\(\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx\) [390]
Optimal result
Integrand size = 25, antiderivative size = 147 \[
\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=-\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {8 d^2 \cos (a+b x)}{b^3}-\frac {4 (c+d x)^2 \cos (a+b x)}{b}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^2 \sec (a+b x)}{b}+\frac {8 d (c+d x) \sin (a+b x)}{b^2}
\]
[Out]
-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*e
xp(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
Rubi [A] (verified)
Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of
steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4516, 3377, 2718, 4492, 4494,
4266, 2317, 2438} \[
\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=-\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {8 d^2 \cos (a+b x)}{b^3}+\frac {8 d (c+d x) \sin (a+b x)}{b^2}-\frac {4 (c+d x)^2 \cos (a+b x)}{b}-\frac {(c+d x)^2 \sec (a+b x)}{b}
\]
[In]
Int[(c + d*x)^2*Sec[a + b*x]^2*Sin[3*a + 3*b*x],x]
[Out]
((-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
Rule 2317
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Rule 2438
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
Rule 2718
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3377
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 4266
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Rule 4492
Int[((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Int[
(c + d*x)^m*Sin[a + b*x]^n*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sin[a + b*x]^(n - 2)*Tan[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Rule 4494
Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[
(c + d*x)^m*(Sec[a + b*x]^n/(b*n)), x] - Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Rule 4516
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] && M
emberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d,
1]
Rubi steps \begin{align*}
\text {integral}& = \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 \\ & = 3 \int (c+d x)^2 \sin (a+b x) \, dx-\int (c+d x)^2 \sin (a+b x) \tan ^2(a+b x) \, dx \\ & = -\frac {3 (c+d x)^2 \cos (a+b x)}{b}+\frac {(6 d) \int (c+d x) \cos (a+b x) \, dx}{b}+\int (c+d x)^2 \sin (a+b x) \, dx-\int (c+d x)^2 \sec (a+b x) \tan (a+b x) \, dx \\ & = -\frac {4 (c+d x)^2 \cos (a+b x)}{b}-\frac {(c+d x)^2 \sec (a+b x)}{b}+\frac {6 d (c+d x) \sin (a+b x)}{b^2}+\frac {(2 d) \int (c+d x) \cos (a+b x) \, dx}{b}+\frac {(2 d) \int (c+d x) \sec (a+b x) \, dx}{b}-\frac {\left (6 d^2\right ) \int \sin (a+b x) \, dx}{b^2} \\ & = -\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {6 d^2 \cos (a+b x)}{b^3}-\frac {4 (c+d x)^2 \cos (a+b x)}{b}-\frac {(c+d x)^2 \sec (a+b x)}{b}+\frac {8 d (c+d x) \sin (a+b x)}{b^2}-\frac {\left (2 d^2\right ) \int \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (2 d^2\right ) \int \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (2 d^2\right ) \int \sin (a+b x) \, dx}{b^2} \\ & = -\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 {(c+d x)^2 \sec (a+b x)}{b}+\frac {8 d (c+d x) \sin (a+b x)}{b^2}+\frac {\left (2 i d^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}-\frac {\left (2 i d^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3} \\ & = -\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} \\
\end{align*}
Mathematica [B] (verified)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of
optimal. \(364\) vs. \(2(147)=294\).
