\(\int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx\) [376]

Optimal result

Integrand size = 25, antiderivative size = 255 \[ \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx=-\frac {6 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {18 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {18 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 \sin (a+b x)}{b^4}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2} \]

[Out]

-6*(d*x+c)^3*arctanh(exp(I*(b*x+a)))/b-24*d^2*(d*x+c)*cos(b*x+a)/b^3+4*(d*x+c)^3*cos(b*x+a)/b+9*I*d*(d*x+c)^2*
polylog(2,-exp(I*(b*x+a)))/b^2-9*I*d*(d*x+c)^2*polylog(2,exp(I*(b*x+a)))/b^2-18*d^2*(d*x+c)*polylog(3,-exp(I*(
b*x+a)))/b^3+18*d^2*(d*x+c)*polylog(3,exp(I*(b*x+a)))/b^3-18*I*d^3*polylog(4,-exp(I*(b*x+a)))/b^4+18*I*d^3*pol
ylog(4,exp(I*(b*x+a)))/b^4+24*d^3*sin(b*x+a)/b^4-12*d*(d*x+c)^2*sin(b*x+a)/b^2

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4516, 4493, 3377, 2717, 4268, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx=-\frac {6 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {18 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {18 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 \sin (a+b x)}{b^4}-\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {4 (c+d x)^3 \cos (a+b x)}{b} \]

[In]

Int[(c + d*x)^3*Csc[a + b*x]^2*Sin[3*a + 3*b*x],x]

[Out]

(-6*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b - (24*d^2*(c + d*x)*Cos[a + b*x])/b^3 + (4*(c + d*x)^3*Cos[a + b*x
])/b + ((9*I)*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^2 - ((9*I)*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x)
)])/b^2 - (18*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (18*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])
/b^3 - ((18*I)*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 + ((18*I)*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4 + (24*d^3*
Sin[a + b*x])/b^4 - (12*d*(c + d*x)^2*Sin[a + b*x])/b^2

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4493

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[
(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \int \left (3 (c+d x)^3 \cos (a+b x) \cot (a+b x)-(c+d x)^3 \sin (a+b x)\right ) \, dx \\ & = 3 \int (c+d x)^3 \cos (a+b x) \cot (a+b x) \, dx-\int (c+d x)^3 \sin (a+b x) \, dx \\ & = \frac {(c+d x)^3 \cos (a+b x)}{b}+3 \int (c+d x)^3 \csc (a+b x) \, dx-3 \int (c+d x)^3 \sin (a+b x) \, dx-\frac {(3 d) \int (c+d x)^2 \cos (a+b x) \, dx}{b} \\ & = -\frac {6 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac {(9 d) \int (c+d x)^2 \cos (a+b x) \, dx}{b}-\frac {(9 d) \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac {(9 d) \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}+\frac {\left (6 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{b^2} \\ & = -\frac {6 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac {\left (18 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (18 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (18 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{b^2}+\frac {\left (6 d^3\right ) \int \cos (a+b x) \, dx}{b^3} \\ & = -\frac {6 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {\left (18 d^3\right ) \int \cos (a+b x) \, dx}{b^3}+\frac {\left (18 d^3\right ) \int \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (18 d^3\right ) \int \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right ) \, dx}{b^3} \\ & = -\frac {6 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 \sin (a+b x)}{b^4}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2}-\frac {\left (18 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (18 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4} \\ & = -\frac {6 (c+d x)^3 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {24 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {4 (c+d x)^3 \cos (a+b x)}{b}+\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {9 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {18 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {18 i d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {18 i d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 \sin (a+b x)}{b^4}-\frac {12 d (c+d x)^2 \sin (a+b x)}{b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.80 \[ \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx=\frac {3 \left (-2 b^3 (c+d x)^3 \text {arctanh}(\cos (a+b x)+i \sin (a+b x))+3 i d \left (b^2 (c+d x)^2 \operatorname {PolyLog}(2,-\cos (a+b x)-i \sin (a+b x))+2 i b d (c+d x) \operatorname {PolyLog}(3,-\cos (a+b x)-i \sin (a+b x))-2 d^2 \operatorname {PolyLog}(4,-\cos (a+b x)-i \sin (a+b x))\right )-3 i d \left (b^2 (c+d x)^2 \operatorname {PolyLog}(2,\cos (a+b x)+i \sin (a+b x))+2 i b d (c+d x) \operatorname {PolyLog}(3,\cos (a+b x)+i \sin (a+b x))-2 d^2 \operatorname {PolyLog}(4,\cos (a+b x)+i \sin (a+b x))\right )\right )}{b^4}+\frac {4 \cos (b x) \left (b^3 c^3 \cos (a)-6 b c d^2 \cos (a)+3 b^3 c^2 d x \cos (a)-6 b d^3 x \cos (a)+3 b^3 c d^2 x^2 \cos (a)+b^3 d^3 x^3 \cos (a)-3 b^2 c^2 d \sin (a)+6 d^3 \sin (a)-6 b^2 c d^2 x \sin (a)-3 b^2 d^3 x^2 \sin (a)\right )}{b^4}-\frac {4 \left (3 b^2 c^2 d \cos (a)-6 d^3 \cos (a)+6 b^2 c d^2 x \cos (a)+3 b^2 d^3 x^2 \cos (a)+b^3 c^3 \sin (a)-6 b c d^2 \sin (a)+3 b^3 c^2 d x \sin (a)-6 b d^3 x \sin (a)+3 b^3 c d^2 x^2 \sin (a)+b^3 d^3 x^3 \sin (a)\right ) \sin (b x)}{b^4} \]

