3.1.33 \(\int (c+d x)^3 \csc ^3(a+b x) \, dx\) [33]

Optimal. Leaf size=309 \[ -\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {3 i d^3 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^4}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac {3 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4} \]

[Out]

-6*d^2*(d*x+c)*arctanh(exp(I*(b*x+a)))/b^3-(d*x+c)^3*arctanh(exp(I*(b*x+a)))/b-3/2*d*(d*x+c)^2*csc(b*x+a)/b^2-
1/2*(d*x+c)^3*cot(b*x+a)*csc(b*x+a)/b+3*I*d^3*polylog(2,-exp(I*(b*x+a)))/b^4+3/2*I*d*(d*x+c)^2*polylog(2,-exp(
I*(b*x+a)))/b^2-3*I*d^3*polylog(2,exp(I*(b*x+a)))/b^4-3/2*I*d*(d*x+c)^2*polylog(2,exp(I*(b*x+a)))/b^2-3*d^2*(d
*x+c)*polylog(3,-exp(I*(b*x+a)))/b^3+3*d^2*(d*x+c)*polylog(3,exp(I*(b*x+a)))/b^3-3*I*d^3*polylog(4,-exp(I*(b*x
+a)))/b^4+3*I*d^3*polylog(4,exp(I*(b*x+a)))/b^4

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4271, 4268, 2317, 2438, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3 i d^3 \text {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {3 i d^3 \text {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}-\frac {3 i d^3 \text {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {3 i d^3 \text {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Csc[a + b*x]^3,x]

[Out]

(-6*d^2*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b^3 - ((c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b - (3*d*(c + d*x)^2*
Csc[a + b*x])/(2*b^2) - ((c + d*x)^3*Cot[a + b*x]*Csc[a + b*x])/(2*b) + ((3*I)*d^3*PolyLog[2, -E^(I*(a + b*x))
])/b^4 + (((3*I)/2)*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^2 - ((3*I)*d^3*PolyLog[2, E^(I*(a + b*x))])/
b^4 - (((3*I)/2)*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^2 - (3*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))
])/b^3 + (3*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 - ((3*I)*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 + ((
3*I)*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4

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 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 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 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 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 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 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*} \int (c+d x)^3 \csc ^3(a+b x) \, dx &=-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^3 \csc (a+b x) \, dx+\frac {\left (3 d^2\right ) \int (c+d x) \csc (a+b x) \, dx}{b^2}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{2 b}+\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{2 b}-\frac {\left (3 d^3\right ) \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (3 d^3\right ) \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac {\left (3 i d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 i d^2\right ) \int (c+d x) \text {Li}_2\left (e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {3 i d^3 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^4}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}+\frac {\left (3 d^3\right ) \int \text {Li}_3\left (-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (3 d^3\right ) \int \text {Li}_3\left (e^{i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {3 i d^3 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^4}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}\\ &=-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {3 i d^3 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^4}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d^3 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^4}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac {3 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.52, size = 478, normalized size = 1.55 \begin {gather*} -\frac {2 b^3 c^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )+12 b c d^2 \tanh ^{-1}\left (e^{i (a+b x)}\right )+b^2 (c+d x)^2 (3 d+b (c+d x) \cot (a+b x)) \csc (a+b x)-3 b^3 c^2 d x \log \left (1-e^{i (a+b x)}\right )-6 b d^3 x \log \left (1-e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1-e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1-e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1+e^{i (a+b x)}\right )+6 b d^3 x \log \left (1+e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1+e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1+e^{i (a+b x)}\right )-3 i d \left (2 d^2+b^2 (c+d x)^2\right ) \text {Li}_2\left (-e^{i (a+b x)}\right )+3 i d \left (2 d^2+b^2 (c+d x)^2\right ) \text {Li}_2\left (e^{i (a+b x)}\right )+6 b c d^2 \text {Li}_3\left (-e^{i (a+b x)}\right )+6 b d^3 x \text {Li}_3\left (-e^{i (a+b x)}\right )-6 b c d^2 \text {Li}_3\left (e^{i (a+b x)}\right )-6 b d^3 x \text {Li}_3\left (e^{i (a+b x)}\right )+6 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )-6 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(2*b^3*c^3*ArcTanh[E^(I*(a + b*x))] + 12*b*c*d^2*ArcTanh[E^(I*(a + b*x))] + b^2*(c + d*x)^2*(3*d + b*(c +
d*x)*Cot[a + b*x])*Csc[a + b*x] - 3*b^3*c^2*d*x*Log[1 - E^(I*(a + b*x))] - 6*b*d^3*x*Log[1 - E^(I*(a + b*x))]
- 3*b^3*c*d^2*x^2*Log[1 - E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 - E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 + E^(
I*(a + b*x))] + 6*b*d^3*x*Log[1 + E^(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 + E^(I*(a + b*x))] + b^3*d^3*x^3*Lo
g[1 + E^(I*(a + b*x))] - (3*I)*d*(2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, -E^(I*(a + b*x))] + (3*I)*d*(2*d^2 + b^2
*(c + d*x)^2)*PolyLog[2, E^(I*(a + b*x))] + 6*b*c*d^2*PolyLog[3, -E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, -E^(
I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, E^(I*(a + b*x))] + (6*I)*d^3*Poly
Log[4, -E^(I*(a + b*x))] - (6*I)*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1055 vs. \(2 (275 ) = 550\).
time = 0.13, size = 1056, normalized size = 3.42

