\(\int x^2 \coth ^{-1}(a+b f^{c+d x}) \, dx\) [291]

Optimal result

Integrand size = 16, antiderivative size = 269 \[ \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (1+\frac {b f^{c+d x}}{1+a}\right )-\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac {\operatorname {PolyLog}\left (4,\frac {b f^{c+d x}}{1-a}\right )}{d^3 \log ^3(f)}-\frac {\operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{1+a}\right )}{d^3 \log ^3(f)} \]

[Out]

1/6*x^3*ln(1-b*f^(d*x+c)/(1-a))-1/6*x^3*ln(1+b*f^(d*x+c)/(1+a))-1/6*x^3*ln(1-1/(a+b*f^(d*x+c)))+1/6*x^3*ln(1+1
/(a+b*f^(d*x+c)))+1/2*x^2*polylog(2,b*f^(d*x+c)/(1-a))/d/ln(f)-1/2*x^2*polylog(2,-b*f^(d*x+c)/(1+a))/d/ln(f)-x
*polylog(3,b*f^(d*x+c)/(1-a))/d^2/ln(f)^2+x*polylog(3,-b*f^(d*x+c)/(1+a))/d^2/ln(f)^2+polylog(4,b*f^(d*x+c)/(1
-a))/d^3/ln(f)^3-polylog(4,-b*f^(d*x+c)/(1+a))/d^3/ln(f)^3

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6349, 2631, 12, 6874, 2221, 2611, 6744, 2320, 6724} \[ \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {\operatorname {PolyLog}\left (4,\frac {b f^{c+d x}}{1-a}\right )}{d^3 \log ^3(f)}-\frac {\operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{a+1}\right )}{d^3 \log ^3(f)}-\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{a+1}\right )}{d^2 \log ^2(f)}+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{a+1}\right )}{2 d \log (f)}+\frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (\frac {b f^{c+d x}}{a+1}+1\right )-\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (\frac {1}{a+b f^{c+d x}}+1\right ) \]

[In]

Int[x^2*ArcCoth[a + b*f^(c + d*x)],x]

[Out]

(x^3*Log[1 - (b*f^(c + d*x))/(1 - a)])/6 - (x^3*Log[1 + (b*f^(c + d*x))/(1 + a)])/6 - (x^3*Log[1 - (a + b*f^(c
+ d*x))^(-1)])/6 + (x^3*Log[1 + (a + b*f^(c + d*x))^(-1)])/6 + (x^2*PolyLog[2, (b*f^(c + d*x))/(1 - a)])/(2*d
*Log[f]) - (x^2*PolyLog[2, -((b*f^(c + d*x))/(1 + a))])/(2*d*Log[f]) - (x*PolyLog[3, (b*f^(c + d*x))/(1 - a)])
/(d^2*Log[f]^2) + (x*PolyLog[3, -((b*f^(c + d*x))/(1 + a))])/(d^2*Log[f]^2) + PolyLog[4, (b*f^(c + d*x))/(1 -
a)]/(d^3*Log[f]^3) - PolyLog[4, -((b*f^(c + d*x))/(1 + a))]/(d^3*Log[f]^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 2631

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[(a + b*x)^(m + 1)*(Log[u]/(b*(m + 1))), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[(a + b*x)^(m + 1)*(D[u, x]/u), x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 6349

