Integrand size = 16, antiderivative size = 126 \[ \int x^2 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx=\frac {b x^4}{12}+\frac {1}{3} x^3 \coth ^{-1}(1+d+d \coth (a+b x))-\frac {1}{6} x^3 \log \left (1-(1+d) e^{2 a+2 b x}\right )-\frac {x^2 \operatorname {PolyLog}\left (2,(1+d) e^{2 a+2 b x}\right )}{4 b}+\frac {x \operatorname {PolyLog}\left (3,(1+d) e^{2 a+2 b x}\right )}{4 b^2}-\frac {\operatorname {PolyLog}\left (4,(1+d) e^{2 a+2 b x}\right )}{8 b^3} \]
1/12*b*x^4+1/3*x^3*arccoth(1+d+d*coth(b*x+a))-1/6*x^3*ln(1-(1+d)*exp(2*b*x +2*a))-1/4*x^2*polylog(2,(1+d)*exp(2*b*x+2*a))/b+1/4*x*polylog(3,(1+d)*exp (2*b*x+2*a))/b^2-1/8*polylog(4,(1+d)*exp(2*b*x+2*a))/b^3
Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.95 \[ \int x^2 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx=\frac {8 b^3 x^3 \coth ^{-1}(1+d+d \coth (a+b x))-4 b^3 x^3 \log \left (1-\frac {e^{-2 (a+b x)}}{1+d}\right )+6 b^2 x^2 \operatorname {PolyLog}\left (2,\frac {e^{-2 (a+b x)}}{1+d}\right )+6 b x \operatorname {PolyLog}\left (3,\frac {e^{-2 (a+b x)}}{1+d}\right )+3 \operatorname {PolyLog}\left (4,\frac {e^{-2 (a+b x)}}{1+d}\right )}{24 b^3} \]
(8*b^3*x^3*ArcCoth[1 + d + d*Coth[a + b*x]] - 4*b^3*x^3*Log[1 - 1/((1 + d) *E^(2*(a + b*x)))] + 6*b^2*x^2*PolyLog[2, 1/((1 + d)*E^(2*(a + b*x)))] + 6 *b*x*PolyLog[3, 1/((1 + d)*E^(2*(a + b*x)))] + 3*PolyLog[4, 1/((1 + d)*E^( 2*(a + b*x)))])/(24*b^3)
Time = 0.85 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6796, 2615, 2620, 3011, 7163, 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.
| \(\displaystyle \int x^2 \coth ^{-1}(d \coth (a+b x)+d+1) \, dx\) |
| \(\Big \downarrow \) 6796 |
| \(\displaystyle \frac {1}{3} b \int \frac {x^3}{1-(d+1) e^{2 a+2 b x}}dx+\frac {1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+d+1)\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle \frac {1}{3} b \left ((d+1) \int \frac {e^{2 a+2 b x} x^3}{1-(d+1) e^{2 a+2 b x}}dx+\frac {x^4}{4}\right )+\frac {1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+d+1)\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {1}{3} b \left ((d+1) \left (\frac {3 \int x^2 \log \left (1-(d+1) e^{2 a+2 b x}\right )dx}{2 b (d+1)}-\frac {x^3 \log \left (1-(d+1) e^{2 a+2 b x}\right )}{2 b (d+1)}\right )+\frac {x^4}{4}\right )+\frac {1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+d+1)\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {1}{3} b \left ((d+1) \left (\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,(d+1) e^{2 a+2 b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,(d+1) e^{2 a+2 b x}\right )}{2 b}\right )}{2 b (d+1)}-\frac {x^3 \log \left (1-(d+1) e^{2 a+2 b x}\right )}{2 b (d+1)}\right )+\frac {x^4}{4}\right )+\frac {1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+d+1)\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {1}{3} b \left ((d+1) \left (\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,(d+1) e^{2 a+2 b x}\right )}{2 b}-\frac {\int \operatorname {PolyLog}\left (3,(d+1) e^{2 a+2 b x}\right )dx}{2 b}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,(d+1) e^{2 a+2 b x}\right )}{2 b}\right )}{2 b (d+1)}-\frac {x^3 \log \left (1-(d+1) e^{2 a+2 b x}\right )}{2 b (d+1)}\right )+\frac {x^4}{4}\right )+\frac {1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+d+1)\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {1}{3} b \left ((d+1) \left (\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,(d+1) e^{2 a+2 b