Time = 2.62 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.48
\[
\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\frac {4 b c d \text {arctanh}\left (\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )\right )+2 d^2 \left (2 \arctan (\cot (a)) \text {arctanh}\left (\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )\right )-\frac {\csc (a) \left ((b x-\arctan (\cot (a))) \left (\log \left (1-e^{i (b x-\arctan (\cot (a)))}\right )-\log \left (1+e^{i (b x-\arctan (\cot (a)))}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i (b x-\arctan (\cot (a)))}\right )-i \operatorname {PolyLog}\left (2,e^{i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {\csc ^2(a)}}\right )-b^2 (c+d x)^2 \sec (a)-4 \cos (b x) \left (\left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a)-2 b d (c+d x) \sin (a)\right )+4 \left (2 b d (c+d x) \cos (a)+\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a)\right ) \sin (b x)-\frac {b^2 (c+d x)^2 \sin \left (\frac {b x}{2}\right )}{\left (\cos \left (\frac {a}{2}\right )-\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )}+\frac {b^2 (c+d x)^2 \sin \left (\frac {b x}{2}\right )}{\left (\cos \left (\frac {a}{2}\right )+\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )}}{b^3}
\]
[In]
Integrate[(c + d*x)^2*Sec[a + b*x]^2*Sin[3*a + 3*b*x],x]
[Out]
(4*b*c*d*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] + 2*d^2*(2*ArcTan[Cot[a]]*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]]
- (Csc[a]*((b*x - ArcTan[Cot[a]])*(Log[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])*(C
os[(a + b*x)/2] + Sin[(a + b*x)/2])))/b^3
Maple [B] (verified)
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 = 4.24 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.27
| | |
method | result | size |
| | |
default |
\(-\frac {4 c^{2} \cos \left (x b +a \right )}{b}-\frac {c^{2}}{b \cos \left (x b +a \right )}+\frac {4 d^{2} \left (-\left (x b +a \right )^{2} \cos \left (x b +a \right )+2 \cos \left (x b +a \right )+2 \left (x b +a \right ) \sin \left (x b +a \right )-2 a \left (\sin \left (x b +a \right )-\left (x b +a \right ) \cos \left (x b +a \right )\right )-a^{2} \cos \left (x b +a \right )\right )}{b^{3}}-\frac {d^{2} \left (\frac {\left (x b +a \right )^{2}}{\cos \left (x b +a \right )}+2 \left (x b +a \right ) \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )-2 \left (x b +a \right ) \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )-2 i \operatorname {dilog}\left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )+2 i \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )-2 