Int[ArcCoth[(a_.) + (b_.)*(f_)^((c_.) + (d_.)*(x_))]*(x_)^(m_.), x_Symbol] :> Dist[1/2, Int[x^m*Log[1 + 1/(a +
b*f^(c + d*x))], x], x] - Dist[1/2, Int[x^m*Log[1 - 1/(a + b*f^(c + d*x))], x], x] /; FreeQ[{a, b, c, d, f},
x] && IGtQ[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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int x^2 \log \left (1-\frac {1}{a+b f^{c+d x}}\right ) \, dx\right )+\frac {1}{2} \int x^2 \log \left (1+\frac {1}{a+b f^{c+d x}}\right ) \, dx \\ & = -\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} \int \frac {b d f^{c+d x} x^3 \log (f)}{\left (-1+a+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx-\frac {1}{6} \int \frac {b d f^{c+d x} x^3 \log (f)}{\left (-a-b f^{c+d x}\right ) \left (1+a+b f^{c+d x}\right )} \, dx \\ & = -\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} (b d \log (f)) \int \frac {f^{c+d x} x^3}{\left (-1+a+b f^{c+d x}\right ) \left (a+b f^{c+d x}\right )} \, dx-\frac {1}{6} (b d \log (f)) \int \frac {f^{c+d x} x^3}{\left (-a-b f^{c+d x}\right ) \left (1+a+b f^{c+d x}\right )} \, dx \\ & = -\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} (b d \log (f)) \int \left (\frac {f^{c+d x} x^3}{-a-b f^{c+d x}}+\frac {f^{c+d x} x^3}{-1+a+b f^{c+d x}}\right ) \, dx-\frac {1}{6} (b d \log (f)) \int \left (\frac {f^{c+d x} x^3}{-a-b f^{c+d x}}+\frac {f^{c+d x} x^3}{1+a+b f^{c+d x}}\right ) \, dx \\ & = -\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} (b d \log (f)) \int \frac {f^{c+d x} x^3}{-1+a+b f^{c+d x}} \, dx-\frac {1}{6} (b d \log (f)) \int \frac {f^{c+d x} x^3}{1+a+b f^{c+d x}} \, dx \\ & = \frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (1+\frac {b f^{c+d x}}{1+a}\right )-\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )-\frac {1}{2} \int x^2 \log \left (1+\frac {b f^{c+d x}}{-1+a}\right ) \, dx+\frac {1}{2} \int x^2 \log \left (1+\frac {b f^{c+d x}}{1+a}\right ) \, dx \\ & = \frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (1+\frac {b f^{c+d x}}{1+a}\right )-\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac {\int x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{-1+a}\right ) \, dx}{d \log (f)}+\frac {\int x \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right ) \, dx}{d \log (f)} \\ & = \frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (1+\frac {b f^{c+d x}}{1+a}\right )-\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac {\int \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{-1+a}\right ) \, dx}{d^2 \log ^2(f)}-\frac {\int \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right ) \, dx}{d^2 \log ^2(f)} \\ & = \frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (1+\frac {b f^{c+d x}}{1+a}\right )-\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {b x}{-1+a}\right )}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {b x}{1+a}\right )}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)} \\ & = \frac {1}{6} x^3 \log \left (1-\frac {b f^{c+d x}}{1-a}\right )-\frac {1}{6} x^3 \log \left (1+\frac {b f^{c+d x}}{1+a}\right )-\frac {1}{6} x^3 \log \left (1-\frac {1}{a+b f^{c+d x}}\right )+\frac {1}{6} x^3 \log \left (1+\frac {1}{a+b f^{c+d x}}\right )+\frac {x^2 \operatorname {PolyLog}\left (2,\frac {b f^{c+d x}}{1-a}\right )}{2 d \log (f)}-\frac {x^2 \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )}{2 d \log (f)}-\frac {x \operatorname {PolyLog}\left (3,\frac {b f^{c+d x}}{1-a}\right )}{d^2 \log ^2(f)}+\frac {x \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )}{d^2 \log ^2(f)}+\frac {\operatorname {PolyLog}\left (4,\frac {b f^{c+d x}}{1-a}\right )}{d^3 \log ^3(f)}-\frac {\operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{1+a}\right )}{d^3 \log ^3(f)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.87 \[ \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {2 d^3 x^3 \coth ^{-1}\left (a+b f^{c+d x}\right ) \log ^3(f)+d^3 x^3 \log ^3(f) \log \left (1+\frac {b f^{c+d x}}{-1+a}\right )-d^3 x^3 \log ^3(f) \log \left (1+\frac {b f^{c+d x}}{1+a}\right )+3 d^2 x^2 \log ^2(f) \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{-1+a}\right )-3 d^2 x^2 \log ^2(f) \operatorname {PolyLog}\left (2,-\frac {b f^{c+d x}}{1+a}\right )-6 d x \log (f) \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{-1+a}\right )+6 d x \log (f) \operatorname {PolyLog}\left (3,-\frac {b f^{c+d x}}{1+a}\right )+6 \operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{-1+a}\right )-6 \operatorname {PolyLog}\left (4,-\frac {b f^{c+d x}}{1+a}\right )}{6 d^3 \log ^3(f)} \]

[In]

