (b*d^2*x + a*d^2)*cos(b*x + a)*log(-I*cos(b *x + a) + sin(b*x + a) + 1) + (b*d^2*x + a*d^2)*cos(b*x + a)*log(-I*cos(b* x + a) - sin(b*x + a) + 1) - (b*c*d - a*d^2)*cos(b*x + a)*log(-cos(b*x + a ) + I*sin(b*x + a) + I) + (b*c*d - a*d^2)*cos(b*x + a)*log(-cos(b*x + a) - I*sin(b*x + a) + I) - 8*(b*d^2*x + b*c*d)*cos(b*x + a)*sin(b*x + a))/(b^3 *cos(b*x + a))
Timed out. \[ \int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**2*sec(b*x+a)**2*sin(3*b*x+3*a),x)
Output:
Timed out
\[ \int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\int { {\left (d x + c\right )}^{2} \sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right ) \,d x } \] Input:
integrate((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="maxima")
Output:
-2*((cos(3*b*x + 3*a) + cos(b*x + a))*cos(4*b*x + 4*a) + (3*cos(2*b*x + 2* a) + 1)*cos(3*b*x + 3*a) + 3*cos(2*b*x + 2*a)*cos(b*x + a) + (sin(3*b*x + 3*a) + sin(b*x + a))*sin(4*b*x + 4*a) + 3*sin(3*b*x + 3*a)*sin(2*b*x + 2*a ) + 3*sin(2*b*x + 2*a)*sin(b*x + a) + cos(b*x + a))*c^2/(b*cos(3*b*x + 3*a )^2 + 2*b*cos(3*b*x + 3*a)*cos(b*x + a) + b*cos(b*x + a)^2 + b*sin(3*b*x + 3*a)^2 + 2*b*sin(3*b*x + 3*a)*sin(b*x + a) + b*sin(b*x + a)^2) - (4*(cos( a)^2 + sin(a)^2)*b*x*cos(b*x + a) + 12*(b*x*cos(2*b*x + 3*a)*cos(b*x + 2*a ) + b*x*cos(b*x + 2*a)*cos(a) + b*x*sin(2*b*x + 3*a)*sin(b*x + 2*a) + b*x* sin(b*x + 2*a)*sin(a))*cos(3*b*x + 3*a)^2 + 4*(b*x*cos(b*x + a) - sin(b*x + a))*cos(2*b*x + 3*a)^2 + 12*(b*x*cos(2*b*x + 3*a)*cos(b*x + 2*a) + b*x*c os(b*x + 2*a)*cos(a) + b*x*sin(2*b*x + 3*a)*sin(b*x + 2*a) + b*x*sin(b*x + 2*a)*sin(a))*sin(3*b*x + 3*a)^2 + 4*(b*x*cos(b*x + a) - sin(b*x + a))*sin (2*b*x + 3*a)^2 + 4*((b*x*cos(2*b*x + 3*a) + b*x*cos(a) + sin(2*b*x + 3*a) + sin(a))*cos(3*b*x + 3*a)^2 + (b*x*cos(a) + sin(a))*cos(b*x + a)^2 + (b* x*cos(2*b*x + 3*a) + b*x*cos(a) + sin(2*b*x + 3*a) + sin(a))*sin(3*b*x + 3 *a)^2 + (b*x*cos(a) + sin(a))*sin(b*x + a)^2 + 2*(b*x*cos(2*b*x + 3*a)*cos (b*x + a) + (b*x*cos(a) + sin(a))*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + (b*x*cos(b*x + a)^2 + b*x*sin(b*x + a)^2)*cos(2*b *x + 3*a) + 2*(b*x*cos(2*b*x + 3*a)*sin(b*x + a) + (b*x*cos(a) + sin(a))*s in(b*x + a) + sin(2*b*x + 3*a)*sin(b*x + a))*sin(3*b*x + 3*a) + (cos(b*...
\[ \int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\int { {\left (d x + c\right )}^{2} \sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right ) \,d x } \] Input:
integrate((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="giac")
Output:
integrate((d*x + c)^2*sec(b*x + a)^2*sin(3*b*x + 3*a), x)
Timed out. \[ \int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\text {Hanged} \] Input:
int((sin(3*a + 3*b*x)*(c + d*x)^2)/cos(a + b*x)^2,x)
Output:
\text{Hanged}
\[ \int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\text {too large to display} \] Input:
int((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x)
Output:
(9801*cos(3*a + 3*b*x)*cos(a + b*x)*tan((3*a + 3*b*x)/2)**2*tan((a + b*x)/ 2)**2*b**2*c**2 + 19602*cos(3*a + 3*b*x)*cos(a + b*x)*tan((3*a + 3*b*x)/2) **2*tan((a + b*x)/2)**2*b**2*c*d*x + 9801*cos(3*a + 3*b*x)*cos(a + b*x)*ta n((3*a + 3*b*x)/2)**2*tan((a + b*x)/2)**2*b**2*d**2*x**2 - 486*cos(3*a + 3 *b*x)*cos(a + b*x)*tan((3*a + 3*b*x)/2)**2*tan((a + b*x)/2)**2*d**2 - 9801 *cos(3*a + 3*b*x)*cos(a + b*x)*tan((3*a + 3*b*x)/2)**2*b**2*c**2 - 19602*c os(3*a + 3*b*x)*cos(a + b*x)*tan((3*a + 3*b*x)/2)**2*b**2*c*d*x - 9801*cos (3*a + 3*b*x)*cos(a + b*x)*tan((3*a + 3*b*x)/2)**2*b**2*d**2*x**2 + 486*co s(3*a + 3*b*x)*cos(a + b*x)*tan((3*a + 3*b*x)/2)**2*d**2 + 9801*cos(3*a + 3*b*x)*cos(a + b*x)*tan((a + b*x)/2)**2*b**2*c**2 + 19602*cos(3*a + 3*b*x) *cos(a + b*x)*tan((a + b*x)/2)**2*b**2*c*d*x + 9801*cos(3*a + 3*b*x)*cos(a + b*x)*tan((a + b*x)/2)**2*b**2*d**2*x**2 - 486*cos(3*a + 3*b*x)*cos(a + b*x)*tan((a + b*x)/2)**2*d**2 - 9801*cos(3*a + 3*b*x)*cos(a + b*x)*b**2*c* *2 - 19602*cos(3*a + 3*b*x)*cos(a + b*x)*b**2*c*d*x - 9801*cos(3*a + 3*b*x )*cos(a + b*x)*b**2*d**2*x**2 + 486*cos(3*a + 3*b*x)*cos(a + b*x)*d**2 + 7 128*cos(3*a + 3*b*x)*sin(a + b*x)*tan((3*a + 3*b*x)/2)**2*tan((a + b*x)/2) **2*b*c*d + 7128*cos(3*a + 3*b*x)*sin(a + b*x)*tan((3*a + 3*b*x)/2)**2*tan ((a + b*x)/2)**2*b*d**2*x - 7128*cos(3*a + 3*b*x)*sin(a + b*x)*tan((3*a + 3*b*x)/2)**2*b*c*d - 7128*cos(3*a + 3*b*x)*sin(a + b*x)*tan((3*a + 3*b*x)/ 2)**2*b*d**2*x + 7128*cos(3*a + 3*b*x)*sin(a + b*x)*tan((a + b*x)/2)**2...