a \left (\frac {x b +a}{\cos \left (x b +a \right )}-\ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )\right )+\frac {a^{2}}{\cos \left (x b +a \right )}\right )}{b^{3}}+\frac {8 c d \left (\sin \left (x b +a \right )-\left (x b +a \right ) \cos \left (x b +a \right )+\cos \left (x b +a \right ) a \right )}{b^{2}}-\frac {2 c d x}{b \cos \left (x b +a \right )}+\frac {2 c d \ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{b^{2}}\) |
\(334\) |
risch |
\(-\frac {2 \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +2 i b \,d^{2} x +b^{2} c^{2}+2 i b c d -2 d^{2}\right ) {\mathrm e}^{i \left (x b +a \right )}}{b^{3}}-\frac {2 \left (x^{2} d^{2} b^{2}+2 b^{2} c d x -2 i b \,d^{2} x +b^{2} c^{2}-2 i b c d -2 d^{2}\right ) {\mathrm e}^{-i \left (x b +a \right )}}{b^{3}}-\frac {2 \,{\mathrm e}^{i \left (x b +a \right )} \left (x^{2} d^{2}+2 c d x +c^{2}\right )}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}-\frac {4 i d c \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {2 d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {2 d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}+\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}+\frac {2 i d^{2} \operatorname {dilog}\left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 i d^{2} \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {4 i d^{2} a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}\) |
\(345\) |
| | |
|
|
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[In]
int((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x,method=_RETURNVERBOSE)
[Out]
-4*c^2*cos(b*x+a)/b-c^2/b/cos(b*x+a)+4*d^2/b^3*(-(b*x+a)^2*cos(b*x+a)+2*cos(b*x+a)+2*(b*x+a)*sin(b*x+a)-2*a*(s
in(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(1+I*exp(I*(b*x+a)))-2
*(b*x+a)*ln(1-I*exp(I*(b*x+a)))-2*I*dilog(1+I*exp(I*(b*x+a)))+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)+cos(b*x+a)*a)-2*c*d/
b/cos(b*x+a)*x+2*c*d/b^2*ln(sec(b*x+a)+tan(b*x+a))
Fricas [B] (verification not implemented)
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 = 0.28 (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 )}
\]
[In]
integrate((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="fricas")
[Out]
-(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
*cos(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)*log(cos(b*x + a) + I*sin(b*x + a) + I) + (b*c*d - a*d^2)*cos(b*x + a)*lo
g(cos(b*x + a) - I*sin(b*x + a) + I) - (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*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))
Sympy [F(-2)]
