3.113 \(\int (c+d x)^3 \cot ^2(a+b x) \csc (a+b x) \, dx\)
Optimal. Leaf size=308 \[ \frac {3 i d^3 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^4}-\frac {3 i d^3 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^4}+\frac {3 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^4}-\frac {3 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^4}+\frac {3 d^2 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^3}-\frac {3 d^2 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 i d (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{2 b^2}+\frac {3 i d (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{2 b^2}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b} \]
[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 = 0.34, antiderivative size = 308, normalized size of antiderivative = 1.00,
number of steps used = 25, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used
= {4415, 4183, 2531, 6609, 2282, 6589, 4186, 2279, 2391} \[ \frac {3 d^2 (c+d x) \text {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}-\frac {3 d^2 (c+d x) \text {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}+\frac {3 i d (c+d x)^2 \text {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}+\frac {3 i d^3 \text {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {3 i d^3 \text {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {3 i d^3 \text {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}-\frac {3 i d^3 \text {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {6 d^2 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \csc (a+b x)}{2 b^2}+\frac {(c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {(c+d x)^3 \cot (a+b x) \csc (a+b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^3*Cot[a + b*x]^2*Csc[a + b*x],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 2279
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 2282
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 2391
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 2531
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 4183
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 4186
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]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Rule 4415
Int[Cot[(a_.) + (b_.)*(x_)]^(p_)*Csc[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*
x)^m*Csc[a + b*x]*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^3*Cot[a + b*x]^(p - 2), x] /; FreeQ[
{a, b, c, d, m}, x] && IGtQ[p/2, 0]
Rule 6589
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 6609
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 \cot ^2(a+b x) \csc (a+b x) \, dx &=-\int (c+d x)^3 \csc (a+b x) \, dx+\int (c+d x)^3 \csc ^3(a+b x) \, dx\\ &=\frac {2 (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 {1}{2} \int (c+d x)^3 \csc (a+b x) \, dx+\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 {\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 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 {(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 (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 {\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 {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 (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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\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 {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 (6 i d^3\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\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 {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}-\frac {\left (3 i d^3\right ) \operatorname {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 ) \operatorname {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 = 4.74, size = 528, normalized size = 1.71 \[ -\frac {b^3 c^3 \log \left (1-e^{i (a+b x)}\right )-b^3 c^3 \log \left (1+e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1-e^{i (a+b x)}\right )-3 b^3 c^2 d x \log \left (1+e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1-e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1+e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1-e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1+e^{i (a+b x)}\right )+3 i d \text {Li}_2\left (-e^{i (a+b x)}\right ) \left (b^2 (c+d x)^2-2 d^2\right )-3 i d \text {Li}_2\left (e^{i (a+b x)}\right ) \left (b^2 (c+d x)^2-2 d^2\right )+b^2 (c+d x)^2 \csc (a+b x) (b (c+d x) \cot (a+b x)+3 d)-6 b c d^2 \text {Li}_3\left (-e^{i (a+b x)}\right )+6 b c d^2 \text {Li}_3\left (e^{i (a+b x)}\right )-6 b c d^2 \log \left (1-e^{i (a+b x)}\right )+6 b c d^2 \log \left (1+e^{i (a+b x)}\right )-6 b d^3 x \text {Li}_3\left (-e^{i (a+b x)}\right )+6 b d^3 x \text {Li}_3\left (e^{i (a+b x)}\right )-6 i d^3 \text {Li}_4\left (-e^{i (a+b x)}\right )+6 i d^3 \text {Li}_4\left (e^{i (a+b x)}\right )-6 b d^3 x \log \left (1-e^{i (a+b x)}\right )+6 b d^3 x \log \left (1+e^{i (a+b x)}\right )}{2 b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^3*Cot[a + b*x]^2*Csc[a + b*x],x]
[Out]
-1/2*(b^2*(c + d*x)^2*(3*d + b*(c + d*x)*Cot[a + b*x])*Csc[a + b*x] + b^3*c^3*Log[1 - E^(I*(a + b*x))] - 6*b*c
*d^2*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*Log[1 - E^(I*(a + b*x))] - b^3*c^3*Log[1 + E^(I*(a + b*
x))] + 6*b*c*d^2*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*Log[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*PolyLog[4, -E^(I*(a + b*x))] + (6*I)*d^3*Po
lyLog[4, E^(I*(a + b*x))])/b^4
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fricas [C] time = 1.19, size = 1734, normalized size = 5.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*cot(b*x+a)^2*csc(b*x+a),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 - 6*I*b^2*c*
d^2*x - 3*I*b^2*c^2*d + 6*I*d^3 + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d - 6*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 + 6*I*b^2*c*d^2*x + 3*I*b^2*c^2*d - 6*I*d^3 + (-3*I*
b^2*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d + 6*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 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d + 6*I*d^3 + (3*I*b^2*d^3*x^2 + 6*I*b^2*c*d^2*x + 3*I*b^2
*c^2*d - 6*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 + 6*I*b^2*c*d^2*x +
3*I*b^2*c^2*d - 6*I*d^3 + (-3*I*b^2*d^3*x^2 - 6*I*b^2*c*d^2*x - 3*I*b^2*c^2*d + 6*I*d^3)*cos(b*x + a)^2)*dilo
g(-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)*l
og(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*s
in(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 + 6*I*d^3)*polylog(4, cos(b*x + a) + I*sin(b*x + a)) + (6*I*d^3*cos(b*x + a
)^2 - 6*I*d^3)*polylog(4, cos(b*x + a) - I*sin(b*x + a)) + (-6*I*d^3*cos(b*x + a)^2 + 6*I*d^3)*polylog(4, -cos
(b*x + a) + I*sin(b*x + a)) + (6*I*d^3*cos(b*x + a)^2 - 6*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)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{3} \cot \left (b x + a\right )^{2} \csc \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*cot(b*x+a)^2*csc(b*x+a),x, algorithm="giac")
[Out]
integrate((d*x + c)^3*cot(b*x + a)^2*csc(b*x + a), x)
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maple [B] time = 0.14, size = 1056, normalized size = 3.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^3*cot(b*x+a)^2*csc(b*x+a),x)
[Out]
