\(\int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx\) [43]

Optimal result

Integrand size = 20, antiderivative size = 227 \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=-\frac {i b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}-\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {a b d^2 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3} \]

[Out]

-I*b^2*(d*x+c)^2/f+1/3*a^2*(d*x+c)^3/d-2/3*I*a*b*(d*x+c)^3/d-1/3*b^2*(d*x+c)^3/d-b^2*(d*x+c)^2*cot(f*x+e)/f+2*
b^2*d*(d*x+c)*ln(1-exp(2*I*(f*x+e)))/f^2+2*a*b*(d*x+c)^2*ln(1-exp(2*I*(f*x+e)))/f-I*b^2*d^2*polylog(2,exp(2*I*
(f*x+e)))/f^3-2*I*a*b*d*(d*x+c)*polylog(2,exp(2*I*(f*x+e)))/f^2+a*b*d^2*polylog(3,exp(2*I*(f*x+e)))/f^3

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3803, 3798, 2221, 2611, 2320, 6724, 3801, 2317, 2438, 32} \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=\frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {2 i a b (c+d x)^3}{3 d}+\frac {a b d^2 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac {i b^2 (c+d x)^2}{f}-\frac {b^2 (c+d x)^3}{3 d}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3} \]

[In]

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

[Out]

((-I)*b^2*(c + d*x)^2)/f + (a^2*(c + d*x)^3)/(3*d) - (((2*I)/3)*a*b*(c + d*x)^3)/d - (b^2*(c + d*x)^3)/(3*d) -
(b^2*(c + d*x)^2*Cot[e + f*x])/f + (2*b^2*d*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^2*
Log[1 - E^((2*I)*(e + f*x))])/f - (I*b^2*d^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^3 - ((2*I)*a*b*d*(c + d*x)*Pol
yLog[2, E^((2*I)*(e + f*x))])/f^2 + (a*b*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/f^3

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2221

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

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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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 \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \cot (e+f x)+b^2 (c+d x)^2 \cot ^2(e+f x)\right ) \, dx \\ & = \frac {a^2 (c+d x)^3}{3 d}+(2 a b) \int (c+d x)^2 \cot (e+f x) \, dx+b^2 \int (c+d x)^2 \cot ^2(e+f x) \, dx \\ & = \frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}-(4 i a b) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{1-e^{2 i (e+f x)}} \, dx-b^2 \int (c+d x)^2 \, dx+\frac {\left (2 b^2 d\right ) \int (c+d x) \cot (e+f x) \, dx}{f} \\ & = -\frac {i b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {(4 a b d) \int (c+d x) \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}-\frac {\left (4 i b^2 d\right ) \int \frac {e^{2 i (e+f x)} (c+d x)}{1-e^{2 i (e+f x)}} \, dx}{f} \\ & = -\frac {i b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {\left (2 i a b d^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac {\left (2 b^2 d^2\right ) \int \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f^2} \\ & = -\frac {i b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {\left (a b d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{f^3}+\frac {\left (i b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{f^3} \\ & = -\frac {i b^2 (c+d x)^2}{f}+\frac {a^2 (c+d x)^3}{3 d}-\frac {2 i a b (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^3}{3 d}-\frac {b^2 (c+d x)^2 \cot (e+f x)}{f}+\frac {2 b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}-\frac {2 i a b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {a b d^2 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(737\) vs. \(2(227)=454\).

