Integrand size = 5, antiderivative size = 74 \[ \int x \arctan (\sinh (x)) \, dx=-x^2 \arctan \left (e^x\right )+\frac {1}{2} x^2 \arctan (\sinh (x))+i x \operatorname {PolyLog}\left (2,-i e^x\right )-i x \operatorname {PolyLog}\left (2,i e^x\right )-i \operatorname {PolyLog}\left (3,-i e^x\right )+i \operatorname {PolyLog}\left (3,i e^x\right ) \]
-x^2*arctan(exp(x))+1/2*x^2*arctan(sinh(x))+I*x*polylog(2,-I*exp(x))-I*x*p olylog(2,I*exp(x))-I*polylog(3,-I*exp(x))+I*polylog(3,I*exp(x))
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int x \arctan (\sinh (x)) \, dx=\frac {1}{2} x^2 \arctan (\sinh (x))-\frac {1}{2} i \left (x^2 \log \left (1-i e^x\right )-x^2 \log \left (1+i e^x\right )-2 x \operatorname {PolyLog}\left (2,-i e^x\right )+2 x \operatorname {PolyLog}\left (2,i e^x\right )+2 \operatorname {PolyLog}\left (3,-i e^x\right )-2 \operatorname {PolyLog}\left (3,i e^x\right )\right ) \]
(x^2*ArcTan[Sinh[x]])/2 - (I/2)*(x^2*Log[1 - I*E^x] - x^2*Log[1 + I*E^x] - 2*x*PolyLog[2, (-I)*E^x] + 2*x*PolyLog[2, I*E^x] + 2*PolyLog[3, (-I)*E^x] - 2*PolyLog[3, I*E^x])
Time = 0.45 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {5728, 3042, 4668, 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.
\(\displaystyle \int x \arctan (\sinh (x)) \, dx\) |
\(\Big \downarrow \) 5728 |
\(\displaystyle \frac {1}{2} x^2 \arctan (\sinh (x))-\frac {1}{2} \int x^2 \text {sech}(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} x^2 \arctan (\sinh (x))-\frac {1}{2} \int x^2 \csc \left (i x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {1}{2} x^2 \arctan (\sinh (x))+\frac {1}{2} \left (2 i \int x \log \left (1-i e^x\right )dx-2 i \int x \log \left (1+i e^x\right )dx-2 x^2 \arctan \left (e^x\right )\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {1}{2} x^2 \arctan (\sinh (x))+\frac {1}{2} \left (-2 i \left (\int \operatorname {PolyLog}\left (2,-i e^x\right )dx-x \operatorname {PolyLog}\left (2,-i e^x\right )\right )+2 i \left (\int \operatorname {PolyLog}\left (2,i e^x\right )dx-x \operatorname {PolyLog}\left (2,i e^x\right )\right )-2 x^2 \arctan \left (e^x\right )\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {1}{2} x^2 \arctan (\sinh (x))+\frac {1}{2} \left (-2 i \left (\int e^{-x} \operatorname {PolyLog}\left (2,-i e^x\right )de^x-x \operatorname {PolyLog}\left (2,-i e^x\right )\right )+2 i \left (\int e^{-x} \operatorname {PolyLog}\left (2,i e^x\right )de^x-x \operatorname {PolyLog}\left (2,i e^x\right )\right )-2 x^2 \arctan \left (e^x\right )\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {1}{2} x^2 \arctan (\sinh (x))+\frac {1}{2} \left (-2 x^2 \arctan \left (e^x\right )-2 i \left (\operatorname {PolyLog}\left (3,-i e^x\right )-x \operatorname {PolyLog}\left (2,-i e^x\right )\right )+2 i \left (\operatorname {PolyLog}\left (3,i e^x\right )-x \operatorname {PolyLog}\left (2,i e^x\right )\right )\right )\) |
(x^2*ArcTan[Sinh[x]])/2 + (-2*x^2*ArcTan[E^x] - (2*I)*(-(x*PolyLog[2, (-I) *E^x]) + PolyLog[3, (-I)*E^x]) + (2*I)*(-(x*PolyLog[2, I*E^x]) + PolyLog[3 , I*E^x]))/2
