\(\int (c+d x)^2 \cot ^2(a+b x) \csc (a+b x) \, dx\) [114]

Optimal result

Integrand size = 22, antiderivative size = 179 \[ \int (c+d x)^2 \cot ^2(a+b x) \csc (a+b x) \, dx=\frac {(c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {d^2 \text {arctanh}(\cos (a+b x))}{b^3}-\frac {d (c+d x) \csc (a+b x)}{b^2}-\frac {(c+d x)^2 \cot (a+b x) \csc (a+b x)}{2 b}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}+\frac {d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}-\frac {d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4500, 4268, 2611, 2320, 6724, 4271, 3855} \[ \int (c+d x)^2 \cot ^2(a+b x) \csc (a+b x) \, dx=-\frac {d^2 \text {arctanh}(\cos (a+b x))}{b^3}+\frac {(c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}-\frac {d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {i d (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {d (c+d x) \csc (a+b x)}{b^2}-\frac {(c+d x)^2 \cot (a+b x) \csc (a+b x)}{2 b} \]

[In]

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

[Out]

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

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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 4500

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 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]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(471\) vs. \(2(179)=358\).

Time = 7.51 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.63 \[ \int (c+d x)^2 \cot ^2(a+b x) \csc (a+b x) \, dx=-\frac {d (c+d x) \csc (a)}{b^2}+\frac {\left (-c^2-2 c d x-d^2 x^2\right ) \csc ^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {-b^2 c^2 \log \left (1-e^{i (a+b x)}\right )+2 d^2 \log \left (1-e^{i (a+b x)}\right )-2 b^2 c d x \log \left (1-e^{i (a+b x)}\right )-b^2 d^2 x^2 \log \left (1-e^{i (a+b x)}\right )+b^2 c^2 \log \left (1+e^{i (a+b x)}\right )-2 d^2 \log \left (1+e^{i (a+b x)}\right )+2 b^2 c d x \log \left (1+e^{i (a+b x)}\right )+b^2 d^2 x^2 \log \left (1+e^{i (a+b x)}\right )-2 i b d (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )+2 i b d (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )+2 d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )-2 d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{2 b^3}+\frac {\left (c^2+2 c d x+d^2 x^2\right ) \sec ^2\left (\frac {a}{2}+\frac {b x}{2}\right )}{8 b}+\frac {\sec \left (\frac {a}{2}\right ) \sec \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (-c d \sin \left (\frac {b x}{2}\right )-d^2 x \sin \left (\frac {b x}{2}\right )\right )}{2 b^2}+\frac {\csc \left (\frac {a}{2}\right ) \csc \left (\frac {a}{2}+\frac {b x}{2}\right ) \left (c d \sin \left (\frac {b x}{2}\right )+d^2 x \sin \left (\frac {b x}{2}\right )\right )}{2 b^2} \]

[In]

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

[Out]

-((d*(c + d*x)*Csc[a])/b^2) + ((-c^2 - 2*c*d*x - d^2*x^2)*Csc[a/2 + (b*x)/2]^2)/(8*b) + (-(b^2*c^2*Log[1 - E^(
I*(a + b*x))]) + 2*d^2*Log[1 - E^(I*(a + b*x))] - 2*b^2*c*d*x*Log[1 - E^(I*(a + b*x))] - b^2*d^2*x^2*Log[1 - E
^(I*(a + b*x))] + b^2*c^2*Log[1 + E^(I*(a + b*x))] - 2*d^2*Log[1 + E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 + E^(I
*(a + b*x))] + b^2*d^2*x^2*Log[1 + E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))] + (2*I)
*b*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))] + 2*d^2*PolyLog[3, -E^(I*(a + b*x))] - 2*d^2*PolyLog[3, E^(I*(a + b
*x))])/(2*b^3) + ((c^2 + 2*c*d*x + d^2*x^2)*Sec[a/2 + (b*x)/2]^2)/(8*b) + (Sec[a/2]*Sec[a/2 + (b*x)/2]*(-(c*d*
Sin[(b*x)/2]) - d^2*x*Sin[(b*x)/2]))/(2*b^2) + (Csc[a/2]*Csc[a/2 + (b*x)/2]*(c*d*Sin[(b*x)/2] + d^2*x*Sin[(b*x
)/2]))/(2*b^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (165 ) = 330\).