Time = 7.15 (sec) , antiderivative size = 737, normalized size of antiderivative = 3.25 \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=-\frac {a b d^2 e^{i e} \csc (e) \left (2 e^{-2 i e} f^3 x^3+3 i \left (1-e^{-2 i e}\right ) f^2 x^2 \log \left (1-e^{-i (e+f x)}\right )+3 i \left (1-e^{-2 i e}\right ) f^2 x^2 \log \left (1+e^{-i (e+f x)}\right )-6 \left (1-e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (2,-e^{-i (e+f x)}\right )-6 \left (1-e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (2,e^{-i (e+f x)}\right )+6 i \left (1-e^{-2 i e}\right ) \operatorname {PolyLog}\left (3,-e^{-i (e+f x)}\right )+6 i \left (1-e^{-2 i e}\right ) \operatorname {PolyLog}\left (3,e^{-i (e+f x)}\right )\right )}{3 f^3}+\frac {1}{3} x \left (3 c^2+3 c d x+d^2 x^2\right ) \csc (e) \left (2 a b \cos (e)+a^2 \sin (e)-b^2 \sin (e)\right )+\frac {2 b^2 c d \csc (e) (-f x \cos (e)+\log (\cos (f x) \sin (e)+\cos (e) \sin (f x)) \sin (e))}{f^2 \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {2 a b c^2 \csc (e) (-f x \cos (e)+\log (\cos (f x) \sin (e)+\cos (e) \sin (f x)) \sin (e))}{f \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {\csc (e) \csc (e+f x) \left (b^2 c^2 \sin (f x)+2 b^2 c d x \sin (f x)+b^2 d^2 x^2 \sin (f x)\right )}{f}-\frac {b^2 d^2 \csc (e) \sec (e) \left (e^{i \arctan (\tan (e))} f^2 x^2+\frac {\left (i f x (-\pi +2 \arctan (\tan (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x+\arctan (\tan (e))) \log \left (1-e^{2 i (f x+\arctan (\tan (e)))}\right )+\pi \log (\cos (f x))+2 \arctan (\tan (e)) \log (\sin (f x+\arctan (\tan (e))))+i \operatorname {PolyLog}\left (2,e^{2 i (f x+\arctan (\tan (e)))}\right )\right ) \tan (e)}{\sqrt {1+\tan ^2(e)}}\right )}{f^3 \sqrt {\sec ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}-\frac {2 a b c d \csc (e) \sec (e) \left (e^{i \arctan (\tan (e))} f^2 x^2+\frac {\left (i f x (-\pi +2 \arctan (\tan (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x+\arctan (\tan (e))) \log \left (1-e^{2 i (f x+\arctan (\tan (e)))}\right )+\pi \log (\cos (f x))+2 \arctan (\tan (e)) \log (\sin (f x+\arctan (\tan (e))))+i \operatorname {PolyLog}\left (2,e^{2 i (f x+\arctan (\tan (e)))}\right )\right ) \tan (e)}{\sqrt {1+\tan ^2(e)}}\right )}{f^2 \sqrt {\sec ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}} \]

[In]

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

[Out]

-1/3*(a*b*d^2*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e +
f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - 6*(1 - E^((-2*I)*e))*f*x*PolyLog[2, -E
^((-I)*(e + f*x))] - 6*(1 - E^((-2*I)*e))*f*x*PolyLog[2, E^((-I)*(e + f*x))] + (6*I)*(1 - E^((-2*I)*e))*PolyLo
g[3, -E^((-I)*(e + f*x))] + (6*I)*(1 - E^((-2*I)*e))*PolyLog[3, E^((-I)*(e + f*x))]))/f^3 + (x*(3*c^2 + 3*c*d*
x + d^2*x^2)*Csc[e]*(2*a*b*Cos[e] + a^2*Sin[e] - b^2*Sin[e]))/3 + (2*b^2*c*d*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f
*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) + (2*a*b*c^2*Csc[e]*(-(f*x*Cos[e]) + Log[Co
s[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (Csc[e]*Csc[e + f*x]*(b^2*c^2*Sin[f*x] +
2*b^2*c*d*x*Sin[f*x] + b^2*d^2*x^2*Sin[f*x]))/f - (b^2*d^2*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*
f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + A
rcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*
(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^3*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (2*a*b*c*d
*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] -
2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[
Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^2
*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 931 vs. \(2 (207 ) = 414\).