[In]

Integrate[(c + d*x)^3*Csc[a + b*x]^2*Sin[3*a + 3*b*x],x]

[Out]

(3*(-2*b^3*(c + d*x)^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]] + (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, -Cos[a + b
*x] - I*Sin[a + b*x]] + (2*I)*b*d*(c + d*x)*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]] - 2*d^2*PolyLog[4, -Cos
[a + b*x] - I*Sin[a + b*x]]) - (3*I)*d*(b^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*x]] + (2*I)*b*d*
(c + d*x)*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]] - 2*d^2*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]])))/b^4 +
(4*Cos[b*x]*(b^3*c^3*Cos[a] - 6*b*c*d^2*Cos[a] + 3*b^3*c^2*d*x*Cos[a] - 6*b*d^3*x*Cos[a] + 3*b^3*c*d^2*x^2*Co
s[a] + b^3*d^3*x^3*Cos[a] - 3*b^2*c^2*d*Sin[a] + 6*d^3*Sin[a] - 6*b^2*c*d^2*x*Sin[a] - 3*b^2*d^3*x^2*Sin[a]))/
b^4 - (4*(3*b^2*c^2*d*Cos[a] - 6*d^3*Cos[a] + 6*b^2*c*d^2*x*Cos[a] + 3*b^2*d^3*x^2*Cos[a] + b^3*c^3*Sin[a] - 6
*b*c*d^2*Sin[a] + 3*b^3*c^2*d*x*Sin[a] - 6*b*d^3*x*Sin[a] + 3*b^3*c*d^2*x^2*Sin[a] + b^3*d^3*x^3*Sin[a])*Sin[b
*x])/b^4

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (237 ) = 474\).

Time = 2.97 (sec) , antiderivative size = 849, normalized size of antiderivative = 3.33

method	result	size
risch	\(\text {Expression too large to display}\)	\(849\)

[In]

int((d*x+c)^3*csc(b*x+a)^2*sin(3*b*x+3*a),x,method=_RETURNVERBOSE)

[Out]

