\(\int x^3 \coth ^3(a+b x) \, dx\) [11]
Optimal result
Integrand size = 12, antiderivative size = 179 \[
\int x^3 \coth ^3(a+b x) \, dx=-\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {x^4}{4}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}
\]
[Out]
-3/2*x^2/b^2+1/2*x^3/b-1/4*x^4-3/2*x^2*coth(b*x+a)/b^2-1/2*x^3*coth(b*x+a)^2/b+3*x*ln(1-exp(2*b*x+2*a))/b^3+x^
3*ln(1-exp(2*b*x+2*a))/b+3/2*polylog(2,exp(2*b*x+2*a))/b^4+3/2*x^2*polylog(2,exp(2*b*x+2*a))/b^2-3/2*x*polylog
(3,exp(2*b*x+2*a))/b^3+3/4*polylog(4,exp(2*b*x+2*a))/b^4
Rubi [A] (verified)
Time = 0.25 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of
steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3801, 3797, 2221, 2317,
2438, 30, 2611, 6744, 2320, 6724} \[
\int x^3 \coth ^3(a+b x) \, dx=\frac {3 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x^2 \coth (a+b x)}{2 b^2}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x^3 \coth ^2(a+b x)}{2 b}-\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {x^4}{4}
\]
[In]
Int[x^3*Coth[a + b*x]^3,x]
[Out]
(-3*x^2)/(2*b^2) + x^3/(2*b) - x^4/4 - (3*x^2*Coth[a + b*x])/(2*b^2) - (x^3*Coth[a + b*x]^2)/(2*b) + (3*x*Log[
1 - E^(2*(a + b*x))])/b^3 + (x^3*Log[1 - E^(2*(a + b*x))])/b + (3*PolyLog[2, E^(2*(a + b*x))])/(2*b^4) + (3*x^
2*PolyLog[2, E^(2*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, E^(2*(a + b
*x))])/(4*b^4)
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
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 2317
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
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 3797
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Rule 3801
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]
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]
Rule 6744
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] - Dist[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]
Rubi steps \begin{align*}
\text {integral}& = -\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {3 \int x^2 \coth ^2(a+b x) \, dx}{2 b}+\int x^3 \coth (a+b x) \, dx \\ & = -\frac {x^4}{4}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}-2 \int \frac {e^{2 (a+b x)} x^3}{1-e^{2 (a+b x)}} \, dx+\frac {3 \int x \coth (a+b x) \, dx}{b^2}+\frac {3 \int x^2 \, dx}{2 b} \\ & = -\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {x^4}{4}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {6 \int \frac {e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx}{b^2}-\frac {3 \int x^2 \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b} \\ & = -\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {x^4}{4}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 \int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^3}-\frac {3 \int x \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {x^4}{4}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}-\frac {3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 \int \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right ) \, dx}{2 b^3} \\ & = -\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {x^4}{4}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{4 b^4} \\ & = -\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {x^4}{4}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(422\) vs. \(2(179)=358\).
