\(\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\) [176]
Optimal result
Integrand size = 19, antiderivative size = 123 \[
\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{8 b^2}
\]
[Out]
-1/6*b*x^3+1/2*x^2*(Pi-arccot(-c-(1-I*c)*cot(b*x+a)))-1/4*I*x^2*ln(1-I*c*exp(2*I*a+2*I*b*x))-1/4*x*polylog(2,I
*c*exp(2*I*a+2*I*b*x))/b-1/8*I*polylog(3,I*c*exp(2*I*a+2*I*b*x))/b^2
Rubi [A] (verified)
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5282, 2215, 2221, 2611, 2320,
6724} \[
\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {i \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{8 b^2}-\frac {x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {b x^3}{6}
\]
[In]
Int[x*ArcCot[c + (1 - I*c)*Cot[a + b*x]],x]
[Out]
-1/6*(b*x^3) + (x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]])/2 - (I/4)*x^2*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)] - (
x*PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)])/(4*b) - ((I/8)*PolyLog[3, I*c*E^((2*I)*a + (2*I)*b*x)])/b^2
Rule 2215
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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 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 5282
Int[ArcCot[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m +
1)*(ArcCot[c + d*Cot[a + b*x]]/(f*(m + 1))), x] + Dist[I*(b/(f*(m + 1))), Int[(e + f*x)^(m + 1)/(c - I*d - c*
E^(2*I*a + 2*I*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - I*d)^2, -1]
Rule 6724
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rubi steps \begin{align*}
\text {integral}& = \frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))+\frac {1}{2} (i b) \int \frac {x^2}{-i (1-i c)+c-c e^{2 i a+2 i b x}} \, dx \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} (b c) \int \frac {e^{2 i a+2 i b x} x^2}{-i (1-i c)+c-c e^{2 i a+2 i b x}} \, dx \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac {1}{2} i \int x \log \left (1-\frac {c e^{2 i a+2 i b x}}{-i (1-i c)+c}\right ) \, dx \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {\int \operatorname {PolyLog}\left (2,\frac {c e^{2 i a+2 i b x}}{-i (1-i c)+c}\right ) \, dx}{4 b} \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i c x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2} \\ & = -\frac {b x^3}{6}+\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{4} i x^2 \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac {x \operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \operatorname {PolyLog}\left (3,i c e^{2 i a+2 i b x}\right )}{8 b^2} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89
\[
\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\frac {1}{2} x^2 \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {i \left (2 b^2 x^2 \log \left (1+\frac {i e^{-2 i (a+b x)}}{c}\right )+2 i b x \operatorname {PolyLog}\left (2,-\frac {i e^{-2 i (a+b x)}}{c}\right )+\operatorname {PolyLog}\left (3,-\frac {i e^{-2 i (a+b x)}}{c}\right )\right )}{8 b^2}
\]
[In]
Integrate[x*ArcCot[c + (1 - I*c)*Cot[a + b*x]],x]
[Out]
(x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]])/2 - ((I/8)*(2*b^2*x^2*Log[1 + I/(c*E^((2*I)*(a + b*x)))] + (2*I)*b*x*
PolyLog[2, (-I)/(c*E^((2*I)*(a + b*x)))] + PolyLog[3, (-I)/(c*E^((2*I)*(a + b*x)))]))/b^2
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.62 (sec) , antiderivative size = 1413, normalized size of antiderivative =
11.49
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| method | result | size |
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| risch |
\(\text {Expression too large to display}\) |
\(1413\) |
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[In]
int(x*(Pi-arccot(-c-(1-I*c)*cot(b*x+a))),x,method=_RETURNVERBOSE)
[Out]
1/2*I/b*a*ln(1-I*exp(I*(b*x+a))*(-I*c)^(1/2))*x-1/8*(Pi*csgn(I*exp(I*(b*x+a)))^2*csgn(I*exp(2*I*(b*x+a)))-2*Pi
*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a)))^2+Pi*csgn(I*exp(2*I*(b*x+a)))^3+Pi*csgn(I*exp(2*I*(b*x+a)))*c
sgn(I*(I+c)/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))-Pi*csgn(I*exp(2*I*(b*x+a
)))*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))^2-Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x
+a))*c+I))*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))+Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(I+c))*c