-6/b*c^3*arctanh(exp(I*(b*x+a)))-18/b^3*c*d^2*a^2*arctanh(exp(I*(b*x+a)))+18/b^2*c^2*d*a*arctanh(exp(I*(b*x+a)
))-9*I/b^2*c^2*d*polylog(2,exp(I*(b*x+a)))+9*I/b^2*c^2*d*polylog(2,-exp(I*(b*x+a)))-9*I/b^2*d^3*polylog(2,exp(
I*(b*x+a)))*x^2+9*I/b^2*d^3*polylog(2,-exp(I*(b*x+a)))*x^2+2*(d^3*x^3*b^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^
3+3*I*b^2*d^3*x^2-6*b*d^3*x+6*I*b^2*c*d^2*x-6*c*d^2*b+3*I*b^2*c^2*d-6*I*d^3)/b^4*exp(I*(b*x+a))+2*(d^3*x^3*b^3
+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3-3*I*b^2*d^3*x^2-6*b*d^3*x-6*I*b^2*c*d^2*x-6*c*d^2*b-3*I*b^2*c^2*d+6*I*d
^3)/b^4*exp(-I*(b*x+a))+6/b^4*d^3*a^3*arctanh(exp(I*(b*x+a)))-9/b^2*c^2*d*ln(exp(I*(b*x+a))+1)*a+9/b^3*d^2*c*l
n(exp(I*(b*x+a))+1)*a^2-3/b^4*d^3*ln(exp(I*(b*x+a))+1)*a^3-18*I*d^3*polylog(4,-exp(I*(b*x+a)))/b^4+9/b^2*d*c^2
*ln(1-exp(I*(b*x+a)))*a-9/b^3*c*d^2*ln(1-exp(I*(b*x+a)))*a^2+9/b*d*c^2*ln(1-exp(I*(b*x+a)))*x-9/b*d*c^2*ln(exp
(I*(b*x+a))+1)*x+9/b*c*d^2*ln(1-exp(I*(b*x+a)))*x^2-9/b*c*d^2*ln(exp(I*(b*x+a))+1)*x^2+3/b^4*d^3*ln(1-exp(I*(b
*x+a)))*a^3+18/b^3*d^3*polylog(3,exp(I*(b*x+a)))*x-18/b^3*d^3*polylog(3,-exp(I*(b*x+a)))*x+3/b*d^3*ln(1-exp(I*
(b*x+a)))*x^3-3/b*d^3*ln(exp(I*(b*x+a))+1)*x^3+18/b^3*c*d^2*polylog(3,exp(I*(b*x+a)))-18/b^3*c*d^2*polylog(3,-
exp(I*(b*x+a)))-18*I/b^2*d^2*c*polylog(2,exp(I*(b*x+a)))*x+18*I/b^2*d^2*c*polylog(2,-exp(I*(b*x+a)))*x+18*I*d^
3*polylog(4,exp(I*(b*x+a)))/b^4

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. 929 vs. \(2 (231) = 462\).

Time = 0.32 (sec) , antiderivative size = 929, normalized size of antiderivative = 3.64 \[ \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^3*csc(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="fricas")

[Out]

1/2*(18*I*d^3*polylog(4, cos(b*x + a) + I*sin(b*x + a)) - 18*I*d^3*polylog(4, cos(b*x + a) - I*sin(b*x + a)) +
18*I*d^3*polylog(4, -cos(b*x + a) + I*sin(b*x + a)) - 18*I*d^3*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) + 8
*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 - 6*b*c*d^2 + 3*(b^3*c^2*d - 2*b*d^3)*x)*cos(b*x + a) - 9*(I*b^2*d^3
*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*dilog(cos(b*x + a) + I*sin(b*x + a)) - 9*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2
*x - I*b^2*c^2*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) - 9*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*dil
og(-cos(b*x + a) + I*sin(b*x + a)) - 9*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*dilog(-cos(b*x + a) -
I*sin(b*x + a)) - 3*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(cos(b*x + a) + I*sin(b*x + a
) + 1) - 3*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(cos(b*x + a) - I*sin(b*x + a) + 1) +
3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) + 3*(b
^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) + 3*(b^3*d
^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*log(-cos(b*x + a) + I*sin(
b*x + a) + 1) + 3*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*lo
g(-cos(b*x + a) - I*sin(b*x + a) + 1) + 18*(b*d^3*x + b*c*d^2)*polylog(3, cos(b*x + a) + I*sin(b*x + a)) + 18*
(b*d^3*x + b*c*d^2)*polylog(3, cos(b*x + a) - I*sin(b*x + a)) - 18*(b*d^3*x + b*c*d^2)*polylog(3, -cos(b*x + a
) + I*sin(b*x + a)) - 18*(b*d^3*x + b*c*d^2)*polylog(3, -cos(b*x + a) - I*sin(b*x + a)) - 24*(b^2*d^3*x^2 + 2*
b^2*c*d^2*x + b^2*c^2*d - 2*d^3)*sin(b*x + a))/b^4