method	result	size
risch	\(\text {Expression too large to display}\)	\(1056\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*I/b^2*d^3*polylog(2,-exp(I*(b*x+a)))*x^2-3/2*I/b^2*d^3*polylog(2,exp(I*(b*x+a)))*x^2-3/2*I/b^2*c^2*d*polyl
og(2,exp(I*(b*x+a)))+3/2*I/b^2*c^2*d*polylog(2,-exp(I*(b*x+a)))+1/b^2/(exp(2*I*(b*x+a))-1)^2*(d^3*x^3*b*exp(3*
I*(b*x+a))+3*c*d^2*x^2*b*exp(3*I*(b*x+a))+3*c^2*d*x*b*exp(3*I*(b*x+a))+d^3*x^3*b*exp(I*(b*x+a))+b*c^3*exp(3*I*
(b*x+a))+3*c*d^2*x^2*b*exp(I*(b*x+a))-3*I*d^3*x^2*exp(3*I*(b*x+a))+3*c^2*d*x*b*exp(I*(b*x+a))-6*I*c*d^2*x*exp(
3*I*(b*x+a))+b*c^3*exp(I*(b*x+a))-3*I*c^2*d*exp(3*I*(b*x+a))+3*I*d^3*x^2*exp(I*(b*x+a))+6*I*c*d^2*x*exp(I*(b*x
+a))+3*I*c^2*d*exp(I*(b*x+a)))+3/b^3*c*d^2*polylog(3,exp(I*(b*x+a)))-3/b^3*c*d^2*polylog(3,-exp(I*(b*x+a)))+1/
b^4*d^3*a^3*arctanh(exp(I*(b*x+a)))+3/b^3*d^3*polylog(3,exp(I*(b*x+a)))*x-3/b^3*d^3*polylog(3,-exp(I*(b*x+a)))
*x-1/b*c^3*arctanh(exp(I*(b*x+a)))-3*I/b^2*polylog(2,exp(I*(b*x+a)))*c*d^2*x+3*I/b^2*polylog(2,-exp(I*(b*x+a))
)*c*d^2*x+3/2/b*c*d^2*ln(1-exp(I*(b*x+a)))*x^2-3/2/b^3*c*d^2*ln(1-exp(I*(b*x+a)))*a^2-3/2/b*c*d^2*ln(exp(I*(b*
x+a))+1)*x^2+3/2/b^3*c*d^2*ln(exp(I*(b*x+a))+1)*a^2-3/2/b*c^2*d*ln(exp(I*(b*x+a))+1)*x-3/2/b^2*c^2*d*ln(exp(I*
(b*x+a))+1)*a+3/2/b*c^2*d*ln(1-exp(I*(b*x+a)))*x+3/2/b^2*c^2*d*ln(1-exp(I*(b*x+a)))*a+1/2/b*d^3*ln(1-exp(I*(b*
x+a)))*x^3+1/2/b^4*d^3*ln(1-exp(I*(b*x+a)))*a^3-1/2/b*d^3*ln(exp(I*(b*x+a))+1)*x^3-1/2/b^4*d^3*ln(exp(I*(b*x+a
))+1)*a^3-3/b^3*c*d^2*a^2*arctanh(exp(I*(b*x+a)))+3/b^2*c^2*d*a*arctanh(exp(I*(b*x+a)))+3/b^4*d^3*ln(1-exp(I*(
b*x+a)))*a-3/b^3*d^3*ln(exp(I*(b*x+a))+1)*x-3/b^4*d^3*ln(exp(I*(b*x+a))+1)*a+3/b^3*d^3*ln(1-exp(I*(b*x+a)))*x-
6/b^3*c*d^2*arctanh(exp(I*(b*x+a)))+6/b^4*d^3*a*arctanh(exp(I*(b*x+a)))-3*I*d^3*polylog(2,exp(I*(b*x+a)))/b^4-
3*I*d^3*polylog(4,-exp(I*(b*x+a)))/b^4+3*I*d^3*polylog(2,-exp(I*(b*x+a)))/b^4+3*I*d^3*polylog(4,exp(I*(b*x+a))
)/b^4