Time = 3.71 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.36
\[
\int x^3 \coth ^3(a+b x) \, dx=\frac {1}{4} \left (x^4 \coth (a)-\frac {2 x^3 \text {csch}^2(a+b x)}{b}-\frac {2 e^{2 a} \left (6 b^2 e^{-2 a} x^2+b^4 e^{-2 a} x^4-6 b \left (1-e^{-2 a}\right ) x \log \left (1-e^{-a-b x}\right )-2 b^3 e^{-2 a} \left (-1+e^{2 a}\right ) x^3 \log \left (1-e^{-a-b x}\right )-6 b \left (1-e^{-2 a}\right ) x \log \left (1+e^{-a-b x}\right )-2 b^3 e^{-2 a} \left (-1+e^{2 a}\right ) x^3 \log \left (1+e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+6 b^2 \left (1-e^{-2 a}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+6 b^2 \left (1-e^{-2 a}\right ) x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+12 b \left (1-e^{-2 a}\right ) x \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )+12 b \left (1-e^{-2 a}\right ) x \operatorname {PolyLog}\left (3,e^{-a-b x}\right )+12 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (4,-e^{-a-b x}\right )+12 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (4,e^{-a-b x}\right )\right )}{b^4 \left (-1+e^{2 a}\right )}+\frac {6 x^2 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b^2}\right )
\]
[In]
Integrate[x^3*Coth[a + b*x]^3,x]
[Out]
(x^4*Coth[a] - (2*x^3*Csch[a + b*x]^2)/b - (2*E^(2*a)*((6*b^2*x^2)/E^(2*a) + (b^4*x^4)/E^(2*a) - 6*b*(1 - E^(-
2*a))*x*Log[1 - E^(-a - b*x)] - (2*b^3*(-1 + E^(2*a))*x^3*Log[1 - E^(-a - b*x)])/E^(2*a) - 6*b*(1 - E^(-2*a))*
x*Log[1 + E^(-a - b*x)] - (2*b^3*(-1 + E^(2*a))*x^3*Log[1 + E^(-a - b*x)])/E^(2*a) + 6*(1 - E^(-2*a))*PolyLog[
2, -E^(-a - b*x)] + 6*b^2*(1 - E^(-2*a))*x^2*PolyLog[2, -E^(-a - b*x)] + 6*(1 - E^(-2*a))*PolyLog[2, E^(-a - b
*x)] + 6*b^2*(1 - E^(-2*a))*x^2*PolyLog[2, E^(-a - b*x)] + 12*b*(1 - E^(-2*a))*x*PolyLog[3, -E^(-a - b*x)] + 1
2*b*(1 - E^(-2*a))*x*PolyLog[3, E^(-a - b*x)] + 12*(1 - E^(-2*a))*PolyLog[4, -E^(-a - b*x)] + 12*(1 - E^(-2*a)
)*PolyLog[4, E^(-a - b*x)]))/(b^4*(-1 + E^(2*a))) + (6*x^2*Csch[a]*Csch[a + b*x]*Sinh[b*x])/b^2)/4
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(161)=322\).
Time = 0.17 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.09
| | |
method | result | size |
| | |
risch |
\(-\frac {3 a^{4}}{2 b^{4}}-\frac {x^{4}}{4}-\frac {a^{3} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {3 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {2 a^{3} x}{b^{3}}-\frac {3 x^{2}}{b^{2}}-\frac {x^{2} \left (2 \,{\mathrm e}^{2 b x +2 a} b x +3 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 a^{2}}{b^{4}}+\frac {3 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 a x}{b^{3}}+\frac {3 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {6 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}\) |
\(375\) |
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[In]
int(x^3*coth(b*x+a)^3,x,method=_RETURNVERBOSE)
[Out]
-3/2/b^4*a^4-1/4*x^4-1/b^4*a^3*ln(exp(b*x+a)-1)+2/b^4*a^3*ln(exp(b*x+a))-6/b^3*polylog(3,exp(b*x+a))*x+3/b^2*p
olylog(2,-exp(b*x+a))*x^2-6/b^3*polylog(3,-exp(b*x+a))*x-2/b^3*a^3*x-3/b^2*x^2-x^2*(2*exp(2*b*x+2*a)*b*x+3*exp
(2*b*x+2*a)-3)/b^2/(exp(2*b*x+2*a)-1)^2-3/b^4*a^2+3/b^4*polylog(2,-exp(b*x+a))+6/b^4*polylog(4,-exp(b*x+a))+3/
b^4*polylog(2,exp(b*x+a))+6/b^4*polylog(4,exp(b*x+a))-6/b^3*a*x+3/b^2*polylog(2,exp(b*x+a))*x^2+3/b^3*ln(exp(b
*x+a)+1)*x+3/b^3*ln(1-exp(b*x+a))*x+1/b*ln(exp(b*x+a)+1)*x^3+1/b*ln(1-exp(b*x+a))*x^3+1/b^4*ln(1-exp(b*x+a))*a
^3+3/b^4*ln(1-exp(b*x+a))*a-3/b^4*a*ln(exp(b*x+a)-1)+6/b^4*a*ln(exp(b*x+a))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1985 vs. \(2 (159) = 318\).