Integrate[x^2*ArcCoth[a + b*f^(c + d*x)],x]

[Out]

(2*d^3*x^3*ArcCoth[a + b*f^(c + d*x)]*Log[f]^3 + d^3*x^3*Log[f]^3*Log[1 + (b*f^(c + d*x))/(-1 + a)] - d^3*x^3*
Log[f]^3*Log[1 + (b*f^(c + d*x))/(1 + a)] + 3*d^2*x^2*Log[f]^2*PolyLog[2, -((b*f^(c + d*x))/(-1 + a))] - 3*d^2
*x^2*Log[f]^2*PolyLog[2, -((b*f^(c + d*x))/(1 + a))] - 6*d*x*Log[f]*PolyLog[3, -((b*f^(c + d*x))/(-1 + a))] +
6*d*x*Log[f]*PolyLog[3, -((b*f^(c + d*x))/(1 + a))] + 6*PolyLog[4, -((b*f^(c + d*x))/(-1 + a))] - 6*PolyLog[4,
-((b*f^(c + d*x))/(1 + a))])/(6*d^3*Log[f]^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(665\) vs. \(2(257)=514\).

Time = 1.04 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.48

method	result	size
risch	\(-\frac {x^{3} \ln \left (b \,f^{d x +c}+a -1\right )}{6}+\frac {x^{3} \ln \left (1+a +b \,f^{d x +c}\right )}{6}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-1-a}\right ) x^{3}}{6}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-1-a}\right ) x \,c^{2}}{2 d^{2}}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{-1-a}\right ) c^{3}}{3 d^{3}}-\frac {\operatorname {polylog}\left (2, \frac {b \,f^{d x} f^{c}}{-1-a}\right ) x^{2}}{2 \ln \left (f \right ) d}+\frac {\operatorname {polylog}\left (2, \frac {b \,f^{d x} f^{c}}{-1-a}\right ) c^{2}}{2 \ln \left (f \right ) d^{3}}+\frac {\operatorname {polylog}\left (3, \frac {b \,f^{d x} f^{c}}{-1-a}\right ) x}{\ln \left (f \right )^{2} d^{2}}-\frac {\operatorname {polylog}\left (4, \frac {b \,f^{d x} f^{c}}{-1-a}\right )}{\ln \left (f \right )^{3} d^{3}}+\frac {c^{3} \ln \left (1+a +f^{d x} f^{c} b \right )}{6 d^{3}}-\frac {c^{2} \operatorname {dilog}\left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right )}{2 \ln \left (f \right ) d^{3}}-\frac {c^{2} \ln \left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right ) x}{2 d^{2}}-\frac {c^{3} \ln \left (\frac {1+a +f^{d x} f^{c} b}{1+a}\right )}{2 d^{3}}+\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) x^{3}}{6}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) x \,c^{2}}{2 d^{2}}-\frac {\ln \left (1-\frac {b \,f^{d x} f^{c}}{1-a}\right ) c^{3}}{3 d^{3}}+\frac {\operatorname {polylog}\left (2, \frac {b \,f^{d x} f^{c}}{1-a}\right ) x^{2}}{2 \ln \left (f \right ) d}-\frac {\operatorname {polylog}\left (2, \frac {b \,f^{d x} f^{c}}{1-a}\right ) c^{2}}{2 \ln \left (f \right ) d^{3}}-\frac {\operatorname {polylog}\left (3, \frac {b \,f^{d x} f^{c}}{1-a}\right ) x}{\ln \left (f \right )^{2} d^{2}}+\frac {\operatorname {polylog}\left (4, \frac {b \,f^{d x} f^{c}}{1-a}\right )}{\ln \left (f \right )^{3} d^{3}}-\frac {c^{3} \ln \left (f^{d x} f^{c} b +a -1\right )}{6 d^{3}}+\frac {c^{2} \operatorname {dilog}\left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right )}{2 \ln \left (f \right ) d^{3}}+\frac {c^{2} \ln \left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right ) x}{2 d^{2}}+\frac {c^{3} \ln \left (\frac {f^{d x} f^{c} b +a -1}{-1+a}\right )}{2 d^{3}}\)	\(666\)

