Integrand size = 13, antiderivative size = 97 \[ \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx=-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}+\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \]
-1/2/(b*x-arccoth(tanh(b*x+a)))/arccoth(tanh(b*x+a))^2+1/(b*x-arccoth(tanh (b*x+a)))^2/arccoth(tanh(b*x+a))-ln(x)/(b*x-arccoth(tanh(b*x+a)))^3+ln(arc coth(tanh(b*x+a)))/(b*x-arccoth(tanh(b*x+a)))^3
Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {b^2 x^2-4 b x \coth ^{-1}(\tanh (a+b x))+\coth ^{-1}(\tanh (a+b x))^2 \left (3+2 \log (b x)-2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 \coth ^{-1}(\tanh (a+b x))^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3} \]
(b^2*x^2 - 4*b*x*ArcCoth[Tanh[a + b*x]] + ArcCoth[Tanh[a + b*x]]^2*(3 + 2* Log[b*x] - 2*Log[ArcCoth[Tanh[a + b*x]]]))/(2*ArcCoth[Tanh[a + b*x]]^2*(-( b*x) + ArcCoth[Tanh[a + b*x]])^3)
Time = 0.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.38, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2594, 2594, 2591, 14, 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.
| \(\displaystyle \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx\) |
| \(\Big \downarrow \) 2594 |
| \(\displaystyle -\frac {\int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
| \(\Big \downarrow \) 2594 |
| \(\displaystyle -\frac {-\frac {\int \frac {1}{x \coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
| \(\Big \downarrow \) 2591 |
| \(\displaystyle -\frac {-\frac {\frac {b \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\int \frac {1}{x}dx}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle -\frac {-\frac {\frac {b \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
| \(\Big \downarrow \) 2588 |
| \(\displaystyle -\frac {-\frac {\frac {\int \frac {1}{\coth ^{-1}(\tanh (a+b x))}d\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle -\frac {1}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}-\frac {\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}\) |
-1/2*1/((b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2) - (-(1/(( b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]])) - (-(Log[x]/(b*x - ArcCoth[Tanh[a + b*x]])) + Log[ArcCoth[Tanh[a + b*x]]]/(b*x - ArcCoth[Tanh [a + b*x]]))/(b*x - ArcCoth[Tanh[a + b*x]]))/(b*x - ArcCoth[Tanh[a + b*x]] )
3.2.81.3.1 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[1/((u_)*(v_)), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D [v, x]]}, Simp[b/(b*u - a*v) Int[1/v, x], x] - Simp[a/(b*u - a*v) 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 + 1)/((n + 1)*(b*u - a*v)), x] - Simp[a*((n + 1)/((n + 1)*(b*u - a*v))) Int[v^(n + 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; Piecew iseLinearQ[u, v, x] && LtQ[n, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.35 (sec) , antiderivative size = 5655, normalized size of antiderivative = 58.30
\[\text {output too large to display}\]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.38 \[ \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {4 \, {\left (8 \, a b x - 3 \, \pi ^{2} + 4 i \, \pi {\left (b x + 3 \, a\right )} + 12 \, a^{2} - 2 \, {\left (4 \, b^{2} x^{2} + 8 \, a b x - \pi ^{2} + 4 i \, \pi {\left (b x + a\right )} + 4 \, a^{2}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right ) + 2 \, {\left (4 \, b^{2} x^{2} + 8 \, a b x - \pi ^{2} + 4 i \, \pi {\left (b x + a\right )} + 4 \, a^{2}\right )} \log \left (x\right )\right )}}{32 \, a^{3} b^{2} x^{2} + 64 \, a^{4} b x + i \, \pi ^{5} + 2 \, \pi ^{4} {\left (2 \, b x + 5 \, a\right )} + 32 \, a^{5} - 4 i \, \pi ^{3} {\left (b^{2} x^{2} + 8 \, a b x + 10 \, a^{2}\right )} - 8 \, \pi ^{2} {\left (3 \, a b^{2} x^{2} + 12 \, a^{2} b x + 10 \, a^{3}\right )} + 16 i \, \pi {\left (3 \, a^{2} b^{2} x^{2} + 8 \, a^{3} b x + 5 \, a^{4}\right )}} \]