x}\right )}{2 b}-\frac {\int e^{-2 a-2 b x} \operatorname {PolyLog}\left (3,(d+1) e^{2 a+2 b x}\right )de^{2 a+2 b x}}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,(d+1) e^{2 a+2 b x}\right )}{2 b}\right )}{2 b (d+1)}-\frac {x^3 \log \left (1-(d+1) e^{2 a+2 b x}\right )}{2 b (d+1)}\right )+\frac {x^4}{4}\right )+\frac {1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+d+1)\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {1}{3} b \left ((d+1) \left (\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,(d+1) e^{2 a+2 b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (4,(d+1) e^{2 a+2 b x}\right )}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,(d+1) e^{2 a+2 b x}\right )}{2 b}\right )}{2 b (d+1)}-\frac {x^3 \log \left (1-(d+1) e^{2 a+2 b x}\right )}{2 b (d+1)}\right )+\frac {x^4}{4}\right )+\frac {1}{3} x^3 \coth ^{-1}(d \coth (a+b x)+d+1)\) |
(x^3*ArcCoth[1 + d + d*Coth[a + b*x]])/3 + (b*(x^4/4 + (1 + d)*(-1/2*(x^3* Log[1 - (1 + d)*E^(2*a + 2*b*x)])/(b*(1 + d)) + (3*(-1/2*(x^2*PolyLog[2, ( 1 + d)*E^(2*a + 2*b*x)])/b + ((x*PolyLog[3, (1 + d)*E^(2*a + 2*b*x)])/(2*b ) - PolyLog[4, (1 + d)*E^(2*a + 2*b*x)]/(4*b^2))/b))/(2*b*(1 + d)))))/3
3.3.22.3.1 Defintions of rubi rules used
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] - Simp[ 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]
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] - Si mp[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]
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[ArcCoth[(c_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcCoth[c + d*Coth[a + b*x]]/(f*( m + 1))), x] + Simp[b/(f*(m + 1)) Int[(e + f*x)^(m + 1)/(c - d - c*E^(2*a + 2*b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[(c - d)^2, 1]
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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.98 (sec) , antiderivative size = 1599, normalized size of antiderivative = 12.69
1/12*b*x^4-1/3*x^3*ln(exp(b*x+a))-1/4/b*d/(1+d)*polylog(2,(1+d)*exp(2*b*x+ 2*a))*x^2+1/3/b^3*d/(1+d)*ln(1-(1+d)*exp(2*b*x+2*a))*a^3+1/4/b^3*d/(1+d)*p olylog(2,(1+d)*exp(2*b*x+2*a))*a^2+1/4/b^2*d/(1+d)*polylog(3,(1+d)*exp(2*b *x+2*a))*x-1/2/b^2*a^2/(1+d)*x*ln(1-exp(b*x+a)*(1+d)^(1/2))-1/2/b^2*a^2/(1 +d)*x*ln(1+exp(b*x+a)*(1+d)^(1/2))+1/6/b^3*d*a^3/(1+d)*ln(d*exp(2*b*x+2*a) +exp(2*b*x+2*a)-1)-1/2/b^3*a^3*d/(1+d)*ln(1-exp(b*x+a)*(1+d)^(1/2))-1/2/b^ 3*a^3*d/(1+d)*ln(1+exp(b*x+a)*(1+d)^(1/2))-1/2/b^3*a^2*d/(1+d)*dilog(1-exp (b*x+a)*(1+d)^(1/2))-1/2/b^3*a^2*d/(1+d)*dilog(1+exp(b*x+a)*(1+d)^(1/2))+1 /2/b^2/(1+d)*ln(1-(1+d)*exp(2*b*x+2*a))*a^2*x-1/8/b^3/(1+d)*polylog(4,(1+d )*exp(2*b*x+2*a))-1/6/(1+d)*ln(1-(1+d)*exp(2*b*x+2*a))*x^3-1/4/b/(1+d)*pol ylog(2,(1+d)*exp(2*b*x+2*a))*x^2+1/3/b^3/(1+d)*ln(1-(1+d)*exp(2*b*x+2*a))* a^3+1/4/b^3/(1+d)*polylog(2,(1+d)*exp(2*b*x+2*a))*a^2+1/4/b^2/(1+d)*polylo g(3,(1+d)*exp(2*b*x+2*a))*x-1/8/b^3*d/(1+d)*polylog(4,(1+d)*exp(2*b*x+2*a) )-1/2/b^3*a^3/(1+d)*ln(1-exp(b*x+a)*(1+d)^(1/2))-1/2/b^3*a^3/(1+d)*ln(1+ex p(b*x+a)*(1+d)^(1/2))-1/2/b^3*a^2/(1+d)*dilog(1-exp(b*x+a)*(1+d)^(1/2))-1/ 2/b^3*a^2/(1+d)*dilog(1+exp(b*x+a)*(1+d)^(1/2))-1/6*d/(1+d)*ln(1-(1+d)*exp (2*b*x+2*a))*x^3+1/6/b^3*a^3/(1+d)*ln(d*exp(2*b*x+2*a)+exp(2*b*x+2*a)-1)-1 /2/b^2*a^2*d/(1+d)*x*ln(1-exp(b*x+a)*(1+d)^(1/2))-1/2/b^2*a^2*d/(1+d)*x*ln (1+exp(b*x+a)*(1+d)^(1/2))+1/2/b^2*d/(1+d)*ln(1-(1+d)*exp(2*b*x+2*a))*a^2* x-1/12*(-I*Pi*csgn(I*d/(exp(2*b*x+2*a)-1)*exp(2*b*x+2*a))^3+I*Pi*csgn(I...