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3886 vs. \(2 (265) = 530\).
time = 1.61, size = 3886, normalized size = 12.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c^3*(2*cos(b*x + a)/(cos(b*x + a)^2 - 1) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1)) - 3*a*c^2*d*(2*
cos(b*x + a)/(cos(b*x + a)^2 - 1) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1))/b + 3*a^2*c*d^2*(2*cos(b*x
+ a)/(cos(b*x + a)^2 - 1) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1))/b^2 - a^3*d^3*(2*cos(b*x + a)/(cos(
b*x + a)^2 - 1) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1))/b^3 - 4*(2*((b*x + a)^3*d^3 + 6*b*c*d^2 - 6*a
*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 2)*d^3)*(b*x + a) + ((b*x + a)^3*
d^3 + 6*b*c*d^2 - 6*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 2)*d^3)*(b*x
+ a))*cos(4*b*x + 4*a) - 2*((b*x + a)^3*d^3 + 6*b*c*d^2 - 6*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*
c^2*d - 2*a*b*c*d^2 + (a^2 + 2)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^3*d^3 - 6*I*b*c*d^2 + 6*I*a*d
^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + 3*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 + (-I*a^2 - 2*I)*d^3)*(b*x + a))*s
in(4*b*x + 4*a) - 2*(I*(b*x + a)^3*d^3 + 6*I*b*c*d^2 - 6*I*a*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 3*(I*
b^2*c^2*d - 2*I*a*b*c*d^2 + (I*a^2 + 2*I)*d^3)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a)
+ 1) - 12*(b*c*d^2 - a*d^3 + (b*c*d^2 - a*d^3)*cos(4*b*x + 4*a) - 2*(b*c*d^2 - a*d^3)*cos(2*b*x + 2*a) + (I*b
*c*d^2 - I*a*d^3)*sin(4*b*x + 4*a) + 2*(-I*b*c*d^2 + I*a*d^3)*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x
+ a) - 1) + 2*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 2)*d^3)
*(b*x + a) + ((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 2)*d^3)*
(b*x + a))*cos(4*b*x + 4*a) - 2*((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^
2 + (a^2 + 2)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^3*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 +
3*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 + (-I*a^2 - 2*I)*d^3)*(b*x + a))*sin(4*b*x + 4*a) - 2*(I*(b*x + a)^3*d^3 + 3*(
I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 3*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + (I*a^2 + 2*I)*d^3)*(b*x + a))*sin(2*b*x +
2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 4*(-I*(b*x + a)^3*d^3 - 3*b^2*c^2*d + 6*a*b*c*d^2 - 3*a^2*d^3
+ 3*(-I*b*c*d^2 + (I*a - 1)*d^3)*(b*x + a)^2 + 3*(-I*b^2*c^2*d + 2*(I*a - 1)*b*c*d^2 + (-I*a^2 + 2*a)*d^3)*(b
*x + a))*cos(3*b*x + 3*a) - 4*(-I*(b*x + a)^3*d^3 + 3*b^2*c^2*d - 6*a*b*c*d^2 + 3*a^2*d^3 + 3*(-I*b*c*d^2 + (I
*a + 1)*d^3)*(b*x + a)^2 + 3*(-I*b^2*c^2*d + 2*(I*a + 1)*b*c*d^2 + (-I*a^2 - 2*a)*d^3)*(b*x + a))*cos(b*x + a)