Time = 0.80 (sec) , antiderivative size = 932, normalized size of antiderivative = 4.11

method	result	size
risch	\(\text {Expression too large to display}\)	\(932\)

[In]

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

[Out]

d*a^2*c*x^2+a^2*c^2*x-d*b^2*c*x^2+4/f^3*b^2*e*d^2*ln(exp(I*(f*x+e)))-2/f^3*b^2*e*d^2*ln(exp(I*(f*x+e))-1)+2/f*
b*a*c^2*ln(exp(I*(f*x+e))+1)-4/f*b*a*c^2*ln(exp(I*(f*x+e)))+2/f*b*a*c^2*ln(exp(I*(f*x+e))-1)+2/f^2*b^2*c*d*ln(
exp(I*(f*x+e))+1)-4/f^2*b^2*c*d*ln(exp(I*(f*x+e)))+2/f^2*b^2*c*d*ln(exp(I*(f*x+e))-1)+4/f^3*b*a*d^2*polylog(3,
exp(I*(f*x+e)))+4/f^3*b*a*d^2*polylog(3,-exp(I*(f*x+e)))+2/f^3*b^2*d^2*ln(1-exp(I*(f*x+e)))*e+2/f^2*b^2*d^2*ln
(1-exp(I*(f*x+e)))*x+2/f^2*b^2*d^2*ln(exp(I*(f*x+e))+1)*x-2*I/f*b^2*d^2*x^2-2*I/f^3*b^2*d^2*e^2-2*I/f^3*b^2*d^
2*polylog(2,exp(I*(f*x+e)))-2*I/f^3*b^2*d^2*polylog(2,-exp(I*(f*x+e)))-2/3*I*d^2*a*b*x^3-2*I*b^2*(d^2*x^2+2*c*
d*x+c^2)/f/(exp(2*I*(f*x+e))-1)-2/f^3*b*a*d^2*ln(1-exp(I*(f*x+e)))*e^2+2/f*b*a*d^2*ln(exp(I*(f*x+e))+1)*x^2+2/
f*b*a*d^2*ln(1-exp(I*(f*x+e)))*x^2-4/f^3*b*e^2*a*d^2*ln(exp(I*(f*x+e)))+2/f^3*b*e^2*a*d^2*ln(exp(I*(f*x+e))-1)
+8/3*I/f^3*b*a*d^2*e^3-4*I/f^2*b^2*d^2*e*x-2*I*d*a*b*c*x^2-8*I/f*b*d*c*a*e*x-4*I/f^2*b*a*d^2*polylog(2,-exp(I*
(f*x+e)))*x+4*I/f^2*b*a*d^2*e^2*x-4*I/f^2*b*a*d^2*polylog(2,exp(I*(f*x+e)))*x-4*I/f^2*b*d*c*a*e^2+8/f^2*b*e*a*
c*d*ln(exp(I*(f*x+e)))-4/f^2*b*e*a*c*d*ln(exp(I*(f*x+e))-1)+4/f*b*d*c*a*ln(1-exp(I*(f*x+e)))*x+4/f*b*d*c*a*ln(
exp(I*(f*x+e))+1)*x+4/f^2*b*d*c*a*ln(1-exp(I*(f*x+e)))*e-4*I/f^2*b*d*c*a*polylog(2,exp(I*(f*x+e)))-4*I/f^2*b*d
*c*a*polylog(2,-exp(I*(f*x+e)))+2*I*a*b*c^2*x+2/3*I/d*a*b*c^3+1/3*d^2*a^2*x^3+1/3/d*a^2*c^3-1/3*d^2*b^2*x^3-b^
2*c^2*x-1/3/d*b^2*c^3

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. 715 vs. \(2 (201) = 402\).

