Integrand size = 10, antiderivative size = 103 \[ \int x \cot ^{-1}\left (e^{a+b x}\right ) \, dx=-\frac {i x \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{2 b}+\frac {i x \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{2 b}-\frac {i \operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )}{2 b^2}+\frac {i \operatorname {PolyLog}\left (3,i e^{-a-b x}\right )}{2 b^2} \]
-1/2*I*x*polylog(2,-I*exp(-b*x-a))/b+1/2*I*x*polylog(2,I*exp(-b*x-a))/b-1/ 2*I*polylog(3,-I*exp(-b*x-a))/b^2+1/2*I*polylog(3,I*exp(-b*x-a))/b^2
Time = 0.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.81 \[ \int x \cot ^{-1}\left (e^{a+b x}\right ) \, dx=-\frac {i \left (b x \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )-b x \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )+\operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )-\operatorname {PolyLog}\left (3,i e^{-a-b x}\right )\right )}{2 b^2} \]
((-1/2*I)*(b*x*PolyLog[2, (-I)*E^(-a - b*x)] - b*x*PolyLog[2, I*E^(-a - b* x)] + PolyLog[3, (-I)*E^(-a - b*x)] - PolyLog[3, I*E^(-a - b*x)]))/b^2
Time = 0.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5667, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \cot ^{-1}\left (e^{a+b x}\right ) \, dx\) |
\(\Big \downarrow \) 5667 |
\(\displaystyle \frac {1}{2} i \int x \log \left (1-i e^{-a-b x}\right )dx-\frac {1}{2} i \int x \log \left (1+i e^{-a-b x}\right )dx\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{2} i \left (\frac {x \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )dx}{b}\right )-\frac {1}{2} i \left (\frac {x \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )dx}{b}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{2} i \left (\frac {\int e^{a+b x} \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )de^{-a-b x}}{b^2}+\frac {x \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{b}\right )-\frac {1}{2} i \left (\frac {\int e^{a+b x} \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )de^{-a-b x}}{b^2}+\frac {x \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{b}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{2} i \left (\frac {\operatorname {PolyLog}\left (3,i e^{-a-b x}\right )}{b^2}+\frac {x \operatorname {PolyLog}\left (2,i e^{-a-b x}\right )}{b}\right )-\frac {1}{2} i \left (\frac {\operatorname {PolyLog}\left (3,-i e^{-a-b x}\right )}{b^2}+\frac {x \operatorname {PolyLog}\left (2,-i e^{-a-b x}\right )}{b}\right )\) |
(-1/2*I)*((x*PolyLog[2, (-I)*E^(-a - b*x)])/b + PolyLog[3, (-I)*E^(-a - b* x)]/b^2) + (I/2)*((x*PolyLog[2, I*E^(-a - b*x)])/b + PolyLog[3, I*E^(-a - b*x)]/b^2)
3.3.22.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[ArcCot[(a_.) + (b_.)*(f_)^((c_.) + (d_.)*(x_))]*(x_)^(m_.), x_Symbol] : > Simp[I/2 Int[x^m*Log[1 - I/(a + b*f^(c + d*x))], x], x] - Simp[I/2 In t[x^m*Log[1 + I/(a + b*f^(c + d*x))], x], x] /; FreeQ[{a, b, c, d, f}, x] & & IntegerQ[m] && m > 0
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (83 ) = 166\).