- 6*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 2)*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) + (b^2*c^2*d -
2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 2)*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 2*(b^2*c^2*
d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 2)*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (I*b^2*c
^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 + (I*a^2 + 2*I)*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(4*b*x +
4*a) + 2*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(b*x + a)^2*d^3 + (-I*a^2 - 2*I)*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*
x + a))*sin(2*b*x + 2*a))*dilog(-e^(I*b*x + I*a)) + 6*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 2)*d
^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) + (b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 2)*d^3 + 2*(b*c*d^2 -
a*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 2*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2 + 2)*d^3 + 2*(b*c*d^
2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*(b*x + a)^2*d^3 + (-I*a^2 - 2*I)*d^
3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) - 2*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3
+ (I*a^2 + 2*I)*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) + (-I*(b*x +
a)^3*d^3 - 6*I*b*c*d^2 + 6*I*a*d^3 - 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 - 3*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + (
I*a^2 + 2*I)*d^3)*(b*x + a) + (-I*(b*x + a)^3*d^3 - 6*I*b*c*d^2 + 6*I*a*d^3 - 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a
)^2 - 3*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + (I*a^2 + 2*I)*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 2*(-I*(b*x + a)^3*d^3
- 6*I*b*c*d^2 + 6*I*a*d^3 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + 3*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 + (-I*a^2 -
2*I)*d^3)*(b*x + a))*cos(2*b*x + 2*a) + ((b*x + a)^3*d^3 + 6*b*c*d^2 - 6*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a
)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 2)*d^3)*(b*x + a))*sin(4*b*x + 4*a) - 2*((b*x + a)^3*d^3 + 6*b*c*d^2
- 6*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 + 2)*d^3)*(b*x + a))*sin(2*b*
x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + (I*(b*x + a)^3*d^3 + 6*I*b*c*d^2 - 6*I*a
*d^3 - 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 - 3*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 + (-I*a^2 - 2*I)*d^3)*(b*x + a)
+ (I*(b*x + a)^3*d^3 + 6*I*b*c*d^2 - 6*I*a*d^3 - 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 - 3*(-I*b^2*c^2*d + 2*I*
a*b*c*d^2 + (-I*a^2 - 2*I)*d^3)*(b*x + a))*cos(4*b*x + 4*a) - 2*(I*(b*x + a)^3*d^3 + 6*I*b*c*d^2 - 6*I*a*d^3 +
3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 3*(I*b^2...