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/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)))-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-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+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))+c^3*b*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))+c^3*b*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)))+1/b*c^3*arctanh(exp(I*(b*x+a)))-3/b^3*d^3*ln(exp(I*(b*x
+a))+1)*x+3/b^3*d^3*ln(1-exp(I*(b*x+a)))*x+3/b^4*d^3*ln(1-exp(I*(b*x+a)))*a-1/b^4*d^3*a^3*arctanh(exp(I*(b*x+a
)))+3/2/b^2*c^2*d*ln(exp(I*(b*x+a))+1)*a-3/2/b^3*c*d^2*a^2*ln(exp(I*(b*x+a))+1)+1/2/b^4*d^3*ln(exp(I*(b*x+a))+
1)*a^3+3/2*I/b^2*d^3*polylog(2,exp(I*(b*x+a)))*x^2+3/b^3*c*d^2*a^2*arctanh(exp(I*(b*x+a)))-3/b^2*c^2*d*a*arcta
nh(exp(I*(b*x+a)))-3/2*I/b^2*d^3*polylog(2,-exp(I*(b*x+a)))*x^2-3/2*I/b^2*c^2*d*polylog(2,-exp(I*(b*x+a)))+3/2
*I/b^2*c^2*d*polylog(2,exp(I*(b*x+a)))+3/2/b*c^2*d*ln(exp(I*(b*x+a))+1)*x-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+3/2/b^3*c*d^2*a^2*ln(1-exp(I*(b*x+a)))-3/2/b*c*d^2*ln(1-exp(I*(b*x+a)))*x^
2+3/2/b*c*d^2*ln(exp(I*(b*x+a))+1)*x^2-1/2/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^3-3/b^4*d^3*ln(exp(I*(b*x+a))+1)*a+6/b^4*d^3*a*arctanh(exp(I*(b*x+a)))-6/b^3
*c*d^2*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
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maxima [B] time = 1.92, size = 3872, normalized size = 12.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^3*cot(b*x+a)^2*csc(b*x+a),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 - 12*b*c*d^2 + 12
*a*d^3 + 6*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 6*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 - 2)*d^3)*(b*x + 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(4*b*x + 4*a) - 4*((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) + (2*I*(b*x + a)^3*d^3 - 12*I*b*c*d^2 + 1
2*I*a*d^3 + (6*I*b*c*d^2 - 6*I*a*d^3)*(b*x + a)^2 + (6*I*b^2*c^2*d - 12*I*a*b*c*d^2 + (6*I*a^2 - 12*I)*d^3)*(b
*x + a))*sin(4*b*x + 4*a) + (-4*I*(b*x + a)^3*d^3 + 24*I*b*c*d^2 - 24*I*a*d^3 + (-12*I*b*c*d^2 + 12*I*a*d^3)*(
b*x + a)^2 + (-12*I*b^2*c^2*d + 24*I*a*b*c*d^2 + (-12*I*a^2 + 24*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 - 12*a*d^3 + 12*(b*c*d^2 - a*d^3)*cos(4*b*x + 4*a) - 24*(b*c*d^2
- a*d^3)*cos(2*b*x + 2*a) + (12*I*b*c*d^2 - 12*I*a*d^3)*sin(4*b*x + 4*a) + (-24*I*b*c*d^2 + 24*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 + 6*(b*c*d^2 - a*d^3)*(b*x + a)^2 +
6*(b^2*c^2*d - 2*a*b*c*d^2 + (a^2 - 2)*d^3)*(b*x + 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(4*b*x + 4*a) - 4*((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) + (2*I*(b*x + a)^3
*d^3 + (6*I*b*c*d^2 - 6*I*a*d^3)*(b*x + a)^2 + (6*I*b^2*c^2*d - 12*I*a*b*c*d^2 + (6*I*a^2 - 12*I)*d^3)*(b*x +
a))*sin(4*b*x + 4*a) + (-4*I*(b*x + a)^3*d^3 + (-12*I*b*c*d^2 + 12*I*a*d^3)*(b*x + a)^2 + (-12*I*b^2*c^2*d + 2
4*I*a*b*c*d^2 + (-12*I*a^2 + 24*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 - 12*b^2*c^2*d + 24*a*b*c*d^2 - 12*a^2*d^3 - 12*(I*b*c*d^2 + (-I*a + 1)*d^3)*(b*x + a)
^2 + (-12*I*b^2*c^2*d - 24*(-I*a + 1)*b*c*d^2 + (-12*I*a^2 + 24*a)*d^3)*(b*x + a))*cos(3*b*x + 3*a) + (-4*I*(b
*x + a)^3*d^3 + 12*b^2*c^2*d - 24*a*b*c*d^2 + 12*a^2*d^3 + (-12*I*b*c*d^2 - 12*(-I*a - 1)*d^3)*(b*x + a)^2 + (
-12*I*b^2*c^2*d - 24*(-I*a - 1)*b*c*d^2 + (-12*I*a^2 - 24*a)*d^3)*(b*x + a))*cos(b*x + a) - (6*b^2*c^2*d - 12*
a*b*c*d^2 + 6*(b*x + a)^2*d^3 + 6*(a^2 - 2)*d^3 + 12*(b*c*d^2 - a*d^3)*(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))*cos(4*b*x + 4*a) - 12*(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) - (-6*I*b^2*c^2*d + 12