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (109) = 218\).
Time = 0.28 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.85 \[ \int x^2 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx=\frac {b^{4} x^{4} + 2 \, b^{3} x^{3} \log \left (\frac {d \cosh \left (b x + a\right ) + {\left (d + 2\right )} \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}\right ) - 6 \, b^{2} x^{2} {\rm Li}_2\left (\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 6 \, b^{2} x^{2} {\rm Li}_2\left (-\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 2 \, a^{3} \log \left (2 \, {\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d + 1\right )} \sinh \left (b x + a\right ) + 2 \, \sqrt {d + 1}\right ) + 2 \, a^{3} \log \left (2 \, {\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d + 1\right )} \sinh \left (b x + a\right ) - 2 \, \sqrt {d + 1}\right ) + 12 \, b x {\rm polylog}\left (3, \sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + 12 \, b x {\rm polylog}\left (3, -\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \log \left (-\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - 12 \, {\rm polylog}\left (4, \sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - 12 \, {\rm polylog}\left (4, -\sqrt {d + 1} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{12 \, b^{3}} \]
1/12*(b^4*x^4 + 2*b^3*x^3*log((d*cosh(b*x + a) + (d + 2)*sinh(b*x + a))/(d *cosh(b*x + a) + d*sinh(b*x + a))) - 6*b^2*x^2*dilog(sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a))) - 6*b^2*x^2*dilog(-sqrt(d + 1)*(cosh(b*x + a) + si nh(b*x + a))) + 2*a^3*log(2*(d + 1)*cosh(b*x + a) + 2*(d + 1)*sinh(b*x + a ) + 2*sqrt(d + 1)) + 2*a^3*log(2*(d + 1)*cosh(b*x + a) + 2*(d + 1)*sinh(b* x + a) - 2*sqrt(d + 1)) + 12*b*x*polylog(3, sqrt(d + 1)*(cosh(b*x + a) + s inh(b*x + a))) + 12*b*x*polylog(3, -sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a))) - 2*(b^3*x^3 + a^3)*log(sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a)) + 1) - 2*(b^3*x^3 + a^3)*log(-sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a)) + 1) - 12*polylog(4, sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a))) - 12*po lylog(4, -sqrt(d + 1)*(cosh(b*x + a) + sinh(b*x + a))))/b^3
\[ \int x^2 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx=\int x^{2} \operatorname {acoth}{\left (d \coth {\left (a + b x \right )} + d + 1 \right )}\, dx \]
Time = 0.77 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98 \[ \int x^2 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + d + 1\right ) + \frac {1}{36} \, {\left (\frac {3 \, x^{4}}{d} - \frac {2 \, {\left (4 \, b^{3} x^{3} \log \left (-{\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left ({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b x {\rm Li}_{3}({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}) + 3 \, {\rm Li}_{4}({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )})\right )}}{b^{4} d}\right )} b d \]
1/3*x^3*arccoth(d*coth(b*x + a) + d + 1) + 1/36*(3*x^4/d - 2*(4*b^3*x^3*lo g(-(d + 1)*e^(2*b*x + 2*a) + 1) + 6*b^2*x^2*dilog((d + 1)*e^(2*b*x + 2*a)) - 6*b*x*polylog(3, (d + 1)*e^(2*b*x + 2*a)) + 3*polylog(4, (d + 1)*e^(2*b *x + 2*a)))/(b^4*d))*b*d
\[ \int x^2 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx=\int { x^{2} \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + d + 1\right ) \,d x } \]
Timed out. \[ \int x^2 \coth ^{-1}(1+d+d \coth (a+b x)) \, dx=\int x^2\,\mathrm {acoth}\left (d+d\,\mathrm {coth}\left (a+b\,x\right )+1\right ) \,d x \]