sgn(I*(I+c)/(exp(2*I*(b*x+a))-1))-Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^2+Pi*csgn
(I/(exp(2*I*(b*x+a))-1))*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^2+Pi*csgn(I*exp(2*I*(b*x+a))*(I+c
)/(exp(2*I*(b*x+a))-1))*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))-Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(
2*I*(b*x+a))-1))^2+Pi*csgn(I*(exp(2*I*(b*x+a))*c+I))*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^2-Pi*
csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))*csgn((exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))-Pi*csgn(
(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^2-Pi*csgn(I*(I+c))*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))^2+Pi*csgn(I
*(I+c)/(exp(2*I*(b*x+a))-1))^3-Pi*csgn(I*(I+c)/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b
*x+a))-1))^2-Pi*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^3+Pi*csgn(I*(exp(2*I*(b*x+a))*c+I)/(exp(2*
I*(b*x+a))-1))*csgn((exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^2+Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(
b*x+a))-1))^3-Pi*csgn(I*exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x+a))-1))*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x
+a))-1))^2+Pi*csgn((exp(2*I*(b*x+a))*c+I)/(exp(2*I*(b*x+a))-1))^3+Pi*csgn(exp(2*I*(b*x+a))*(I+c)/(exp(2*I*(b*x
+a))-1))^3+2*I*ln(I+c))*x^2-1/4*I/b^2*ln(1-I*exp(2*I*(b*x+a))*c)*a^2-1/2*I*x^2*ln(exp(I*(b*x+a)))+1/2*I/b^2*a^
2*ln(1-I*exp(I*(b*x+a))*(-I*c)^(1/2))-1/4*I*ln(1-I*exp(2*I*(b*x+a))*c)*x^2-1/4/b*polylog(2,I*exp(2*I*(b*x+a))*
c)*x-1/4/b^2*polylog(2,I*exp(2*I*(b*x+a))*c)*a-1/4*I/b^2*a^2*ln(exp(2*I*(b*x+a))*c+I)+1/2*I/b^2*a^2*ln(1+I*exp
(I*(b*x+a))*(-I*c)^(1/2))+1/4*I*x^2*ln(exp(2*I*(b*x+a))*c+I)-1/2*I/b*ln(1-I*exp(2*I*(b*x+a))*c)*a*x-1/8*I/b^2*
polylog(3,I*exp(2*I*(b*x+a))*c)+1/2*I/b*a*ln(1+I*exp(I*(b*x+a))*(-I*c)^(1/2))*x+1/2/b^2*a*dilog(1-I*exp(I*(b*x
+a))*(-I*c)^(1/2))+1/2/b^2*a*dilog(1+I*exp(I*(b*x+a))*(-I*c)^(1/2))-1/6*b*x^3
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.24
\[
\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {4 \, b^{3} x^{3} - 12 \, \pi b^{2} x^{2} - 6 i \, b^{2} x^{2} \log \left (\frac {{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c + i}\right ) + 4 \, a^{3} + 6 \, b x {\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 6 i \, a^{2} \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} + i}{c}\right ) + 6 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \log \left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) + 3 i \, {\rm polylog}\left (3, i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{2}}
\]
[In]
integrate(x*(pi-arccot(-c-(1-I*c)*cot(b*x+a))),x, algorithm="fricas")
[Out]
-1/24*(4*b^3*x^3 - 12*pi*b^2*x^2 - 6*I*b^2*x^2*log((c*e^(2*I*b*x + 2*I*a) + I)*e^(-2*I*b*x - 2*I*a)/(c + I)) +
4*a^3 + 6*b*x*dilog(I*c*e^(2*I*b*x + 2*I*a)) + 6*I*a^2*log((c*e^(2*I*b*x + 2*I*a) + I)/c) + 6*(I*b^2*x^2 - I*
a^2)*log(-I*c*e^(2*I*b*x + 2*I*a) + 1) + 3*I*polylog(3, I*c*e^(2*I*b*x + 2*I*a)))/b^2
Sympy [F(-2)]
Exception generated. \[
\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\text {Exception raised: CoercionFailed}
\]
[In]
integrate(x*(pi-acot(-c-(1-I*c)*cot(b*x+a))),x)
[Out]
Exception raised: CoercionFailed >> Cannot convert -_t0**4 + 3*_t0**2*I*c*exp(2*I*a) - _t0**2*exp(2*I*a) + 2*c
**2*exp(4*I*a) + I*c*exp(4*I*a) of type <class 'sympy.core.add.Add'> to QQ_I[x,b,c,_t0,exp(I*a)]
Maxima [F(-2)]
Exception generated. \[
\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate(x*(pi-arccot(-c-(1-I*c)*cot(b*x+a))),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c-1>0)', see `assume?` for mor
e details)Is
Giac [F]
\[
\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\int { {\left (\pi - \operatorname {arccot}\left (-{\left (-i \, c + 1\right )} \cot \left (b x + a\right ) - c\right )\right )} x \,d x }
\]
[In]
integrate(x*(pi-arccot(-c-(1-I*c)*cot(b*x+a))),x, algorithm="giac")
[Out]
integrate((pi - arccot(-(-I*c + 1)*cot(b*x + a) - c))*x, x)
Mupad [F(-1)]
Timed out. \[
\int x \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\int x\,\left (\Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (-1+c\,1{}\mathrm {i}\right )\right )\right ) \,d x
\]
[In]
int(x*(Pi + acot(c - cot(a + b*x)*(c*1i - 1))),x)
[Out]
int(x*(Pi + acot(c - cot(a + b*x)*(c*1i - 1))), x)