3.1.74.3.1 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[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(c + d*x)^(m + 1)*((a + b*ArcTan[u])/(d*(m + 1))), x] - Simp[b/(d*(m + 1) ) Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 + u^2)), x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] & & !FunctionOfQ[(c + d*x)^(m + 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.56 (sec) , antiderivative size = 632, normalized size of antiderivative = 8.54
1/2*I*x^2*ln(exp(x)+I)-1/2*I*x^2*ln(exp(x)-I)+1/2*I*x^2*ln(1+I*exp(x))+I*x *polylog(2,-I*exp(x))-I*polylog(3,-I*exp(x))-1/8*Pi*(csgn(I*(exp(x)-I))^2* csgn(I*(exp(x)-I)^2)-2*csgn(I*(exp(x)-I))*csgn(I*(exp(x)-I)^2)^2+csgn(I*(e xp(x)-I)^2)^3+csgn(I*(exp(x)-I)^2)*csgn(I*exp(-x))*csgn(I*exp(-x)*(exp(x)- I)^2)-csgn(I*(exp(x)-I)^2)*csgn(I*exp(-x)*(exp(x)-I)^2)^2-csgn(I*(exp(x)+I ))^2*csgn(I*(exp(x)+I)^2)+2*csgn(I*(exp(x)+I))*csgn(I*(exp(x)+I)^2)^2-csgn (I*(exp(x)+I)^2)^3-csgn(I*(exp(x)+I)^2)*csgn(I*exp(-x))*csgn(I*exp(-x)*(ex p(x)+I)^2)+csgn(I*(exp(x)+I)^2)*csgn(I*exp(-x)*(exp(x)+I)^2)^2-csgn(I*exp( -x))*csgn(I*exp(-x)*(exp(x)-I)^2)^2+csgn(I*exp(-x))*csgn(I*exp(-x)*(exp(x) +I)^2)^2-csgn(I*exp(-x)*(exp(x)+I)^2)*csgn(exp(-x)*(exp(x)+I)^2)+csgn(exp( -x)*(exp(x)+I)^2)^2+csgn(I*exp(-x)*(exp(x)-I)^2)*csgn(exp(-x)*(exp(x)-I)^2 )+csgn(exp(-x)*(exp(x)-I)^2)^2+csgn(I*exp(-x)*(exp(x)-I)^2)^3-csgn(I*exp(- x)*(exp(x)-I)^2)*csgn(exp(-x)*(exp(x)-I)^2)^2-csgn(I*exp(-x)*(exp(x)+I)^2) ^3+csgn(I*exp(-x)*(exp(x)+I)^2)*csgn(exp(-x)*(exp(x)+I)^2)^2-csgn(exp(-x)* (exp(x)+I)^2)^3-csgn(exp(-x)*(exp(x)-I)^2)^3-2)*x^2-1/2*I*x^2*ln(1-I*exp(x ))-I*x*polylog(2,I*exp(x))+I*polylog(3,I*exp(x))
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.26 \[ \int x \arctan (\sinh (x)) \, dx=\frac {1}{2} \, x^{2} \arctan \left (\sinh \left (x\right )\right ) + \frac {1}{2} i \, x^{2} \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - \frac {1}{2} i \, x^{2} \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) - i \, x {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + i \, x {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + i \, {\rm polylog}\left (3, i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - i \, {\rm polylog}\left (3, -i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) \]
1/2*x^2*arctan(sinh(x)) + 1/2*I*x^2*log(I*cosh(x) + I*sinh(x) + 1) - 1/2*I *x^2*log(-I*cosh(x) - I*sinh(x) + 1) - I*x*dilog(I*cosh(x) + I*sinh(x)) + I*x*dilog(-I*cosh(x) - I*sinh(x)) + I*polylog(3, I*cosh(x) + I*sinh(x)) - I*polylog(3, -I*cosh(x) - I*sinh(x))
\[ \int x \arctan (\sinh (x)) \, dx=\int x \operatorname {atan}{\left (\sinh {\left (x \right )} \right )}\, dx \]
\[ \int x \arctan (\sinh (x)) \, dx=\int { x \arctan \left (\sinh \left (x\right )\right ) \,d x } \]
\[ \int x \arctan (\sinh (x)) \, dx=\int { x \arctan \left (\sinh \left (x\right )\right ) \,d x } \]
Timed out. \[ \int x \arctan (\sinh (x)) \, dx=\int x\,\mathrm {atan}\left (\mathrm {sinh}\left (x\right )\right ) \,d x \]