Exception generated. \[
\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\text {Exception raised: HeuristicGCDFailed}
\]
[In]
integrate((d*x+c)**2*sec(b*x+a)**2*sin(3*b*x+3*a),x)
[Out]
Exception raised: HeuristicGCDFailed >> no luck
Maxima [F]
\[
\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\int { {\left (d x + c\right )}^{2} \sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right ) \,d x }
\]
[In]
integrate((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="maxima")
[Out]
-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) - si
n(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*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))*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))*sin(b*x + a) + sin(2*b*x
+ 3*a)*sin(b*x + a))*sin(3*b*x + 3*a) + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 3*a))*cos(3*b*x + 4*a) +
4*(6*b*x*cos(b*x + 2*a)*cos(b*x + a)*cos(a) + 6*b*x*cos(b*x + a)*sin(b*x + 2*a)*sin(a) + b*x*cos(2*b*x + 3*a)
^2 + b*x*sin(2*b*x + 3*a)^2 + (cos(a)^2 + sin(a)^2)*b*x + 2*(3*b*x*cos(b*x + 2*a)*cos(b*x + a) + b*x*cos(a))*c
os(2*b*x + 3*a) + 2*(3*b*x*cos(b*x + a)*sin(b*x + 2*a) + b*x*sin(a))*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + 4*(2
*b*x*cos(b*x + a)*cos(a) + 3*(b*x*cos(b*x + a)^2 + b*x*sin(b*x + a)^2)*cos(b*x + 2*a) - 2*cos(a)*sin(b*x + a))
*cos(2*b*x + 3*a) + 12*(b*x*cos(b*x + a)^2*cos(a) + b*x*cos(a)*sin(b*x + a)^2)*cos(b*x + 2*a) - ((cos(2*b*x +
3*a)^2 + 2*cos(2*b*x + 3*a)*cos(a) + cos(a)^2 + sin(2*b*x + 3*a)^2 + 2*sin(2*b*x + 3*a)*sin(a) + sin(a)^2)*cos
(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 3*a)^2 + (cos(a)^2 + sin(a)^2)*cos(b*x + a)^2
+ (cos(2*b*x + 3*a)^2 + 2*cos(2*b*x + 3*a)*cos(a) + cos(a)^2 + sin(2*b*x + 3*a)^2 + 2*sin(2*b*x + 3*a)*sin(a)
+ sin(a)^2)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 3*a)^2 + (cos(a)^2 + sin(a)^2)*
sin(b*x + a)^2 + 2*(cos(2*b*x + 3*a)^2*cos(b*x + a) + 2*cos(2*b*x + 3*a)*cos(b*x + a)*cos(a) + cos(b*x + a)*si
n(2*b*x + 3*a)^2 + 2*cos(b*x + a)*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*cos(b*x + a))*cos(3*b*x + 3*
a) + 2*(cos(b*x + a)^2*cos(a) + cos(a)*sin(b*x + a)^2)*cos(2*b*x + 3*a) + 2*(cos(2*b*x + 3*a)^2*sin(b*x + a) +
2*cos(2*b*x + 3*a)*cos(a)*sin(b*x + a) + sin(2*b*x + 3*a)^2*sin(b*x + a) + 2*sin(2*b*x + 3*a)*sin(b*x + a)*si
n(a) + (cos(a)^2 + sin(a)^2)*sin(b*x + a))*sin(3*b*x + 3*a) + 2*(cos(b*x + a)^2*sin(a) + sin(b*x + a)^2*sin(a)
)*sin(2*b*x + 3*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + ((cos(2*b*x + 3*a)^2 + 2*cos(2