Time = 1.20 (sec) , antiderivative size = 546, normalized size of antiderivative = 3.05

method	result	size
risch	\(\frac {x^{2} d^{2} b \,{\mathrm e}^{3 i \left (x b +a \right )}+2 c d x b \,{\mathrm e}^{3 i \left (x b +a \right )}+c^{2} b \,{\mathrm e}^{3 i \left (x b +a \right )}+x^{2} d^{2} b \,{\mathrm e}^{i \left (x b +a \right )}+2 c d x b \,{\mathrm e}^{i \left (x b +a \right )}-2 i d^{2} x \,{\mathrm e}^{3 i \left (x b +a \right )}+c^{2} b \,{\mathrm e}^{i \left (x b +a \right )}-2 i d c \,{\mathrm e}^{3 i \left (x b +a \right )}+2 i d^{2} x \,{\mathrm e}^{i \left (x b +a \right )}+2 i d c \,{\mathrm e}^{i \left (x b +a \right )}}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}+\frac {i d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x^{2}}{2 b}+\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a^{2}}{2 b^{3}}-\frac {d^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a^{2}}{2 b^{3}}-\frac {i c d \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {i c d \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {2 c d a \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {c^{2} \operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {d^{2} a^{2} \operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{2 b}-\frac {d c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}-\frac {d c \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {d c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b}+\frac {c d \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) a}{b^{2}}-\frac {d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 d^{2} \operatorname {arctanh}\left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}\)	\(546\)

[In]

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

[Out]

1/b^2/(exp(2*I*(b*x+a))-1)^2*(x^2*d^2*b*exp(3*I*(b*x+a))+2*c*d*x*b*exp(3*I*(b*x+a))+c^2*b*exp(3*I*(b*x+a))+x^2
*d^2*b*exp(I*(b*x+a))+2*c*d*x*b*exp(I*(b*x+a))-2*I*d^2*x*exp(3*I*(b*x+a))+c^2*b*exp(I*(b*x+a))-2*I*d*c*exp(3*I
*(b*x+a))+2*I*d^2*x*exp(I*(b*x+a))+2*I*d*c*exp(I*(b*x+a)))+I/b^2*d^2*polylog(2,exp(I*(b*x+a)))*x+1/2/b*d^2*ln(
exp(I*(b*x+a))+1)*x^2+1/2/b^3*d^2*ln(1-exp(I*(b*x+a)))*a^2-1/2/b^3*d^2*ln(exp(I*(b*x+a))+1)*a^2-I/b^2*c*d*poly
log(2,-exp(I*(b*x+a)))+I/b^2*c*d*polylog(2,exp(I*(b*x+a)))-I/b^2*d^2*polylog(2,-exp(I*(b*x+a)))*x-2/b^2*c*d*a*
arctanh(exp(I*(b*x+a)))+1/b*c^2*arctanh(exp(I*(b*x+a)))+1/b^3*d^2*a^2*arctanh(exp(I*(b*x+a)))-1/2/b*d^2*ln(1-e
xp(I*(b*x+a)))*x^2-1/b*d*c*ln(1-exp(I*(b*x+a)))*x-1/b^2*d*c*ln(1-exp(I*(b*x+a)))*a+1/b*d*c*ln(exp(I*(b*x+a))+1
)*x+1/b^2*c*d*ln(exp(I*(b*x+a))+1)*a-d^2*polylog(3,exp(I*(b*x+a)))/b^3+d^2*polylog(3,-exp(I*(b*x+a)))/b^3-2/b^
3*d^2*arctanh(exp(I*(b*x+a)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 970 vs. \(2 (161) = 322\).

Time = 0.31 (sec) , antiderivative size = 970, normalized size of antiderivative = 5.42 \[ \int (c+d x)^2 \cot ^2(a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x)**2*cot(a + b*x)**2*csc(a + b*x), x)

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. 1938 vs. \(2 (161) = 322\).