Time = 0.29 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.15 \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=-\frac {6 \, b^{2} d^{2} f^{2} x^{2} + 12 \, b^{2} c d f^{2} x + 6 \, b^{2} c^{2} f^{2} - 3 \, a b d^{2} {\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) - 3 \, a b d^{2} {\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (2 i \, a b d^{2} f x + 2 i \, a b c d f + i \, b^{2} d^{2}\right )} {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (-2 i \, a b d^{2} f x - 2 i \, a b c d f - i \, b^{2} d^{2}\right )} {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) - 6 \, {\left (a b d^{2} e^{2} + a b c^{2} f^{2} - b^{2} d^{2} e - {\left (2 \, a b c d e - b^{2} c d\right )} f\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) + \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) \sin \left (2 \, f x + 2 \, e\right ) - 6 \, {\left (a b d^{2} e^{2} + a b c^{2} f^{2} - b^{2} d^{2} e - {\left (2 \, a b c d e - b^{2} c d\right )} f\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) - \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) \sin \left (2 \, f x + 2 \, e\right ) - 6 \, {\left (a b d^{2} f^{2} x^{2} - a b d^{2} e^{2} + 2 \, a b c d e f + b^{2} d^{2} e + {\left (2 \, a b c d f^{2} + b^{2} d^{2} f\right )} x\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right ) \sin \left (2 \, f x + 2 \, e\right ) - 6 \, {\left (a b d^{2} f^{2} x^{2} - a b d^{2} e^{2} + 2 \, a b c d e f + b^{2} d^{2} e + {\left (2 \, a b c d f^{2} + b^{2} d^{2} f\right )} x\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right ) \sin \left (2 \, f x + 2 \, e\right ) + 6 \, {\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 2 \, {\left ({\left (a^{2} - b^{2}\right )} d^{2} f^{3} x^{3} + 3 \, {\left (a^{2} - b^{2}\right )} c d f^{3} x^{2} + 3 \, {\left (a^{2} - b^{2}\right )} c^{2} f^{3} x\right )} \sin \left (2 \, f x + 2 \, e\right )}{6 \, f^{3} \sin \left (2 \, f x + 2 \, e\right )} \]

[In]

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

[Out]

-1/6*(6*b^2*d^2*f^2*x^2 + 12*b^2*c*d*f^2*x + 6*b^2*c^2*f^2 - 3*a*b*d^2*polylog(3, cos(2*f*x + 2*e) + I*sin(2*f
*x + 2*e))*sin(2*f*x + 2*e) - 3*a*b*d^2*polylog(3, cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) + 3
*(2*I*a*b*d^2*f*x + 2*I*a*b*c*d*f + I*b^2*d^2)*dilog(cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) +
3*(-2*I*a*b*d^2*f*x - 2*I*a*b*c*d*f - I*b^2*d^2)*dilog(cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e
) - 6*(a*b*d^2*e^2 + a*b*c^2*f^2 - b^2*d^2*e - (2*a*b*c*d*e - b^2*c*d)*f)*log(-1/2*cos(2*f*x + 2*e) + 1/2*I*si
n(2*f*x + 2*e) + 1/2)*sin(2*f*x + 2*e) - 6*(a*b*d^2*e^2 + a*b*c^2*f^2 - b^2*d^2*e - (2*a*b*c*d*e - b^2*c*d)*f)
*log(-1/2*cos(2*f*x + 2*e) - 1/2*I*sin(2*f*x + 2*e) + 1/2)*sin(2*f*x + 2*e) - 6*(a*b*d^2*f^2*x^2 - a*b*d^2*e^2
+ 2*a*b*c*d*e*f + b^2*d^2*e + (2*a*b*c*d*f^2 + b^2*d^2*f)*x)*log(-cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e) + 1)*
sin(2*f*x + 2*e) - 6*(a*b*d^2*f^2*x^2 - a*b*d^2*e^2 + 2*a*b*c*d*e*f + b^2*d^2*e + (2*a*b*c*d*f^2 + b^2*d^2*f)*
x)*log(-cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e) + 6*(b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^2*x + b
^2*c^2*f^2)*cos(2*f*x + 2*e) - 2*((a^2 - b^2)*d^2*f^3*x^3 + 3*(a^2 - b^2)*c*d*f^3*x^2 + 3*(a^2 - b^2)*c^2*f^3*
x)*sin(2*f*x + 2*e))/(f^3*sin(2*f*x + 2*e))

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*cot(e + f*x))**2*(c + d*x)**2, 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. 1948 vs. \(2 (201) = 402\).

Time = 0.59 (sec) , antiderivative size = 1948, normalized size of antiderivative = 8.58 \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 (a+b \cot (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \]

[In]

int((a + b*cot(e + f*x))^2*(c + d*x)^2,x)

[Out]

int((a + b*cot(e + f*x))^2*(c + d*x)^2, x)