[In]

int(x^2*arccoth(a+b*f^(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/6*x^3*ln(b*f^(d*x+c)+a-1)+1/6*x^3*ln(1+a+b*f^(d*x+c))-1/6*ln(1-b*f^(d*x)*f^c/(-1-a))*x^3+1/2/d^2*ln(1-b*f^(
d*x)*f^c/(-1-a))*x*c^2+1/3/d^3*ln(1-b*f^(d*x)*f^c/(-1-a))*c^3-1/2/ln(f)/d*polylog(2,b*f^(d*x)*f^c/(-1-a))*x^2+
1/2/ln(f)/d^3*polylog(2,b*f^(d*x)*f^c/(-1-a))*c^2+1/ln(f)^2/d^2*polylog(3,b*f^(d*x)*f^c/(-1-a))*x-1/ln(f)^3/d^
3*polylog(4,b*f^(d*x)*f^c/(-1-a))+1/6/d^3*c^3*ln(1+a+f^(d*x)*f^c*b)-1/2/ln(f)/d^3*c^2*dilog((1+a+f^(d*x)*f^c*b
)/(1+a))-1/2/d^2*c^2*ln((1+a+f^(d*x)*f^c*b)/(1+a))*x-1/2/d^3*c^3*ln((1+a+f^(d*x)*f^c*b)/(1+a))+1/6*ln(1-b*f^(d
*x)*f^c/(1-a))*x^3-1/2/d^2*ln(1-b*f^(d*x)*f^c/(1-a))*x*c^2-1/3/d^3*ln(1-b*f^(d*x)*f^c/(1-a))*c^3+1/2/ln(f)/d*p
olylog(2,b*f^(d*x)*f^c/(1-a))*x^2-1/2/ln(f)/d^3*polylog(2,b*f^(d*x)*f^c/(1-a))*c^2-1/ln(f)^2/d^2*polylog(3,b*f
^(d*x)*f^c/(1-a))*x+1/ln(f)^3/d^3*polylog(4,b*f^(d*x)*f^c/(1-a))-1/6/d^3*c^3*ln(f^(d*x)*f^c*b+a-1)+1/2/ln(f)/d
^3*c^2*dilog((f^(d*x)*f^c*b+a-1)/(-1+a))+1/2/d^2*c^2*ln((f^(d*x)*f^c*b+a-1)/(-1+a))*x+1/2/d^3*c^3*ln((f^(d*x)*
f^c*b+a-1)/(-1+a))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.78 \[ \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {d^{3} x^{3} \log \left (f\right )^{3} \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}\right ) - 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{a + 1} + 1\right ) \log \left (f\right )^{2} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}{a - 1} + 1\right ) \log \left (f\right )^{2} + c^{3} \log \left (b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1\right ) \log \left (f\right )^{3} - c^{3} \log \left (b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1\right ) \log \left (f\right )^{3} - {\left (d^{3} x^{3} + c^{3}\right )} \log \left (f\right )^{3} \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a + 1}{a + 1}\right ) + {\left (d^{3} x^{3} + c^{3}\right )} \log \left (f\right )^{3} \log \left (\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + a - 1}{a - 1}\right ) + 6 \, d x \log \left (f\right ) {\rm polylog}\left (3, -\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right )}{a + 1}\right ) - 6 \, d x \log \left (f\right ) {\rm polylog}\left (3, -\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right )}{a - 1}\right ) - 6 \, {\rm polylog}\left (4, -\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right )}{a + 1}\right ) + 6 \, {\rm polylog}\left (4, -\frac {b \cosh \left ({\left (d x + c\right )} \log \left (f\right )\right ) + b \sinh \left ({\left (d x + c\right )} \log \left (f\right )\right )}{a - 1}\right )}{6 \, d^{3} \log \left (f\right )^{3}} \]

[In]

integrate(x^2*arccoth(a+b*f^(d*x+c)),x, algorithm="fricas")

[Out]