*b*x + 3*a)*cos(a) + cos(a)^2 + sin(2*b*x + 3*a)^2 + 2*sin(2*b*x + 3*a)*sin(a) + sin(a)^2)*cos(3*b*x + 3*a)^2
+ (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 3*a)^2 + (cos(a)^2 + sin(a)^2)*cos(b*x + a)^2 + (cos(2*b*x + 3
*a)^2 + 2*cos(2*b*x + 3*a)*cos(a) + cos(a)^2 + sin(2*b*x + 3*a)^2 + 2*sin(2*b*x + 3*a)*sin(a) + sin(a)^2)*sin(
3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 3*a)^2 + (cos(a)^2 + sin(a)^2)*sin(b*x + a)^2 +
2*(cos(2*b*x + 3*a)^2*cos(b*x + a) + 2*cos(2*b*x + 3*a)*cos(b*x + a)*cos(a) + cos(b*x + a)*sin(2*b*x + 3*a)^2
+ 2*cos(b*x + a)*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*cos(b*x + a))*cos(3*b*x + 3*a) + 2*(cos(b*x
+ a)^2*cos(a) + cos(a)*sin(b*x + a)^2)*cos(2*b*x + 3*a) + 2*(cos(2*b*x + 3*a)^2*sin(b*x + a) + 2*cos(2*b*x + 3
*a)*cos(a)*sin(b*x + a) + sin(2*b*x + 3*a)^2*sin(b*x + a) + 2*sin(2*b*x + 3*a)*sin(b*x + a)*sin(a) + (cos(a)^2
+ sin(a)^2)*sin(b*x + a))*sin(3*b*x + 3*a) + 2*(cos(b*x + a)^2*sin(a) + sin(b*x + a)^2*sin(a))*sin(2*b*x + 3*
a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) + 4*((b*x*sin(2*b*x + 3*a) + b*x*sin(a) - cos(2*
b*x + 3*a) - cos(a))*cos(3*b*x + 3*a)^2 + (b*x*sin(a) - cos(a))*cos(b*x + a)^2 + (b*x*sin(2*b*x + 3*a) + b*x*s
in(a) - cos(2*b*x + 3*a) - cos(a))*sin(3*b*x + 3*a)^2 + (b*x*sin(a) - cos(a))*sin(b*x + a)^2 + 2*(b*x*cos(b*x
+ a)*sin(2*b*x + 3*a) + (b*x*sin(a) - cos(a))*cos(b*x + a) - cos(2*b*x + 3*a)*cos(b*x + a))*cos(3*b*x + 3*a) -
(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 3*a) + 2*(b*x*sin(2*b*x + 3*a)*sin(b*x + a) + (b*x*sin(a) - cos
(a))*sin(b*x + a) - cos(2*b*x + 3*a)*sin(b*x + a))*sin(3*b*x + 3*a) + (b*x*cos(b*x + a)^2 + b*x*sin(b*x + a)^2
)*sin(2*b*x + 3*a))*sin(3*b*x + 4*a) + 4*(6*b*x*cos(b*x + 2*a)*cos(a)*sin(b*x + a) + 6*b*x*sin(b*x + 2*a)*sin(
b*x + a)*sin(a) + 2*(3*b*x*cos(b*x + 2*a)*sin(b*x + a) - cos(a))*cos(2*b*x + 3*a) - cos(2*b*x + 3*a)^2 - cos(a
)^2 + 2*(3*b*x*sin(b*x + 2*a)*sin(b*x + a) - sin(a))*sin(2*b*x + 3*a) - sin(2*b*x + 3*a)^2 - sin(a)^2)*sin(3*b
*x + 3*a) + 4*(2*b*x*cos(b*x + a)*sin(a) + 3*(b*x*cos(b*x + a)^2 + b*x*sin(b*x + a)^2)*sin(b*x + 2*a) - 2*sin(
b*x + a)*sin(a))*sin(2*b*x + 3*a) + 12*(b*x*cos(b*x + a)^2*sin(a) + b*x*sin(b*x + a)^2*sin(a))*sin(b*x + 2*a)
- 4*(cos(a)^2 + sin(a)^2)*sin(b*x + a))*c*d/((cos(a)^2 + sin(a)^2)*b^2*cos(b*x + a)^2 + (cos(a)^2 + sin(a)^2)*
b^2*sin(b*x + a)^2 + (b^2*cos(2*b*x + 3*a)^2 + 2*b^2*cos(2*b*x + 3*a)*cos(a) + b^2*sin(2*b*x + 3*a)^2 + 2*b^2*
sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^2)*cos(3*b*x + 3*a)^2 + (b^2*cos(b*x + a)^2 + b^2*sin(b*x +
a)^2)*cos(2*b*x + 3*a)^2 + (b^2*cos(2*b*x + 3*a)^2 + 2*b^2*cos(2*b*x + 3*a)*cos(a) + b^2*sin(2*b*x + 3*a)^2 +
