3.1.23 \(\int (c+d x)^3 \csc (a+b x) \, dx\) [23]

Optimal. Leaf size=185 \[ -\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4268, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {6 i d^3 \text {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \text {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {6 d^2 (c+d x) \text {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {6 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 )}{b^2}-\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Csc[a + b*x],x]

[Out]

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

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 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 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 (a+b x) \, dx &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {(3 d) \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac {(3 d) \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {\left (6 i d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (6 i d^2\right ) \int (c+d x) \text {Li}_2\left (e^{i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}+\frac {\left (6 d^3\right ) \int \text {Li}_3\left (-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (6 d^3\right ) \int \text {Li}_3\left (e^{i (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {\left (6 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 (6 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 {2 (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 364, normalized size = 1.97 \begin {gather*} \frac {-2 b^3 c^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )+3 b^3 c^2 d 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 )-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 b^2 d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )-3 i b^2 d (c+d x)^2 \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 )}{b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*Csc[a + b*x],x]

[Out]

(-2*b^3*c^3*ArcTanh[E^(I*(a + b*x))] + 3*b^3*c^2*d*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))] - 3*b^3*c*d^2*x^2*L
og[1 + E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 + E^(I*(a + b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, -E^(I*(a +
b*x))] - (3*I)*b^2*d*(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*PolyLog[4, -E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (167 ) = 334\).
time = 0.07, size = 633, normalized size = 3.42

method	result	size
risch	\(-\frac {2 c^{3} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}+\frac {6 c \,d^{2} \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {6 c \,d^{2} \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {2 d^{3} a^{3} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 d^{3} \polylog \left (3, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 i d^{3} \polylog \left (4, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 i d^{3} \polylog \left (4, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b}+\frac {d^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{3}}{b^{4}}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{3}}{b}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a^{3}}{b^{4}}-\frac {3 c^{2} d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}-\frac {3 c^{2} d \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{b^{2}}+\frac {3 c^{2} d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {3 i d^{3} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}+\frac {3 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}-\frac {3 i c^{2} d \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {3 i c^{2} d \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {6 c^{2} d a \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {3 c^{2} d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}-\frac {6 i c \,d^{2} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {6 i c \,d^{2} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {6 c \,d^{2} a^{2} \arctanh \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {3 c \,d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {3 c \,d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{3}}-\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b}+\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) a^{2}}{b^{3}}\)	\(633\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (161) = 322\).
time = 0.39, size = 716, normalized size = 3.87 \begin {gather*} -\frac {2 \, c^{3} \log \left (\cot \left (b x + a\right ) + \csc \left (b x + a\right )\right ) - \frac {6 \, a c^{2} d \log \left (\cot \left (b x + a\right ) + \csc \left (b x + a\right )\right )}{b} + \frac {6 \, a^{2} c d^{2} \log \left (\cot \left (b x + a\right ) + \csc \left (b x + a\right )\right )}{b^{2}} - \frac {2 \, a^{3} d^{3} \log \left (\cot \left (b x + a\right ) + \csc \left (b x + a\right )\right )}{b^{3}} + \frac {12 i \, d^{3} {\rm Li}_{4}(-e^{\left (i \, b x + i \, a\right )}) - 12 i \, d^{3} {\rm Li}_{4}(e^{\left (i \, b x + i \, a\right )}) - 2 \, {\left (-i \, {\left (b x + a\right )}^{3} d^{3} + 3 \, {\left (-i \, b c d^{2} + i \, a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (-i \, b^{2} c^{2} d + 2 i \, a b c d^{2} - i \, a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 2 \, {\left (-i \, {\left (b x + a\right )}^{3} d^{3} + 3 \, {\left (-i \, b c d^{2} + i \, a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (-i \, b^{2} c^{2} d + 2 i \, a b c d^{2} - i \, a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 6 \, {\left (i \, b^{2} c^{2} d - 2 i \, a b c d^{2} + i \, {\left (b x + a\right )}^{2} d^{3} + i \, a^{2} d^{3} + 2 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) - 6 \, {\left (-i \, b^{2} c^{2} d + 2 i \, a b c d^{2} - i \, {\left (b x + a\right )}^{2} d^{3} - i \, a^{2} d^{3} + 2 \, {\left (-i \, b c d^{2} + i \, a d^{3}\right )} {\left (b x + a\right )}\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + {\left ({\left (b x + a\right )}^{3} d^{3} + 3 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (b x + a\right )}\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 ) - {\left ({\left (b x + a\right )}^{3} d^{3} + 3 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (b x + a\right )}\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 ) + 12 \, {\left (b c d^{2} + {\left (b x + a\right )} d^{3} - a d^{3}\right )} {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 12 \, {\left (b c d^{2} + {\left (b x + a\right )} d^{3} - a d^{3}\right )} {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )})}{b^{3}}}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (161) = 322\).
time = 0.40, size = 820, normalized size = 4.43 \begin {gather*} \frac {6 i \, d^{3} {\rm polylog}\left (4, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 6 i \, d^{3} {\rm polylog}\left (4, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 6 i \, d^{3} {\rm polylog}\left (4, -\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 6 i \, d^{3} {\rm polylog}\left (4, -\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\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 (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\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 (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\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 (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\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 (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - {\left (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}\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - {\left (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}\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (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}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + {\left (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}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, -\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 6 \, {\left (b d^{3} x + b c d^{2}\right )} {\rm polylog}\left (3, -\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{2 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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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),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{\sin \left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/sin(a + b*x),x)

[Out]

int((c + d*x)^3/sin(a + b*x), x)

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