Integrand size = 13, antiderivative size = 71 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {3 x}{b^3}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}-\frac {3 x^2}{2 b^2 \coth ^{-1}(\tanh (a+b x))}+\frac {3 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4} \]
3*x/b^3-1/2*x^3/b/arccoth(tanh(b*x+a))^2-3/2*x^2/b^2/arccoth(tanh(b*x+a))+ 3*(b*x-arccoth(tanh(b*x+a)))*ln(arccoth(tanh(b*x+a)))/b^4
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=-\frac {b^3 x^3+3 b^2 x^2 \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^3 \left (5+6 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )-b x \coth ^{-1}(\tanh (a+b x))^2 \left (11+6 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 b^4 \coth ^{-1}(\tanh (a+b x))^2} \]
-1/2*(b^3*x^3 + 3*b^2*x^2*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^ 3*(5 + 6*Log[ArcCoth[Tanh[a + b*x]]]) - b*x*ArcCoth[Tanh[a + b*x]]^2*(11 + 6*Log[ArcCoth[Tanh[a + b*x]]]))/(b^4*ArcCoth[Tanh[a + b*x]]^2)
Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2599, 2599, 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.
\(\displaystyle \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx\) |
\(\Big \downarrow \) 2599 |
\(\displaystyle \frac {3 \int \frac {x^2}{\coth ^{-1}(\tanh (a+b x))^2}dx}{2 b}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 2599 |
\(\displaystyle \frac {3 \left (\frac {2 \int \frac {x}{\coth ^{-1}(\tanh (a+b x))}dx}{b}-\frac {x^2}{b \coth ^{-1}(\tanh (a+b x))}\right )}{2 b}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 2589 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}dx}{b}+\frac {x}{b}\right )}{b}-\frac {x^2}{b \coth ^{-1}(\tanh (a+b x))}\right )}{2 b}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 2588 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}d\coth ^{-1}(\tanh (a+b x))}{b^2}+\frac {x}{b}\right )}{b}-\frac {x^2}{b \coth ^{-1}(\tanh (a+b x))}\right )}{2 b}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {x}{b}\right )}{b}-\frac {x^2}{b \coth ^{-1}(\tanh (a+b x))}\right )}{2 b}-\frac {x^3}{2 b \coth ^{-1}(\tanh (a+b x))^2}\) |
-1/2*x^3/(b*ArcCoth[Tanh[a + b*x]]^2) + (3*(-(x^2/(b*ArcCoth[Tanh[a + b*x] ])) + (2*(x/b + ((b*x - ArcCoth[Tanh[a + b*x]])*Log[ArcCoth[Tanh[a + b*x]] ])/b^2))/b))/(2*b)
3.2.77.3.1 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[(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 = 0.24 (sec) , antiderivative size = 4977, normalized size of antiderivative = 70.10
-2*I*(3*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*ex p(2*b*x+2*a)/(exp(2*b*x+2*a)+1))-3*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1))*csgn( I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+6*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1)) ^3-6*Pi*x^2*csgn(I/(exp(2*b*x+2*a)+1))^2+3*Pi*x^2*csgn(I*exp(b*x+a))^2*csg n(I*exp(2*b*x+2*a))-6*Pi*x^2*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2+3 *Pi*x^2*csgn(I*exp(2*b*x+2*a))^3-3*Pi*x^2*csgn(I*exp(2*b*x+2*a))*csgn(I*ex p(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+3*Pi*x^2*csgn(I*exp(2*b*x+2*a)/(exp(2*b *x+2*a)+1))^3+12*I*x^2*ln(exp(b*x+a))+6*Pi*x^2+4*I*x^3*b)/b^2/(Pi*csgn(I*e xp(2*b*x+2*a))^3-2*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2-Pi*csgn( I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+Pi*csgn(I/(e xp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+ 2*a)+1))+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+Pi*csgn(I*exp(2*b* x+2*a)/(exp(2*b*x+2*a)+1))^3-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b* x+2*a)/(exp(2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I /(exp(2*b*x+2*a)+1))^2+4*I*ln(exp(b*x+a))+2*Pi)^2+3/b^3*x+3/2*I/b^4*ln(Pi* csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(e xp(2*b*x+2*a)+1))-Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp (2*b*x+2*a)+1))^2+2*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3-2*Pi*csgn(I/(exp(2*b*x +2*a)+1))^2+Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))-2*Pi*csgn(I*exp (b*x+a))*csgn(I*exp(2*b*x+2*a))^2+Pi*csgn(I*exp(2*b*x+2*a))^3-Pi*csgn(I...