2*b^2*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^2)*sin(3*b*x + 3*a)^2 + (b^2*cos(b*x + a)^2 + b^2*sin(
b*x + a)^2)*sin(2*b*x + 3*a)^2 + 2*(b^2*cos(2*b*x + 3*a)^2*cos(b*x + a) + 2*b^2*cos(2*b*x + 3*a)*cos(b*x + a)*
cos(a) + b^2*cos(b*x + a)*sin(2*b*x + 3*a)^2 + 2*b^2*cos(b*x + a)*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)
^2)*b^2*cos(b*x + a))*cos(3*b*x + 3*a) + 2*(b^2*cos(b*x + a)^2*cos(a) + b^2*cos(a)*sin(b*x + a)^2)*cos(2*b*x +
3*a) + 2*(b^2*cos(2*b*x + 3*a)^2*sin(b*x + a) + 2*b^2*cos(2*b*x + 3*a)*cos(a)*sin(b*x + a) + b^2*sin(2*b*x +
3*a)^2*sin(b*x + a) + 2*b^2*sin(2*b*x + 3*a)*sin(b*x + a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^2*sin(b*x + a))*sin
(3*b*x + 3*a) + 2*(b^2*cos(b*x + a)^2*sin(a) + b^2*sin(b*x + a)^2*sin(a))*sin(2*b*x + 3*a)) + 2*(2*(cos(a)^2 +
sin(a)^2)*b*x*sin(b*x + a) - ((3*b^2*x^2 - 4)*cos(2*b*x + 3*a)*cos(b*x + 2*a) + (3*b^2*x^2 - 4)*sin(2*b*x + 3
*a)*sin(b*x + 2*a) + (3*b^2*x^2*cos(a) - 4*cos(a))*cos(b*x + 2*a) + (3*b^2*x^2*sin(a) - 4*sin(a))*sin(b*x + 2*
a))*cos(3*b*x + 3*a)^2 + (2*b*x*sin(b*x + a) - (b^2*x^2 - 2)*cos(b*x + a))*cos(2*b*x + 3*a)^2 - ((3*b^2*x^2 -
4)*cos(2*b*x + 3*a)*cos(b*x + 2*a) + (3*b^2*x^2 - 4)*sin(2*b*x + 3*a)*sin(b*x + 2*a) + (3*b^2*x^2*cos(a) - 4*c
os(a))*cos(b*x + 2*a) + (3*b^2*x^2*sin(a) - 4*sin(a))*sin(b*x + 2*a))*sin(3*b*x + 3*a)^2 + (2*b*x*sin(b*x + a)
- (b^2*x^2 - 2)*cos(b*x + a))*sin(2*b*x + 3*a)^2 - ((b^2*x^2*cos(a) + 2*b*x*sin(2*b*x + 3*a) + 2*b*x*sin(a) +
(b^2*x^2 - 2)*cos(2*b*x + 3*a) - 2*cos(a))*cos(3*b*x + 3*a)^2 + (b^2*x^2*cos(a) + 2*b*x*sin(a) - 2*cos(a))*co
s(b*x + a)^2 + (b^2*x^2*cos(a) + 2*b*x*sin(2*b*x + 3*a) + 2*b*x*sin(a) + (b^2*x^2 - 2)*cos(2*b*x + 3*a) - 2*co
s(a))*sin(3*b*x + 3*a)^2 + (b^2*x^2*cos(a) + 2*b*x*sin(a) - 2*cos(a))*sin(b*x + a)^2 + 2*(2*b*x*cos(b*x + a)*s
in(2*b*x + 3*a) + (b^2*x^2 - 2)*cos(2*b*x + 3*a)*cos(b*x + a) + (b^2*x^2*cos(a) + 2*b*x*sin(a) - 2*cos(a))*cos
(b*x + a))*cos(3*b*x + 3*a) + ((b^2*x^2 - 2)*cos(b*x + a)^2 + (b^2*x^2 - 2)*sin(b*x + a)^2)*cos(2*b*x + 3*a) +
2*(2*b*x*sin(2*b*x + 3*a)*sin(b*x + a) + (b^2*x^2 - 2)*cos(2*b*x + 3*a)*sin(b*x + a) + (b^2*x^2*cos(a) + 2*b*
x*sin(a) - 2*cos(a))*sin(b*x + a))*sin(3*b*x + 3*a) + 2*(b*x*cos(b*x + a)^2 + b*x*sin(b*x + a)^2)*sin(2*b*x +
3*a))*cos(3*b*x + 4*a) - ((cos(a)^2 + sin(a)^2)*b^2*x^2 + (b^2*x^2 - 2)*cos(2*b*x + 3*a)^2 + 2*(3*b^2*x^2*cos(
a) - 4*cos(a))*cos(b*x + 2*a)*cos(b*x + a) + (b^2*x^2 - 2)*sin(2*b*x + 3*a)^2 + 2*(3*b^2*x^2*sin(a) - 4*sin(a)
)*cos(b*x + a)*sin(b*x + 2*a) + 2*(b^2*x^2*cos(a) + (3*b^2*x^2 - 4)*cos(b*x + 2*a)*cos(b*x + a) - 2*cos(a))*co