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**3*csc(b*x+a)**2*sin(3*b*x+3*a),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (231) = 462\).

Time = 0.42 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.38 \[ \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx=\frac {c^{3} {\left (8 \, \cos \left (b x + a\right ) - 3 \, \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + 3 \, \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right )\right )}}{2 \, b} - \frac {36 i \, d^{3} {\rm Li}_{4}(-e^{\left (i \, b x + i \, a\right )}) - 36 i \, d^{3} {\rm Li}_{4}(e^{\left (i \, b x + i \, a\right )}) - 6 \, {\left (-i \, b^{3} d^{3} x^{3} - 3 i \, b^{3} c d^{2} x^{2} - 3 i \, b^{3} c^{2} d x\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 6 \, {\left (-i \, b^{3} d^{3} x^{3} - 3 i \, b^{3} c d^{2} x^{2} - 3 i \, b^{3} c^{2} d x\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 8 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} - 6 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right ) - 18 \, {\left (i \, b^{2} d^{3} x^{2} + 2 i \, b^{2} c d^{2} x + i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) - 18 \, {\left (-i \, b^{2} d^{3} x^{2} - 2 i \, b^{2} c d^{2} x - i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + 3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - 3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) + 36 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 36 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )}) + 24 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (b x + a\right )}{2 \, b^{4}} \]

[In]

integrate((d*x+c)^3*csc(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="maxima")

[Out]

1/2*c^3*(8*cos(b*x + a) - 3*log(cos(b*x)^2 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + s
in(a)^2) + 3*log(cos(b*x)^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2))/b - 1
/2*(36*I*d^3*polylog(4, -e^(I*b*x + I*a)) - 36*I*d^3*polylog(4, e^(I*b*x + I*a)) - 6*(-I*b^3*d^3*x^3 - 3*I*b^3
*c*d^2*x^2 - 3*I*b^3*c^2*d*x)*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 6*(-I*b^3*d^3*x^3 - 3*I*b^3*c*d^2*x^2
- 3*I*b^3*c^2*d*x)*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 8*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 - 6*b*c*d^2 + 3
*(b^3*c^2*d - 2*b*d^3)*x)*cos(b*x + a) - 18*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*dilog(-e^(I*b*x +
I*a)) - 18*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*dilog(e^(I*b*x + I*a)) + 3*(b^3*d^3*x^3 + 3*b^3*c*
d^2*x^2 + 3*b^3*c^2*d*x)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - 3*(b^3*d^3*x^3 + 3*b^3*c*
d^2*x^2 + 3*b^3*c^2*d*x)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 36*(b*d^3*x + b*c*d^2)*po
lylog(3, -e^(I*b*x + I*a)) - 36*(b*d^3*x + b*c*d^2)*polylog(3, e^(I*b*x + I*a)) + 24*(b^2*d^3*x^2 + 2*b^2*c*d^
2*x + b^2*c^2*d - 2*d^3)*sin(b*x + a))/b^4

Giac [F]

\[ \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx=\int { {\left (d x + c\right )}^{3} \csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right ) \,d x } \]

[In]

integrate((d*x+c)^3*csc(b*x+a)^2*sin(3*b*x+3*a),x, algorithm="giac")

[Out]

integrate((d*x + c)^3*csc(b*x + a)^2*sin(3*b*x + 3*a), x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx=\text {Hanged} \]

[In]

int((sin(3*a + 3*b*x)*(c + d*x)^3)/sin(a + b*x)^2,x)

[Out]

\text{Hanged}