Time = 0.31 (sec) , antiderivative size = 1985, normalized size of antiderivative = 11.09
\[
\int x^3 \coth ^3(a+b x) \, dx=\text {Too large to display}
\]
[In]
integrate(x^3*coth(b*x+a)^3,x, algorithm="fricas")
[Out]
-1/4*(b^4*x^4 + (b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12*a^2)*cosh(b*x + a)^4 + 4*(b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12
*a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12*a^2)*sinh(b*x + a)^4 - 2*a^4 - 2*(b^4
*x^4 - 4*b^3*x^3 - 2*a^4 + 6*b^2*x^2 - 12*a^2)*cosh(b*x + a)^2 - 2*(b^4*x^4 - 4*b^3*x^3 - 2*a^4 + 6*b^2*x^2 -
3*(b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12*a^2)*cosh(b*x + a)^2 - 12*a^2)*sinh(b*x + a)^2 - 12*a^2 - 12*((b^2*x^2 +
1)*cosh(b*x + a)^4 + 4*(b^2*x^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 + 1)*sinh(b*x + a)^4 + b^2*x^2 -
2*(b^2*x^2 + 1)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 + 1)*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 4*((b^2
*x^2 + 1)*cosh(b*x + a)^3 - (b^2*x^2 + 1)*cosh(b*x + a))*sinh(b*x + a) + 1)*dilog(cosh(b*x + a) + sinh(b*x + a
)) - 12*((b^2*x^2 + 1)*cosh(b*x + a)^4 + 4*(b^2*x^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 + 1)*sinh(b*
x + a)^4 + b^2*x^2 - 2*(b^2*x^2 + 1)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 + 1)*cosh(b*x + a)^2 + 1)*sinh(
b*x + a)^2 + 4*((b^2*x^2 + 1)*cosh(b*x + a)^3 - (b^2*x^2 + 1)*cosh(b*x + a))*sinh(b*x + a) + 1)*dilog(-cosh(b*
x + a) - sinh(b*x + a)) - 4*(b^3*x^3 + (b^3*x^3 + 3*b*x)*cosh(b*x + a)^4 + 4*(b^3*x^3 + 3*b*x)*cosh(b*x + a)*s
inh(b*x + a)^3 + (b^3*x^3 + 3*b*x)*sinh(b*x + a)^4 - 2*(b^3*x^3 + 3*b*x)*cosh(b*x + a)^2 - 2*(b^3*x^3 - 3*(b^3
*x^3 + 3*b*x)*cosh(b*x + a)^2 + 3*b*x)*sinh(b*x + a)^2 + 3*b*x + 4*((b^3*x^3 + 3*b*x)*cosh(b*x + a)^3 - (b^3*x
^3 + 3*b*x)*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 4*((a^3 + 3*a)*cosh(b*x + a
)^4 + 4*(a^3 + 3*a)*cosh(b*x + a)*sinh(b*x + a)^3 + (a^3 + 3*a)*sinh(b*x + a)^4 + a^3 - 2*(a^3 + 3*a)*cosh(b*x
+ a)^2 - 2*(a^3 - 3*(a^3 + 3*a)*cosh(b*x + a)^2 + 3*a)*sinh(b*x + a)^2 + 4*((a^3 + 3*a)*cosh(b*x + a)^3 - (a^
3 + 3*a)*cosh(b*x + a))*sinh(b*x + a) + 3*a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 4*(b^3*x^3 + (b^3*x^3 +
a^3 + 3*b*x + 3*a)*cosh(b*x + a)^4 + 4*(b^3*x^3 + a^3 + 3*b*x + 3*a)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^3*x^3
+ a^3 + 3*b*x + 3*a)*sinh(b*x + a)^4 + a^3 - 2*(b^3*x^3 + a^3 + 3*b*x + 3*a)*cosh(b*x + a)^2 - 2*(b^3*x^3 + a^
3 - 3*(b^3*x^3 + a^3 + 3*b*x + 3*a)*cosh(b*x + a)^2 + 3*b*x + 3*a)*sinh(b*x + a)^2 + 3*b*x + 4*((b^3*x^3 + a^3
+ 3*b*x + 3*a)*cosh(b*x + a)^3 - (b^3*x^3 + a^3 + 3*b*x + 3*a)*cosh(b*x + a))*sinh(b*x + a) + 3*a)*log(-cosh(
b*x + a) - sinh(b*x + a) + 1) - 24*(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3
*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)