s(2*b*x + 3*a) - 2*cos(a)^2 + 2*(b^2*x^2*sin(a) + (3*b^2*x^2 - 4)*cos(b*x + a)*sin(b*x + 2*a) - 2*sin(a))*sin(
2*b*x + 3*a) - 2*sin(a)^2)*cos(3*b*x + 3*a) + (4*b*x*cos(a)*sin(b*x + a) - ((3*b^2*x^2 - 4)*cos(b*x + a)^2 + (
3*b^2*x^2 - 4)*sin(b*x + a)^2)*cos(b*x + 2*a) - 2*(b^2*x^2*cos(a) - 2*cos(a))*cos(b*x + a))*cos(2*b*x + 3*a) -
((3*b^2*x^2*cos(a) - 4*cos(a))*cos(b*x + a)^2 + (3*b^2*x^2*cos(a) - 4*cos(a))*sin(b*x + a)^2)*cos(b*x + 2*a)
- ((cos(a)^2 + sin(a)^2)*b^2*x^2 - 2*cos(a)^2 - 2*sin(a)^2)*cos(b*x + a) + 2*((cos(a)^2 + sin(a)^2)*b^3*cos(b*
x + a)^2 + (cos(a)^2 + sin(a)^2)*b^3*sin(b*x + a)^2 + (b^3*cos(2*b*x + 3*a)^2 + 2*b^3*cos(2*b*x + 3*a)*cos(a)
+ b^3*sin(2*b*x + 3*a)^2 + 2*b^3*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3)*cos(3*b*x + 3*a)^2 + (b^
3*cos(b*x + a)^2 + b^3*sin(b*x + a)^2)*cos(2*b*x + 3*a)^2 + (b^3*cos(2*b*x + 3*a)^2 + 2*b^3*cos(2*b*x + 3*a)*c
os(a) + b^3*sin(2*b*x + 3*a)^2 + 2*b^3*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3)*sin(3*b*x + 3*a)^2
+ (b^3*cos(b*x + a)^2 + b^3*sin(b*x + a)^2)*sin(2*b*x + 3*a)^2 + 2*(b^3*cos(2*b*x + 3*a)^2*cos(b*x + a) + 2*b
^3*cos(2*b*x + 3*a)*cos(b*x + a)*cos(a) + b^3*cos(b*x + a)*sin(2*b*x + 3*a)^2 + 2*b^3*cos(b*x + a)*sin(2*b*x +
3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3*cos(b*x + a))*cos(3*b*x + 3*a) + 2*(b^3*cos(b*x + a)^2*cos(a) + b^3*c
os(a)*sin(b*x + a)^2)*cos(2*b*x + 3*a) + 2*(b^3*cos(2*b*x + 3*a)^2*sin(b*x + a) + 2*b^3*cos(2*b*x + 3*a)*cos(a
)*sin(b*x + a) + b^3*sin(2*b*x + 3*a)^2*sin(b*x + a) + 2*b^3*sin(2*b*x + 3*a)*sin(b*x + a)*sin(a) + (cos(a)^2
+ sin(a)^2)*b^3*sin(b*x + a))*sin(3*b*x + 3*a) + 2*(b^3*cos(b*x + a)^2*sin(a) + b^3*sin(b*x + a)^2*sin(a))*sin
(2*b*x + 3*a))*integrate((x*cos(2*b*x + 2*a)*cos(b*x + a) + x*sin(2*b*x + 2*a)*sin(b*x + a) + x*cos(b*x + a))/
(b*cos(2*b*x + 2*a)^2 + b*sin(2*b*x + 2*a)^2 + 2*b*cos(2*b*x + 2*a) + b), x) - ((b^2*x^2*sin(a) - 2*b*x*cos(2*
b*x + 3*a) - 2*b*x*cos(a) + (b^2*x^2 - 2)*sin(2*b*x + 3*a) - 2*sin(a))*cos(3*b*x + 3*a)^2 + (b^2*x^2*sin(a) -
2*b*x*cos(a) - 2*sin(a))*cos(b*x + a)^2 + (b^2*x^2*sin(a) - 2*b*x*cos(2*b*x + 3*a) - 2*b*x*cos(a) + (b^2*x^2 -
2)*sin(2*b*x + 3*a) - 2*sin(a))*sin(3*b*x + 3*a)^2 + (b^2*x^2*sin(a) - 2*b*x*cos(a) - 2*sin(a))*sin(b*x + a)^
2 - 2*(2*b*x*cos(2*b*x + 3*a)*cos(b*x + a) - (b^2*x^2 - 2)*cos(b*x + a)*sin(2*b*x + 3*a) - (b^2*x^2*sin(a) - 2
*b*x*cos(a) - 2*sin(a))*cos(b*x + a))*cos(3*b*x + 3*a) - 2*(b*x*cos(b*x + a)^2 + b*x*sin(b*x + a)^2)*cos(2*b*x
+ 3*a) - 2*(2*b*x*cos(2*b*x + 3*a)*sin(b*x + a) - (b^2*x^2 - 2)*sin(2*b*x + 3*a)*sin(b*x + a) - (b^2*x^2*sin(