Time = 0.80 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.45
method | result | size |
risch | \(-\frac {i \operatorname {polylog}\left (3, i {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {\pi \,x^{2}}{4}+\frac {i \operatorname {polylog}\left (3, -i {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {i \ln \left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) a^{2}}{2 b^{2}}+\frac {i a^{2} \ln \left (1-i {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {i x \operatorname {polylog}\left (2, i {\mathrm e}^{b x +a}\right )}{2 b}+\frac {i \ln \left (1-i {\mathrm e}^{b x +a}\right ) a x}{2 b}-\frac {i \operatorname {dilog}\left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) a}{2 b^{2}}-\frac {i \operatorname {polylog}\left (2, -i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {i \operatorname {polylog}\left (2, i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {i \ln \left (1+i {\mathrm e}^{b x +a}\right ) a x}{2 b}-\frac {i \operatorname {dilog}\left (-i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {i \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a x}{2 b}-\frac {i \ln \left (-i \left ({\mathrm e}^{b x +a}+i\right )\right ) a x}{2 b}-\frac {i a^{2} \ln \left (1+i {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {i \ln \left (-i {\mathrm e}^{b x +a}\right ) \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a}{2 b^{2}}-\frac {i x \operatorname {polylog}\left (2, -i {\mathrm e}^{b x +a}\right )}{2 b}+\frac {i \ln \left (-i \left (-{\mathrm e}^{b x +a}+i\right )\right ) a^{2}}{2 b^{2}}\) | \(355\) |
-1/2*I/b^2*polylog(3,I*exp(b*x+a))+1/4*Pi*x^2+1/2*I/b^2*polylog(3,-I*exp(b *x+a))-1/2*I/b^2*ln(-I*(exp(b*x+a)+I))*a^2+1/2*I/b^2*a^2*ln(1-I*exp(b*x+a) )+1/2*I/b*polylog(2,I*exp(b*x+a))*x+1/2*I/b*ln(1-I*exp(b*x+a))*a*x-1/2*I/b ^2*dilog(-I*(exp(b*x+a)+I))*a-1/2*I/b^2*polylog(2,-I*exp(b*x+a))*a+1/2*I/b ^2*polylog(2,I*exp(b*x+a))*a-1/2*I/b*ln(1+I*exp(b*x+a))*a*x-1/2*I/b^2*dilo g(-I*exp(b*x+a))*a+1/2*I/b*ln(-I*(-exp(b*x+a)+I))*a*x-1/2*I/b*ln(-I*(exp(b *x+a)+I))*a*x-1/2*I/b^2*a^2*ln(1+I*exp(b*x+a))-1/2*I/b^2*ln(-I*exp(b*x+a)) *ln(-I*(-exp(b*x+a)+I))*a-1/2*I/b*polylog(2,-I*exp(b*x+a))*x+1/2*I/b^2*ln( -I*(-exp(b*x+a)+I))*a^2
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (73) = 146\).
Time = 0.36 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.47 \[ \int x \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\frac {2 \, b^{2} x^{2} \operatorname {arccot}\left (e^{\left (b x + a\right )}\right ) + 2 i \, b x {\rm Li}_2\left (i \, e^{\left (b x + a\right )}\right ) - 2 i \, b x {\rm Li}_2\left (-i \, e^{\left (b x + a\right )}\right ) + i \, a^{2} \log \left (e^{\left (b x + a\right )} + i\right ) - i \, a^{2} \log \left (e^{\left (b x + a\right )} - i\right ) + {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (i \, e^{\left (b x + a\right )} + 1\right ) + {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \log \left (-i \, e^{\left (b x + a\right )} + 1\right ) - 2 i \, {\rm polylog}\left (3, i \, e^{\left (b x + a\right )}\right ) + 2 i \, {\rm polylog}\left (3, -i \, e^{\left (b x + a\right )}\right )}{4 \, b^{2}} \]
1/4*(2*b^2*x^2*arccot(e^(b*x + a)) + 2*I*b*x*dilog(I*e^(b*x + a)) - 2*I*b* x*dilog(-I*e^(b*x + a)) + I*a^2*log(e^(b*x + a) + I) - I*a^2*log(e^(b*x + a) - I) + (-I*b^2*x^2 + I*a^2)*log(I*e^(b*x + a) + 1) + (I*b^2*x^2 - I*a^2 )*log(-I*e^(b*x + a) + 1) - 2*I*polylog(3, I*e^(b*x + a)) + 2*I*polylog(3, -I*e^(b*x + a)))/b^2
\[ \int x \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\int x \operatorname {acot}{\left (e^{a} e^{b x} \right )}\, dx \]
\[ \int x \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\int { x \operatorname {arccot}\left (e^{\left (b x + a\right )}\right ) \,d x } \]
\[ \int x \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\int { x \operatorname {arccot}\left (e^{\left (b x + a\right )}\right ) \,d x } \]
Timed out. \[ \int x \cot ^{-1}\left (e^{a+b x}\right ) \, dx=\int x\,\mathrm {acot}\left ({\mathrm {e}}^{a+b\,x}\right ) \,d x \]