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.75 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {16 \, b^{3} x^{3} + 32 \, a b^{2} x^{2} - 32 \, a^{2} b x + 5 i \, \pi ^{3} + 2 \, \pi ^{2} {\left (4 \, b x + 15 \, a\right )} - 40 \, a^{3} + 4 i \, \pi {\left (4 \, b^{2} x^{2} - 8 \, a b x - 15 \, a^{2}\right )} - 6 \, {\left (8 \, a b^{2} x^{2} + 16 \, a^{2} b x - i \, \pi ^{3} - 2 \, \pi ^{2} {\left (2 \, b x + 3 \, a\right )} + 8 \, a^{3} + 4 i \, \pi {\left (b^{2} x^{2} + 4 \, a b x + 3 \, a^{2}\right )}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, {\left (4 \, b^{6} x^{2} + 8 \, a b^{5} x - \pi ^{2} b^{4} + 4 \, a^{2} b^{4} + 4 i \, \pi {\left (b^{5} x + a b^{4}\right )}\right )}} \]
1/4*(16*b^3*x^3 + 32*a*b^2*x^2 - 32*a^2*b*x + 5*I*pi^3 + 2*pi^2*(4*b*x + 1 5*a) - 40*a^3 + 4*I*pi*(4*b^2*x^2 - 8*a*b*x - 15*a^2) - 6*(8*a*b^2*x^2 + 1 6*a^2*b*x - I*pi^3 - 2*pi^2*(2*b*x + 3*a) + 8*a^3 + 4*I*pi*(b^2*x^2 + 4*a* b*x + 3*a^2))*log(I*pi + 2*b*x + 2*a))/(4*b^6*x^2 + 8*a*b^5*x - pi^2*b^4 + 4*a^2*b^4 + 4*I*pi*(b^5*x + a*b^4))
\[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\int \frac {x^{3}}{\operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.06 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {16 \, b^{3} x^{3} - 5 i \, \pi ^{3} + 30 \, \pi ^{2} a + 60 i \, \pi a^{2} - 40 \, a^{3} - 16 \, {\left (i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{2} + 8 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x}{4 \, {\left (4 \, b^{6} x^{2} - \pi ^{2} b^{4} - 4 i \, \pi a b^{4} + 4 \, a^{2} b^{4} - 4 \, {\left (i \, \pi b^{5} - 2 \, a b^{5}\right )} x\right )}} - \frac {3 \, {\left (-i \, \pi + 2 \, a\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{4}} \]
1/4*(16*b^3*x^3 - 5*I*pi^3 + 30*pi^2*a + 60*I*pi*a^2 - 40*a^3 - 16*(I*pi*b ^2 - 2*a*b^2)*x^2 + 8*(pi^2*b + 4*I*pi*a*b - 4*a^2*b)*x)/(4*b^6*x^2 - pi^2 *b^4 - 4*I*pi*a*b^4 + 4*a^2*b^4 - 4*(I*pi*b^5 - 2*a*b^5)*x) - 3/2*(-I*pi + 2*a)*log(-I*pi + 2*b*x + 2*a)/b^4
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.73 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {12 \, \pi ^{2} b x - 48 i \, \pi a b x - 48 \, a^{2} b x + 5 i \, \pi ^{3} + 30 \, \pi ^{2} a - 60 i \, \pi a^{2} - 40 \, a^{3}}{4 \, {\left (4 \, b^{6} x^{2} + 4 i \, \pi b^{5} x + 8 \, a b^{5} x - \pi ^{2} b^{4} + 4 i \, \pi a b^{4} + 4 \, a^{2} b^{4}\right )}} + \frac {x}{b^{3}} + \frac {3 \, {\left (-i \, \pi - 2 \, a\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{4}} \]
1/4*(12*pi^2*b*x - 48*I*pi*a*b*x - 48*a^2*b*x + 5*I*pi^3 + 30*pi^2*a - 60* I*pi*a^2 - 40*a^3)/(4*b^6*x^2 + 4*I*pi*b^5*x + 8*a*b^5*x - pi^2*b^4 + 4*I* pi*a*b^4 + 4*a^2*b^4) + x/b^3 + 3/2*(-I*pi - 2*a)*log(I*pi + 2*b*x + 2*a)/ b^4
Time = 4.10 (sec) , antiderivative size = 620, normalized size of antiderivative = 8.73 \[ \int \frac {x^3}{\coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {x}{b^3}-\frac {x\,\left (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 )}^2-12\,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 )+12\,a^2\right )-\frac {5\,\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 )}{4\,b}}{b^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+x\,\left (8\,a\,b^4-4\,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 )+4\,a^2\,b^3+4\,b^5\,x^2-4\,a\,b^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 )}+\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 (3\,\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-3\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+6\,b\,x\right )}{2\,b^4} \]
x/b^3 - (x*(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 - 12*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) + 12*a^2) - (5*((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)*e xp(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*b))/(b^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 + x*(8*a*b^4 - 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)) + 4*a^2*b^3 + 4*b^5*x^2 - 4*a*b^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)) + (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)))*(3*log(-2/(exp(2 *a)*exp(2*b*x) - 1)) - 3*log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 6*b*x))/(2*b^4)