*I*a*b*c*d^2 - 6*I*(b*x + a)^2*d^3 + (-6*I*a^2 + 12*I)*d^3 + (-12*I*b*c*d^2 + 12*I*a*d^3)*(b*x + a))*sin(4*b*x
+ 4*a) - (12*I*b^2*c^2*d - 24*I*a*b*c*d^2 + 12*I*(b*x + a)^2*d^3 + (12*I*a^2 - 24*I)*d^3 + (24*I*b*c*d^2 - 24
*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 - 12*a*b*c*d^2 + 6*(b*x + a)^2*d
^3 + 6*(a^2 - 2)*d^3 + 12*(b*c*d^2 - a*d^3)*(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))*cos(4*b*x + 4*a) - 12*(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) + (6*I*b^2*c^2*d - 12*I*a*b*c*d^2 + 6*I*(b*x + a)
^2*d^3 + (6*I*a^2 - 12*I)*d^3 + (12*I*b*c*d^2 - 12*I*a*d^3)*(b*x + a))*sin(4*b*x + 4*a) + (-12*I*b^2*c^2*d + 2
4*I*a*b*c*d^2 - 12*I*(b*x + a)^2*d^3 + (-12*I*a^2 + 24*I)*d^3 + (-24*I*b*c*d^2 + 24*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 + 3*I*a*d^3
)*(b*x + a)^2 + (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 + (-3*I*a^2 + 6*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 + 3*I*a*d^3)*(b*x + a)^2 + (-3*I*b^2*c^2*d + 6*I*a*b*c*d^2 + (-3*I*a^2 + 6
*I)*d^3)*(b*x + a))*cos(4*b*x + 4*a) + (2*I*(b*x + a)^3*d^3 - 12*I*b*c*d^2 + 12*I*a*d^3 + (6*I*b*c*d^2 - 6*I*a
*d^3)*(b*x + a)^2 + (6*I*b^2*c^2*d - 12*I*a*b*c*d^2 + (6*I*a^2 - 12*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 - 3*I*a*d^3)*(b*x + a)^2
+ (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 - 6*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 - 3*I*a*d^3)*(b*x + a)^2 + (3*I*b^2*c^2*d - 6*I*a*b*c*d^2 + (3*I*a^2 - 6*I)*d^3)*(b*x + a))*
cos(4*b*x + 4*a) + (-2*I*(b*x + a)^3*d^3 + 12*I*b*c*d^2 - 12*I*a*d^3 + (-6*I*b*c*d^2 + 6*I*a*d^3)*(b*x + a)^2
+ (-6*I*b^2*c^2*d + 12*I*a*b*c*d^2 + (-6*I*a^2 + 12*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) + (12*d^3*cos(4*b*x + 4*a) - 24*d^3*cos(2*b*x + 2*a) + 12*I*d^3*sin(4*b*x + 4*a) - 24*I*d^3*sin(2*b*x +
2*a) + 12*d^3)*polylog(4, -e^(I*b*x + I*a)) - (12*d^3*cos(4*b*x + 4*a) - 24*d^3*cos(2*b*x + 2*a) + 12*I*d^3*s
in(4*b*x + 4*a) - 24*I*d^3*sin(2*b*x + 2*a) + 12*d^3)*polylog(4, e^(I*b*x + I*a)) + (-12*I*b*c*d^2 - 12*I*(b*x
+ a)*d^3 + 12*I*a*d^3 + (-12*I*b*c*d^2 - 12*I*(b*x + a)*d^3 + 12*I*a*d^3)*cos(4*b*x + 4*a) + (24*I*b*c*d^2 +
24*I*(b*x + a)*d^3 - 24*I*a*d^3)*cos(2*b*x + 2*a) + 12*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(4*b*x + 4*a) - 24
*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*polylog(3, -e^(I*b*x + I*a)) + (12*I*b*c*d^2 + 12*I*(b*x
+ a)*d^3 - 12*I*a*d^3 + (12*I*b*c*d^2 + 12*I*(b*x + a)*d^3 - 12*I*a*d^3)*cos(4*b*x + 4*a) + (-24*I*b*c*d^2 - 2
4*I*(b*x + a)*d^3 + 24*I*a*d^3)*cos(2*b*x + 2*a) - 12*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(4*b*x + 4*a) + 24*
(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*polylog(3, e^(I*b*x + I*a)) + (4*(b*x + a)^3*d^3 - 12*I*b^
2*c^2*d + 24*I*a*b*c*d^2 - 12*I*a^2*d^3 + (12*b*c*d^2 - (12*a + 12*I)*d^3)*(b*x + a)^2 + (12*b^2*c^2*d - (24*a
+ 24*I)*b*c*d^2 + 12*(a^2 + 2*I*a)*d^3)*(b*x + a))*sin(3*b*x + 3*a) + (4*(b*x + a)^3*d^3 + 12*I*b^2*c^2*d - 2
4*I*a*b*c*d^2 + 12*I*a^2*d^3 + (12*b*c*d^2 - (12*a - 12*I)*d^3)*(b*x + a)^2 + (12*b^2*c^2*d - (24*a - 24*I)*b*
c*d^2 + 12*(a^2 - 2*I*a)*d^3)*(b*x + a))*sin(b*x + a))/(-4*I*b^3*cos(4*b*x + 4*a) + 8*I*b^3*cos(2*b*x + 2*a) +
4*b^3*sin(4*b*x + 4*a) - 8*b^3*sin(2*b*x + 2*a) - 4*I*b^3))/b
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cot(a + b*x)^2*(c + d*x)^3)/sin(a + b*x),x)
[Out]
\text{Hanged}
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{3} \cot ^{2}{\left (a + b x \right )} \csc {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**3*cot(b*x+a)**2*csc(b*x+a),x)
[Out]
Integral((c + d*x)**3*cot(a + b*x)**2*csc(a + b*x), x)
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