Integrand size = 12, antiderivative size = 87 \[ \int x^3 \coth ^2(a+b x) \, dx=-\frac {x^3}{b}+\frac {x^4}{4}-\frac {x^3 \coth (a+b x)}{b}+\frac {3 x^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 x \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}-\frac {3 \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^4} \]
-x^3/b+1/4*x^4-x^3*coth(b*x+a)/b+3*x^2*ln(1-exp(2*b*x+2*a))/b^2+3*x*polylo g(2,exp(2*b*x+2*a))/b^3-3/2*polylog(3,exp(2*b*x+2*a))/b^4
Leaf count is larger than twice the leaf count of optimal. \(222\) vs. \(2(87)=174\).
Time = 1.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.55 \[ \int x^3 \coth ^2(a+b x) \, dx=\frac {x^4}{4}-\frac {e^{2 a} \left (2 b^3 e^{-2 a} x^3-3 b^2 \left (1-e^{-2 a}\right ) x^2 \log \left (1-e^{-a-b x}\right )-3 b^2 \left (1-e^{-2 a}\right ) x^2 \log \left (1+e^{-a-b x}\right )+6 b \left (1-e^{-2 a}\right ) x \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+6 b \left (1-e^{-2 a}\right ) x \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (3,e^{-a-b x}\right )\right )}{b^4 \left (-1+e^{2 a}\right )}+\frac {x^3 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b} \]
x^4/4 - (E^(2*a)*((2*b^3*x^3)/E^(2*a) - 3*b^2*(1 - E^(-2*a))*x^2*Log[1 - E ^(-a - b*x)] - 3*b^2*(1 - E^(-2*a))*x^2*Log[1 + E^(-a - b*x)] + 6*b*(1 - E ^(-2*a))*x*PolyLog[2, -E^(-a - b*x)] + 6*b*(1 - E^(-2*a))*x*PolyLog[2, E^( -a - b*x)] + 6*(1 - E^(-2*a))*PolyLog[3, -E^(-a - b*x)] + 6*(1 - E^(-2*a)) *PolyLog[3, E^(-a - b*x)]))/(b^4*(-1 + E^(2*a))) + (x^3*Csch[a]*Csch[a + b *x]*Sinh[b*x])/b
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 25, 4203, 15, 26, 3042, 26, 4201, 2620, 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.
\(\displaystyle \int x^3 \coth ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -x^3 \tan \left (i a+i b x+\frac {\pi }{2}\right )^2dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int x^3 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2dx\) |
\(\Big \downarrow \) 4203 |
\(\displaystyle -\frac {3 i \int i x^2 \coth (a+b x)dx}{b}+\int x^3dx-\frac {x^3 \coth (a+b x)}{b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {3 i \int i x^2 \coth (a+b x)dx}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {3 \int x^2 \coth (a+b x)dx}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \int -i x^2 \tan \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {3 i \int x^2 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {3 i \left (2 i \int \frac {e^{2 a+2 b x-i \pi } x^2}{1+e^{2 a+2 b x-i \pi }}dx-\frac {i x^3}{3}\right )}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {3 i \left (2 i \left (\frac {x^2 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int x \log \left (1+e^{2 a+2 b x-i \pi }\right )dx}{b}\right )-\frac {i x^3}{3}\right )}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {3 i \left (2 i \left (\frac {x^2 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\frac {\int \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{2 b}-\frac {x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}}{b}\right )-\frac {i x^3}{3}\right )}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {3 i \left (2 i \left (\frac {x^2 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\frac {\int e^{-2 a-2 b x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )de^{2 a+2 b x-i \pi }}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}}{b}\right )-\frac {i x^3}{3}\right )}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {3 i \left (2 i \left (\frac {x^2 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\frac {\operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}}{b}\right )-\frac {i x^3}{3}\right )}{b}-\frac {x^3 \coth (a+b x)}{b}+\frac {x^4}{4}\) |
x^4/4 - (x^3*Coth[a + b*x])/b - ((3*I)*((-1/3*I)*x^3 + (2*I)*((x^2*Log[1 + E^(2*a - I*Pi + 2*b*x)])/(2*b) - (-1/2*(x*PolyLog[2, -E^(2*a - I*Pi + 2*b *x)])/b + PolyLog[3, -E^(2*a - I*Pi + 2*b*x)]/(4*b^2))/b)))/b
3.1.6.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs. \(2(83)=166\).