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1744 vs. \(2 (265) = 530\).
time = 0.42, size = 1744, normalized size = 5.64 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(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)*cos(b*x + a) - 3*(-I*b^2*d^3*x^2 - 2*I*b^2*c*
d^2*x - I*b^2*c^2*d - 2*I*d^3 + (I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d + 2*I*d^3)*cos(b*x + a)^2)*dilo
g(cos(b*x + a) + I*sin(b*x + a)) - 3*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d + 2*I*d^3 + (-I*b^2*d^3*x^
2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d - 2*I*d^3)*cos(b*x + a)^2)*dilog(cos(b*x + a) - I*sin(b*x + a)) - 3*(-I*b^2*
d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d - 2*I*d^3 + (I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d + 2*I*d^3)*
cos(b*x + a)^2)*dilog(-cos(b*x + a) + I*sin(b*x + a)) - 3*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d + 2*I
*d^3 + (-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d - 2*I*d^3)*cos(b*x + a)^2)*dilog(-cos(b*x + a) - I*sin(
b*x + a)) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 + 6*b*c*d^2 - (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)^2 + 3*(b^3*c^2*d + 2*b*d^3)*x)*log(cos(b*x + a) + I*sin(b*
x + a) + 1) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 + 6*b*c*d^2 - (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)^2 + 3*(b^3*c^2*d + 2*b*d^3)*x)*log(cos(b*x + a) - I*sin(
b*x + a) + 1) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*(a^2 + 2)*b*c*d^2 - (a^3 + 6*a)*d^3 - (b^3*c^3 - 3*a*b^2*c^2*d +
3*(a^2 + 2)*b*c*d^2 - (a^3 + 6*a)*d^3)*cos(b*x + a)^2)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) - (b^
3*c^3 - 3*a*b^2*c^2*d + 3*(a^2 + 2)*b*c*d^2 - (a^3 + 6*a)*d^3 - (b^3*c^3 - 3*a*b^2*c^2*d + 3*(a^2 + 2)*b*c*d^2
- (a^3 + 6*a)*d^3)*cos(b*x + a)^2)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) - (b^3*d^3*x^3 + 3*b^3*c
*d^2*x^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + (a^3 + 6*a)*d^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d -
3*a^2*b*c*d^2 + (a^3 + 6*a)*d^3 + 3*(b^3*c^2*d + 2*b*d^3)*x)*cos(b*x + a)^2 + 3*(b^3*c^2*d + 2*b*d^3)*x)*log(-
cos(b*x + a) + I*sin(b*x + a) + 1) - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + (a^3 + 6
*a)*d^3 - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + (a^3 + 6*a)*d^3 + 3*(b^3*c^2*d + 2*
b*d^3)*x)*cos(b*x + a)^2 + 3*(b^3*c^2*d + 2*b*d^3)*x)*log(-cos(b*x + a) - I*sin(b*x + a) + 1) - 6*(-I*d^3*cos(
b*x + a)^2 + I*d^3)*polylog(4, cos(b*x + a) + I*sin(b*x + a)) - 6*(I*d^3*cos(b*x + a)^2 - I*d^3)*polylog(4, co
s(b*x + a) - I*sin(b*x + a)) - 6*(-I*d^3*cos(b*x + a)^2 + I*d^3)*polylog(4, -cos(b*x + a) + I*sin(b*x + a)) -
6*(I*d^3*cos(b*x + a)^2 - I*d^3)*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) - 6*(b*d^3*x + b*c*d^2 - (b*d^3*x
+ b*c*d^2)*cos(b*x + a)^2)*polylog(3, cos(b*x + a) + I*sin(b*x + a)) - 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d
^2)*cos(b*x + a)^2)*polylog(3, cos(b*x + a) - I*sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cos
(b*x + a)^2)*polylog(3, -cos(b*x + a) + I*sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cos(b*x +
a)^2)*polylog(3, -cos(b*x + a) - I*sin(b*x + a)) + 6*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*sin(b*x + a))/
(b^4*cos(b*x + a)^2 - b^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c + d x\right )^{3} \csc ^{3}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*x)**3*csc(a + b*x)**3, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/sin(a + b*x)^3,x)

[Out]

\text{Hanged}

________________________________________________________________________________________