Integrand size = 13, antiderivative size = 98 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {4 x^3}{3 b^2}+\frac {2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {4 x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}+\frac {4 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5} \]
4/3*x^3/b^2+2*x^2*(b*x-arccoth(tanh(b*x+a)))/b^3+4*x*(b*x-arccoth(tanh(b*x +a)))^2/b^4-x^4/b/arccoth(tanh(b*x+a))+4*(b*x-arccoth(tanh(b*x+a)))^3*ln(a rccoth(tanh(b*x+a)))/b^5
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {x^3}{3 b^2}-\frac {x^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {3 x \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}{b^4}-\frac {\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^4}{b^5 \coth ^{-1}(\tanh (a+b x))}-\frac {4 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^5} \]
x^3/(3*b^2) - (x^2*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/b^3 + (3*x*(-(b*x) + ArcCoth[Tanh[a + b*x]])^2)/b^4 - (-(b*x) + ArcCoth[Tanh[a + b*x]])^4/(b^5 *ArcCoth[Tanh[a + b*x]]) - (4*(-(b*x) + ArcCoth[Tanh[a + b*x]])^3*Log[ArcC oth[Tanh[a + b*x]]])/b^5
Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2599, 2590, 2590, 2589, 2588, 14}
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.
\(\displaystyle \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx\) |
\(\Big \downarrow \) 2599 |
\(\displaystyle \frac {4 \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))}dx}{b}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2590 |
\(\displaystyle \frac {4 \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))}dx}{b}+\frac {x^3}{3 b}\right )}{b}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2590 |
\(\displaystyle \frac {4 \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {x}{\coth ^{-1}(\tanh (a+b x))}dx}{b}+\frac {x^2}{2 b}\right )}{b}+\frac {x^3}{3 b}\right )}{b}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2589 |
\(\displaystyle \frac {4 \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}dx}{b}+\frac {x}{b}\right )}{b}+\frac {x^2}{2 b}\right )}{b}+\frac {x^3}{3 b}\right )}{b}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2588 |
\(\displaystyle \frac {4 \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}d\coth ^{-1}(\tanh (a+b x))}{b^2}+\frac {x}{b}\right )}{b}+\frac {x^2}{2 b}\right )}{b}+\frac {x^3}{3 b}\right )}{b}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {4 \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {x}{b}\right )}{b}+\frac {x^2}{2 b}\right )}{b}+\frac {x^3}{3 b}\right )}{b}-\frac {x^4}{b \coth ^{-1}(\tanh (a+b x))}\) |
-(x^4/(b*ArcCoth[Tanh[a + b*x]])) + (4*(x^3/(3*b) + ((b*x - ArcCoth[Tanh[a + b*x]])*(x^2/(2*b) + ((b*x - ArcCoth[Tanh[a + b*x]])*(x/b + ((b*x - ArcC oth[Tanh[a + b*x]])*Log[ArcCoth[Tanh[a + b*x]]])/b^2))/b))/b))/b
3.2.67.3.1 Defintions of rubi rules used
Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Simp[1/c Subst [Int[x^m, x], x, u], x]] /; FreeQ[m, x] && PiecewiseLinearQ[u, x]
Int[(v_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[b*(x/a), x] - Simp[(b*u - a*v)/a Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]
Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[ D[v, x]]}, Simp[v^n/(a*n), x] - Simp[(b*u - a*v)/a Int[v^(n - 1)/u, x], x ] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && GtQ[n, 0] && NeQ[n, 1]
Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Sim plify[D[v, x]]}, Simp[u^(m + 1)*(v^n/(a*(m + 1))), x] - Simp[b*(n/(a*(m + 1 ))) Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n} , x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0 ] && !(ILtQ[m + n, -2] && (FractionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ [n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] && !IntegerQ[m]) || (ILt Q[m, 0] && !IntegerQ[n]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.56 (sec) , antiderivative size = 131085, normalized size of antiderivative = 1337.60
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.16 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {16 \, b^{4} x^{4} - 32 \, a b^{3} x^{3} + 96 \, a^{2} b^{2} x^{2} + 144 \, a^{3} b x - 3 \, \pi ^{4} - 6 i \, \pi ^{3} {\left (3 \, b x - 4 \, a\right )} - 48 \, a^{4} - 12 \, \pi ^{2} {\left (2 \, b^{2} x^{2} + 9 \, a b x - 6 \, a^{2}\right )} - 8 i \, \pi {\left (2 \, b^{3} x^{3} - 12 \, a b^{2} x^{2} - 27 \, a^{2} b x + 12 \, a^{3}\right )} - 12 \, {\left (16 \, a^{3} b x + \pi ^{4} - 2 i \, \pi ^{3} {\left (b x + 4 \, a\right )} + 16 \, a^{4} - 12 \, \pi ^{2} {\left (a b x + 2 \, a^{2}\right )} + 8 i \, \pi {\left (3 \, a^{2} b x + 4 \, a^{3}\right )}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{24 \, {\left (2 \, b^{6} x + i \, \pi b^{5} + 2 \, a b^{5}\right )}} \]