Time = 0.11 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.28
method | result | size |
risch | \(\frac {x^{4}}{4}-\frac {2 x^{3}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {4 a^{3}}{b^{4}}+\frac {6 x \,a^{2}}{b^{3}}-\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}+\frac {6 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {2 x^{3}}{b}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {6 a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}\) | \(198\) |
1/4*x^4-2*x^3/b/(exp(2*b*x+2*a)-1)+4/b^4*a^3+6*x/b^3*a^2-6/b^4*polylog(3,e xp(b*x+a))-6/b^4*polylog(3,-exp(b*x+a))+6/b^3*polylog(2,exp(b*x+a))*x+3/b^ 2*ln(exp(b*x+a)+1)*x^2+6/b^3*polylog(2,-exp(b*x+a))*x-2*x^3/b+3/b^2*ln(1-e xp(b*x+a))*x^2-6/b^4*a^2*ln(exp(b*x+a))-3/b^4*ln(1-exp(b*x+a))*a^2+3/b^4*a ^2*ln(exp(b*x+a)-1)
Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (82) = 164\).
Time = 0.26 (sec) , antiderivative size = 632, normalized size of antiderivative = 7.26 \[ \int x^3 \coth ^2(a+b x) \, dx=-\frac {b^{4} x^{4} - 8 \, a^{3} - {\left (b^{4} x^{4} - 8 \, b^{3} x^{3} - 8 \, a^{3}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{4} x^{4} - 8 \, b^{3} x^{3} - 8 \, a^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{4} x^{4} - 8 \, b^{3} x^{3} - 8 \, a^{3}\right )} \sinh \left (b x + a\right )^{2} - 24 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 24 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 12 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )^{2} - b^{2} x^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 12 \, {\left (a^{2} \cosh \left (b x + a\right )^{2} + 2 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 12 \, {\left (b^{2} x^{2} - {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 24 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 24 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{4 \, {\left (b^{4} \cosh \left (b x + a\right )^{2} + 2 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )^{2} - b^{4}\right )}} \]
-1/4*(b^4*x^4 - 8*a^3 - (b^4*x^4 - 8*b^3*x^3 - 8*a^3)*cosh(b*x + a)^2 - 2* (b^4*x^4 - 8*b^3*x^3 - 8*a^3)*cosh(b*x + a)*sinh(b*x + a) - (b^4*x^4 - 8*b ^3*x^3 - 8*a^3)*sinh(b*x + a)^2 - 24*(b*x*cosh(b*x + a)^2 + 2*b*x*cosh(b*x + a)*sinh(b*x + a) + b*x*sinh(b*x + a)^2 - b*x)*dilog(cosh(b*x + a) + sin h(b*x + a)) - 24*(b*x*cosh(b*x + a)^2 + 2*b*x*cosh(b*x + a)*sinh(b*x + a) + b*x*sinh(b*x + a)^2 - b*x)*dilog(-cosh(b*x + a) - sinh(b*x + a)) - 12*(b ^2*x^2*cosh(b*x + a)^2 + 2*b^2*x^2*cosh(b*x + a)*sinh(b*x + a) + b^2*x^2*s inh(b*x + a)^2 - b^2*x^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 12*(a^2 *cosh(b*x + a)^2 + 2*a^2*cosh(b*x + a)*sinh(b*x + a) + a^2*sinh(b*x + a)^2 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 12*(b^2*x^2 - (b^2*x^2 - a^2)*cosh(b*x + a)^2 - 2*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a) - (b^ 2*x^2 - a^2)*sinh(b*x + a)^2 - a^2)*log(-cosh(b*x + a) - sinh(b*x + a) + 1 ) + 24*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*polylog(3, cosh(b*x + a) + sinh(b*x + a)) + 24*(cosh(b*x + a)^2 + 2*c osh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*polylog(3, -cosh(b*x + a ) - sinh(b*x + a)))/(b^4*cosh(b*x + a)^2 + 2*b^4*cosh(b*x + a)*sinh(b*x + a) + b^4*sinh(b*x + a)^2 - b^4)
\[ \int x^3 \coth ^2(a+b x) \, dx=\int x^{3} \coth ^{2}{\left (a + b x \right )}\, dx \]
Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.68 \[ \int x^3 \coth ^2(a+b x) \, dx=-\frac {2 \, x^{3}}{b} + \frac {b x^{4} e^{\left (2 \, b x + 2 \, a\right )} - b x^{4} - 8 \, x^{3}}{4 \, {\left (b e^{\left (2 \, b x + 2 \, a\right )} - b\right )}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )}}{b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )}}{b^{4}} \]
-2*x^3/b + 1/4*(b*x^4*e^(2*b*x + 2*a) - b*x^4 - 8*x^3)/(b*e^(2*b*x + 2*a) - b) + 3*(b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*dilog(-e^(b*x + a)) - 2*pol ylog(3, -e^(b*x + a)))/b^4 + 3*(b^2*x^2*log(-e^(b*x + a) + 1) + 2*b*x*dilo g(e^(b*x + a)) - 2*polylog(3, e^(b*x + a)))/b^4
\[ \int x^3 \coth ^2(a+b x) \, dx=\int { x^{3} \coth \left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x^3 \coth ^2(a+b x) \, dx=\int x^3\,{\mathrm {coth}\left (a+b\,x\right )}^2 \,d x \]