a) - 2*b*x*cos(a) - 2*sin(a))*sin(b*x + a))*sin(3*b*x + 3*a) + ((b^2*x^2 - 2)*cos(b*x + a)^2 + (b^2*x^2 - 2)*s
in(b*x + a)^2)*sin(2*b*x + 3*a))*sin(3*b*x + 4*a) + 2*(b*x*cos(2*b*x + 3*a)^2 + b*x*sin(2*b*x + 3*a)^2 + (cos(
a)^2 + sin(a)^2)*b*x - (3*b^2*x^2*cos(a) - 4*cos(a))*cos(b*x + 2*a)*sin(b*x + a) - (3*b^2*x^2*sin(a) - 4*sin(a
))*sin(b*x + 2*a)*sin(b*x + a) + (2*b*x*cos(a) - (3*b^2*x^2 - 4)*cos(b*x + 2*a)*sin(b*x + a))*cos(2*b*x + 3*a)
- ((3*b^2*x^2 - 4)*sin(b*x + 2*a)*sin(b*x + a) - 2*b*x*sin(a))*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) + (4*b*x*si
n(b*x + a)*sin(a) - 2*(b^2*x^2*sin(a) - 2*sin(a))*cos(b*x + a) - ((3*b^2*x^2 - 4)*cos(b*x + a)^2 + (3*b^2*x^2
- 4)*sin(b*x + a)^2)*sin(b*x + 2*a))*sin(2*b*x + 3*a) - ((3*b^2*x^2*sin(a) - 4*sin(a))*cos(b*x + a)^2 + (3*b^2
*x^2*sin(a) - 4*sin(a))*sin(b*x + a)^2)*sin(b*x + 2*a))*d^2/((cos(a)^2 + sin(a)^2)*b^3*cos(b*x + a)^2 + (cos(a
)^2 + sin(a)^2)*b^3*sin(b*x + a)^2 + (b^3*cos(2*b*x + 3*a)^2 + 2*b^3*cos(2*b*x + 3*a)*cos(a) + b^3*sin(2*b*x +
3*a)^2 + 2*b^3*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3)*cos(3*b*x + 3*a)^2 + (b^3*cos(b*x + a)^2
+ b^3*sin(b*x + a)^2)*cos(2*b*x + 3*a)^2 + (b^3*cos(2*b*x + 3*a)^2 + 2*b^3*cos(2*b*x + 3*a)*cos(a) + b^3*sin(2
*b*x + 3*a)^2 + 2*b^3*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3)*sin(3*b*x + 3*a)^2 + (b^3*cos(b*x +
a)^2 + b^3*sin(b*x + a)^2)*sin(2*b*x + 3*a)^2 + 2*(b^3*cos(2*b*x + 3*a)^2*cos(b*x + a) + 2*b^3*cos(2*b*x + 3*
a)*cos(b*x + a)*cos(a) + b^3*cos(b*x + a)*sin(2*b*x + 3*a)^2 + 2*b^3*cos(b*x + a)*sin(2*b*x + 3*a)*sin(a) + (c
os(a)^2 + sin(a)^2)*b^3*cos(b*x + a))*cos(3*b*x + 3*a) + 2*(b^3*cos(b*x + a)^2*cos(a) + b^3*cos(a)*sin(b*x + a
)^2)*cos(2*b*x + 3*a) + 2*(b^3*cos(2*b*x + 3*a)^2*sin(b*x + a) + 2*b^3*cos(2*b*x + 3*a)*cos(a)*sin(b*x + a) +
b^3*sin(2*b*x + 3*a)^2*sin(b*x + a) + 2*b^3*sin(2*b*x + 3*a)*sin(b*x + a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3*s
in(b*x + a))*sin(3*b*x + 3*a) + 2*(b^3*cos(b*x + a)^2*sin(a) + b^3*sin(b*x + a)^2*sin(a))*sin(2*b*x + 3*a))
Giac [F]
\[
\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\int { {\left (d x + c\right )}^{2} \sec \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right ) \,d x }
\]
[In]
integrate((d*x+c)^2*sec(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="giac")
[Out]
integrate((d*x + c)^2*sec(b*x + a)^2*sin(3*b*x + 3*a), x)
Mupad [F(-1)]
Timed out. \[
\int (c+d x)^2 \sec ^2(a+b x) \sin (3 a+3 b x) \, dx=\text {Hanged}
\]
[In]
int((sin(3*a + 3*b*x)*(c + d*x)^2)/cos(a + b*x)^2,x)
[Out]
\text{Hanged}