1/24*(16*b^4*x^4 - 32*a*b^3*x^3 + 96*a^2*b^2*x^2 + 144*a^3*b*x - 3*pi^4 - 6*I*pi^3*(3*b*x - 4*a) - 48*a^4 - 12*pi^2*(2*b^2*x^2 + 9*a*b*x - 6*a^2) - 8*I*pi*(2*b^3*x^3 - 12*a*b^2*x^2 - 27*a^2*b*x + 12*a^3) - 12*(16*a^3*b*x + pi^4 - 2*I*pi^3*(b*x + 4*a) + 16*a^4 - 12*pi^2*(a*b*x + 2*a^2) + 8*I*pi*( 3*a^2*b*x + 4*a^3))*log(I*pi + 2*b*x + 2*a))/(2*b^6*x + I*pi*b^5 + 2*a*b^5 )
\[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\int \frac {x^{4}}{\operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.83 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {16 \, b^{4} x^{4} - 3 \, \pi ^{4} - 24 i \, \pi ^{3} a + 72 \, \pi ^{2} a^{2} + 96 i \, \pi a^{3} - 48 \, a^{4} - 16 \, {\left (-i \, \pi b^{3} + 2 \, a b^{3}\right )} x^{3} - 24 \, {\left (\pi ^{2} b^{2} + 4 i \, \pi a b^{2} - 4 \, a^{2} b^{2}\right )} x^{2} - 18 \, {\left (-i \, \pi ^{3} b + 6 \, \pi ^{2} a b + 12 i \, \pi a^{2} b - 8 \, a^{3} b\right )} x}{24 \, {\left (2 \, b^{6} x - i \, \pi b^{5} + 2 \, a b^{5}\right )}} - \frac {{\left (i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{5}} \]
1/24*(16*b^4*x^4 - 3*pi^4 - 24*I*pi^3*a + 72*pi^2*a^2 + 96*I*pi*a^3 - 48*a ^4 - 16*(-I*pi*b^3 + 2*a*b^3)*x^3 - 24*(pi^2*b^2 + 4*I*pi*a*b^2 - 4*a^2*b^ 2)*x^2 - 18*(-I*pi^3*b + 6*pi^2*a*b + 12*I*pi*a^2*b - 8*a^3*b)*x)/(2*b^6*x - I*pi*b^5 + 2*a*b^5) - 1/2*(I*pi^3 - 6*pi^2*a - 12*I*pi*a^2 + 8*a^3)*log (-I*pi + 2*b*x + 2*a)/b^5
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {x^{3}}{3 \, b^{2}} - \frac {\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}}{8 \, {\left (2 \, b^{6} x + i \, \pi b^{5} + 2 \, a b^{5}\right )}} - \frac {{\left (i \, \pi + 2 \, a\right )} x^{2}}{2 \, b^{3}} - \frac {3 \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} x}{4 \, b^{4}} + \frac {{\left (i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{5}} \]
1/3*x^3/b^2 - 1/8*(pi^4 - 8*I*pi^3*a - 24*pi^2*a^2 + 32*I*pi*a^3 + 16*a^4) /(2*b^6*x + I*pi*b^5 + 2*a*b^5) - 1/2*(I*pi + 2*a)*x^2/b^3 - 3/4*(pi^2 - 4 *I*pi*a - 4*a^2)*x/b^4 + 1/2*(I*pi^3 + 6*pi^2*a - 12*I*pi*a^2 - 8*a^3)*log (I*pi + 2*b*x + 2*a)/b^5
Time = 3.90 (sec) , antiderivative size = 669, normalized size of antiderivative = 6.83 \[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {x^3}{3\,b^2}-\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^4+24\,a^2\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+16\,a^4-8\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3-32\,a^3\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{2\,b\,\left (8\,a\,b^4+8\,b^5\,x-4\,b^4\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )}+\frac {x^2\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{2\,b^3}+\frac {3\,x\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2}{4\,b^4}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3-8\,a^3-6\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2+12\,a^2\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )}{2\,b^5} \]
x^3/(3*b^2) - ((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1 )) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^4 + 24*a^2*(2*a - log((2*e xp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b* x) - 1)) + 2*b*x)^2 + 16*a^4 - 8*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp (2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^3 - 32 *a^3*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(- 2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x))/(2*b*(8*a*b^4 + 8*b^5*x - 4*b^4*(2* a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2 *a)*exp(2*b*x) - 1)) + 2*b*x))) + (x^2*(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x))/(2*b^3) + (3*x*(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/( exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^2)/(4*b^4) + (log(log((2*exp(2*a)*exp(2 *b*x))/(exp(2*a)*exp(2*b*x) - 1)) - log(-2/(exp(2*a)*exp(2*b*x) - 1)))*((2 *a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp( 2*a)*exp(2*b*x) - 1)) + 2*b*x)^3 - 8*a^3 - 6*a*(2*a - log((2*exp(2*a)*exp( 2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2 *b*x)^2 + 12*a^2*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)))/(2*b^5)