1/6*(d^3*x^3*log(f)^3*log((b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a + 1)/(b*cosh((d*x + c)*log(
f)) + b*sinh((d*x + c)*log(f)) + a - 1)) - 3*d^2*x^2*dilog(-(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f
)) + a + 1)/(a + 1) + 1)*log(f)^2 + 3*d^2*x^2*dilog(-(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a
- 1)/(a - 1) + 1)*log(f)^2 + c^3*log(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a + 1)*log(f)^3 - c
^3*log(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a - 1)*log(f)^3 - (d^3*x^3 + c^3)*log(f)^3*log((b
*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a + 1)/(a + 1)) + (d^3*x^3 + c^3)*log(f)^3*log((b*cosh((d
*x + c)*log(f)) + b*sinh((d*x + c)*log(f)) + a - 1)/(a - 1)) + 6*d*x*log(f)*polylog(3, -(b*cosh((d*x + c)*log(
f)) + b*sinh((d*x + c)*log(f)))/(a + 1)) - 6*d*x*log(f)*polylog(3, -(b*cosh((d*x + c)*log(f)) + b*sinh((d*x +
c)*log(f)))/(a - 1)) - 6*polylog(4, -(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)))/(a + 1)) + 6*polylo
g(4, -(b*cosh((d*x + c)*log(f)) + b*sinh((d*x + c)*log(f)))/(a - 1)))/(d^3*log(f)^3)

Sympy [F]

\[ \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int x^{2} \operatorname {acoth}{\left (a + b f^{c + d x} \right )}\, dx \]

[In]

integrate(x**2*acoth(a+b*f**(d*x+c)),x)

[Out]

Integral(x**2*acoth(a + b*f**(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.94 \[ \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (b f^{d x + c} + a\right ) - \frac {1}{6} \, b d {\left (\frac {d^{3} x^{3} \log \left (\frac {b f^{d x} f^{c}}{a + 1} + 1\right ) \log \left (f\right )^{3} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b f^{d x} f^{c}}{a + 1}\right ) \log \left (f\right )^{2} - 6 \, d x \log \left (f\right ) {\rm Li}_{3}(-\frac {b f^{d x} f^{c}}{a + 1}) + 6 \, {\rm Li}_{4}(-\frac {b f^{d x} f^{c}}{a + 1})}{b d^{4} \log \left (f\right )^{4}} - \frac {d^{3} x^{3} \log \left (\frac {b f^{d x} f^{c}}{a - 1} + 1\right ) \log \left (f\right )^{3} + 3 \, d^{2} x^{2} {\rm Li}_2\left (-\frac {b f^{d x} f^{c}}{a - 1}\right ) \log \left (f\right )^{2} - 6 \, d x \log \left (f\right ) {\rm Li}_{3}(-\frac {b f^{d x} f^{c}}{a - 1}) + 6 \, {\rm Li}_{4}(-\frac {b f^{d x} f^{c}}{a - 1})}{b d^{4} \log \left (f\right )^{4}}\right )} \log \left (f\right ) \]

[In]

integrate(x^2*arccoth(a+b*f^(d*x+c)),x, algorithm="maxima")

[Out]

1/3*x^3*arccoth(b*f^(d*x + c) + a) - 1/6*b*d*((d^3*x^3*log(b*f^(d*x)*f^c/(a + 1) + 1)*log(f)^3 + 3*d^2*x^2*dil
og(-b*f^(d*x)*f^c/(a + 1))*log(f)^2 - 6*d*x*log(f)*polylog(3, -b*f^(d*x)*f^c/(a + 1)) + 6*polylog(4, -b*f^(d*x
)*f^c/(a + 1)))/(b*d^4*log(f)^4) - (d^3*x^3*log(b*f^(d*x)*f^c/(a - 1) + 1)*log(f)^3 + 3*d^2*x^2*dilog(-b*f^(d*
x)*f^c/(a - 1))*log(f)^2 - 6*d*x*log(f)*polylog(3, -b*f^(d*x)*f^c/(a - 1)) + 6*polylog(4, -b*f^(d*x)*f^c/(a -
1)))/(b*d^4*log(f)^4))*log(f)

Giac [F]

\[ \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int { x^{2} \operatorname {arcoth}\left (b f^{d x + c} + a\right ) \,d x } \]

[In]

integrate(x^2*arccoth(a+b*f^(d*x+c)),x, algorithm="giac")

[Out]

integrate(x^2*arccoth(b*f^(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \coth ^{-1}\left (a+b f^{c+d x}\right ) \, dx=\int x^2\,\mathrm {acoth}\left (a+b\,f^{c+d\,x}\right ) \,d x \]

[In]

int(x^2*acoth(a + b*f^(c + d*x)),x)

[Out]

int(x^2*acoth(a + b*f^(c + d*x)), x)