4*(8*a*b*x - 3*pi^2 + 4*I*pi*(b*x + 3*a) + 12*a^2 - 2*(4*b^2*x^2 + 8*a*b*x - pi^2 + 4*I*pi*(b*x + a) + 4*a^2)*log(I*pi + 2*b*x + 2*a) + 2*(4*b^2*x^2 + 8*a*b*x - pi^2 + 4*I*pi*(b*x + a) + 4*a^2)*log(x))/(32*a^3*b^2*x^2 + 64 *a^4*b*x + I*pi^5 + 2*pi^4*(2*b*x + 5*a) + 32*a^5 - 4*I*pi^3*(b^2*x^2 + 8* a*b*x + 10*a^2) - 8*pi^2*(3*a*b^2*x^2 + 12*a^2*b*x + 10*a^3) + 16*I*pi*(3* a^2*b^2*x^2 + 8*a^3*b*x + 5*a^4))
\[ \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx=\int \frac {1}{x \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.76 \[ \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {4 \, {\left (-3 i \, \pi + 4 \, b x + 6 \, a\right )}}{\pi ^{4} + 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} + 16 \, a^{4} - 4 \, {\left (\pi ^{2} b^{2} + 4 i \, \pi a b^{2} - 4 \, a^{2} b^{2}\right )} x^{2} - 4 \, {\left (-i \, \pi ^{3} b + 6 \, \pi ^{2} a b + 12 i \, \pi a^{2} b - 8 \, a^{3} b\right )} x} + \frac {8 \, \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} - \frac {8 \, \log \left (x\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} \]
4*(-3*I*pi + 4*b*x + 6*a)/(pi^4 + 8*I*pi^3*a - 24*pi^2*a^2 - 32*I*pi*a^3 + 16*a^4 - 4*(pi^2*b^2 + 4*I*pi*a*b^2 - 4*a^2*b^2)*x^2 - 4*(-I*pi^3*b + 6*p i^2*a*b + 12*I*pi*a^2*b - 8*a^3*b)*x) + 8*log(-I*pi + 2*b*x + 2*a)/(-I*pi^ 3 + 6*pi^2*a + 12*I*pi*a^2 - 8*a^3) - 8*log(x)/(-I*pi^3 + 6*pi^2*a + 12*I* pi*a^2 - 8*a^3)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.78 \[ \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx=\frac {4 \, {\left (-3 i \, \pi - 4 \, b x - 6 \, a\right )}}{4 \, \pi ^{2} b^{2} x^{2} - 16 i \, \pi a b^{2} x^{2} - 16 \, a^{2} b^{2} x^{2} + 4 i \, \pi ^{3} b x + 24 \, \pi ^{2} a b x - 48 i \, \pi a^{2} b x - 32 \, a^{3} b x - \pi ^{4} + 8 i \, \pi ^{3} a + 24 \, \pi ^{2} a^{2} - 32 i \, \pi a^{3} - 16 \, a^{4}} - \frac {8 i \, \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{3} - 6 i \, \pi ^{2} a - 12 \, \pi a^{2} + 8 i \, a^{3}} + \frac {8 i \, \log \left (x\right )}{\pi ^{3} - 6 i \, \pi ^{2} a - 12 \, \pi a^{2} + 8 i \, a^{3}} \]
4*(-3*I*pi - 4*b*x - 6*a)/(4*pi^2*b^2*x^2 - 16*I*pi*a*b^2*x^2 - 16*a^2*b^2 *x^2 + 4*I*pi^3*b*x + 24*pi^2*a*b*x - 48*I*pi*a^2*b*x - 32*a^3*b*x - pi^4 + 8*I*pi^3*a + 24*pi^2*a^2 - 32*I*pi*a^3 - 16*a^4) - 8*I*log(I*pi + 2*b*x + 2*a)/(pi^3 - 6*I*pi^2*a - 12*pi*a^2 + 8*I*a^3) + 8*I*log(x)/(pi^3 - 6*I* pi^2*a - 12*pi*a^2 + 8*I*a^3)
Time = 9.79 (sec) , antiderivative size = 902, normalized size of antiderivative = 9.30 \[ \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx=\text {Too large to display} \]
- (16*atanh((16*(4*b*x - ((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/(e xp(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*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 - 4*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) + 4*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)^2/16 - (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))/4 + a^2/4))/(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)^3))/(log(-2/(ex p(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)^3 - (12/(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) - (16*b*x)/((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 - 4*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) + 4*a^2))/( (2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/...