Time = 0.53 (sec) , antiderivative size = 1938, normalized size of antiderivative = 10.83 \[ \int (c+d x)^2 \cot ^2(a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(c^2*(2*cos(b*x + a)/(cos(b*x + a)^2 - 1) + log(cos(b*x + a) + 1) - log(cos(b*x + a) - 1)) - 2*a*c*d*(2*co
s(b*x + a)/(cos(b*x + a)^2 - 1) + log(cos(b*x + a) + 1) - log(cos(b*x + a) - 1))/b + a^2*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 + 4*(2*((b*x + a)^2*d^2 + 2*(b*c*d -
a*d^2)*(b*x + a) - 2*d^2 + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*cos(4*b*x + 4*a) - 2*((b*x
+ a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*cos(2*b*x + 2*a) - (-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + I*a*d
^2)*(b*x + a) + 2*I*d^2)*sin(4*b*x + 4*a) - 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a) - 2*I*d^2)*
sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 4*(d^2*cos(4*b*x + 4*a) - 2*d^2*cos(2*b*x + 2*a) +
I*d^2*sin(4*b*x + 4*a) - 2*I*d^2*sin(2*b*x + 2*a) + d^2)*arctan2(sin(b*x + a), cos(b*x + a) - 1) + 2*((b*x +
a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*cos(4*b*x + 4*a) - 2*
((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + I*a*d^2
)*(b*x + a))*sin(4*b*x + 4*a) - 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a))*sin(2*b*x + 2*a))*arct
an2(sin(b*x + a), -cos(b*x + a) + 1) - 4*(I*(b*x + a)^2*d^2 + 2*b*c*d - 2*a*d^2 + 2*(I*b*c*d + (-I*a + 1)*d^2)
*(b*x + a))*cos(3*b*x + 3*a) - 4*(I*(b*x + a)^2*d^2 - 2*b*c*d + 2*a*d^2 + 2*(I*b*c*d + (-I*a - 1)*d^2)*(b*x +
a))*cos(b*x + a) - 4*(b*c*d + (b*x + a)*d^2 - a*d^2 + (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(4*b*x + 4*a) - 2*(b*
c*d + (b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) + (I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*sin(4*b*x + 4*a) + 2*(-I
*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(2*b*x + 2*a))*dilog(-e^(I*b*x + I*a)) + 4*(b*c*d + (b*x + a)*d^2 - a*d
^2 + (b*c*d + (b*x + a)*d^2 - a*d^2)*cos(4*b*x + 4*a) - 2*(b*c*d + (b*x + a)*d^2 - a*d^2)*cos(2*b*x + 2*a) - (
-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*sin(4*b*x + 4*a) - 2*(I*b*c*d + I*(b*x + a)*d^2 - I*a*d^2)*sin(2*b*x + 2
*a))*dilog(e^(I*b*x + I*a)) + (-I*(b*x + a)^2*d^2 - 2*(I*b*c*d - I*a*d^2)*(b*x + a) + 2*I*d^2 + (-I*(b*x + a)^
2*d^2 - 2*(I*b*c*d - I*a*d^2)*(b*x + a) + 2*I*d^2)*cos(4*b*x + 4*a) - 2*(-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + I*
a*d^2)*(b*x + a) + 2*I*d^2)*cos(2*b*x + 2*a) + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*sin(4*b
*x + 4*a) - 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + s
in(b*x + a)^2 + 2*cos(b*x + a) + 1) + (I*(b*x + a)^2*d^2 - 2*(-I*b*c*d + I*a*d^2)*(b*x + a) - 2*I*d^2 + (I*(b*
x + a)^2*d^2 - 2*(-I*b*c*d + I*a*d^2)*(b*x + a) - 2*I*d^2)*cos(4*b*x + 4*a) - 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*
d - I*a*d^2)*(b*x + a) - 2*I*d^2)*cos(2*b*x + 2*a) - ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*s
in(4*b*x + 4*a) + 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d^2)*sin(2*b*x + 2*a))*log(cos(b*x + a)
^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 4*(I*d^2*cos(4*b*x + 4*a) - 2*I*d^2*cos(2*b*x + 2*a) - d^2*sin(4*b
*x + 4*a) + 2*d^2*sin(2*b*x + 2*a) + I*d^2)*polylog(3, -e^(I*b*x + I*a)) - 4*(-I*d^2*cos(4*b*x + 4*a) + 2*I*d^
2*cos(2*b*x + 2*a) + d^2*sin(4*b*x + 4*a) - 2*d^2*sin(2*b*x + 2*a) - I*d^2)*polylog(3, e^(I*b*x + I*a)) + 4*((
b*x + a)^2*d^2 - 2*I*b*c*d + 2*I*a*d^2 + 2*(b*c*d - (a + I)*d^2)*(b*x + a))*sin(3*b*x + 3*a) + 4*((b*x + a)^2*
d^2 + 2*I*b*c*d - 2*I*a*d^2 + 2*(b*c*d - (a - I)*d^2)*(b*x + a))*sin(b*x + a))/(-4*I*b^2*cos(4*b*x + 4*a) + 8*
I*b^2*cos(2*b*x + 2*a) + 4*b^2*sin(4*b*x + 4*a) - 8*b^2*sin(2*b*x + 2*a) - 4*I*b^2))/b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}