+ 1)*polylog(4, cosh(b*x + a) + sinh(b*x + a)) - 24*(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(
b*x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 - cosh(b*x + a
))*sinh(b*x + a) + 1)*polylog(4, -cosh(b*x + a) - sinh(b*x + a)) + 24*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x +
a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x +
a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*polylog(3, cosh(b*x + a) + sinh(b*x +
a)) + 24*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - 2*b*x*cosh(b*x + a
)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh
(b*x + a))*polylog(3, -cosh(b*x + a) - sinh(b*x + a)) + 4*((b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12*a^2)*cosh(b*x +
a)^3 - (b^4*x^4 - 4*b^3*x^3 - 2*a^4 + 6*b^2*x^2 - 12*a^2)*cosh(b*x + a))*sinh(b*x + a))/(b^4*cosh(b*x + a)^4 +
4*b^4*cosh(b*x + a)*sinh(b*x + a)^3 + b^4*sinh(b*x + a)^4 - 2*b^4*cosh(b*x + a)^2 + b^4 + 2*(3*b^4*cosh(b*x +
a)^2 - b^4)*sinh(b*x + a)^2 + 4*(b^4*cosh(b*x + a)^3 - b^4*cosh(b*x + a))*sinh(b*x + a))
Sympy [F]
\[
\int x^3 \coth ^3(a+b x) \, dx=\int x^{3} \coth ^{3}{\left (a + b x \right )}\, dx
\]
[In]
integrate(x**3*coth(b*x+a)**3,x)
[Out]
Integral(x**3*coth(a + b*x)**3, x)
Maxima [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.69
\[
\int x^3 \coth ^3(a+b x) \, dx=\frac {b^{2} x^{4} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{4} + 12 \, x^{2} - 2 \, {\left (b^{2} x^{4} e^{\left (2 \, a\right )} + 4 \, b x^{3} e^{\left (2 \, a\right )} + 6 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{4 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} - \frac {b^{4} x^{4} + 6 \, b^{2} x^{2}}{2 \, b^{4}} + \frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} + \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}}
\]
[In]
integrate(x^3*coth(b*x+a)^3,x, algorithm="maxima")
[Out]
1/4*(b^2*x^4*e^(4*b*x + 4*a) + b^2*x^4 + 12*x^2 - 2*(b^2*x^4*e^(2*a) + 4*b*x^3*e^(2*a) + 6*x^2*e^(2*a))*e^(2*b
*x))/(b^2*e^(4*b*x + 4*a) - 2*b^2*e^(2*b*x + 2*a) + b^2) - 1/2*(b^4*x^4 + 6*b^2*x^2)/b^4 + (b^3*x^3*log(e^(b*x
+ a) + 1) + 3*b^2*x^2*dilog(-e^(b*x + a)) - 6*b*x*polylog(3, -e^(b*x + a)) + 6*polylog(4, -e^(b*x + a)))/b^4
+ (b^3*x^3*log(-e^(b*x + a) + 1) + 3*b^2*x^2*dilog(e^(b*x + a)) - 6*b*x*polylog(3, e^(b*x + a)) + 6*polylog(4,
e^(b*x + a)))/b^4 + 3*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^4 + 3*(b*x*log(-e^(b*x + a) + 1) + d
ilog(e^(b*x + a)))/b^4
Giac [F]
\[
\int x^3 \coth ^3(a+b x) \, dx=\int { x^{3} \coth \left (b x + a\right )^{3} \,d x }
\]
[In]
integrate(x^3*coth(b*x+a)^3,x, algorithm="giac")
[Out]
integrate(x^3*coth(b*x + a)^3, x)
Mupad [F(-1)]
Timed out. \[
\int x^3 \coth ^3(a+b x) \, dx=\int x^3\,{\mathrm {coth}\left (a+b\,x\right )}^3 \,d x
\]
[In]
int(x^3*coth(a + b*x)^3,x)
[Out]
int(x^3*coth(a + b*x)^3, x)