Integrand size = 13, antiderivative size = 102 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {2 b}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {2 b \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3}-\frac {2 b \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3} \]
-2*b/(b*x-arccoth(tanh(b*x+a)))^2/arccoth(tanh(b*x+a))+1/x/(b*x-arccoth(ta nh(b*x+a)))/arccoth(tanh(b*x+a))+2*b*ln(x)/(b*x-arccoth(tanh(b*x+a)))^3-2* b*ln(arccoth(tanh(b*x+a)))/(b*x-arccoth(tanh(b*x+a)))^3
Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {-b^2 x^2+\coth ^{-1}(\tanh (a+b x))^2+2 b x \coth ^{-1}(\tanh (a+b x)) \left (\log (x)-\log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))} \]
(-(b^2*x^2) + ArcCoth[Tanh[a + b*x]]^2 + 2*b*x*ArcCoth[Tanh[a + b*x]]*(Log [x] - Log[ArcCoth[Tanh[a + b*x]]]))/(x*(b*x - ArcCoth[Tanh[a + b*x]])^3*Ar cCoth[Tanh[a + b*x]])
Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.32, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2602, 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.
\(\displaystyle \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx\) |
\(\Big \downarrow \) 2602 |
\(\displaystyle \frac {2 b \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2}dx}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2594 |
\(\displaystyle \frac {2 b \left (-\frac {\int \frac {1}{x \coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2591 |
\(\displaystyle \frac {2 b \left (-\frac {\frac {b \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\int \frac {1}{x}dx}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {2 b \left (-\frac {\frac {b \int \frac {1}{\coth ^{-1}(\tanh (a+b x))}dx}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 2588 |
\(\displaystyle \frac {2 b \left (-\frac {\frac {\int \frac {1}{\coth ^{-1}(\tanh (a+b x))}d\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}+\frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {1}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}+\frac {2 b \left (-\frac {1}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))}-\frac {\frac {\log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b x-\coth ^{-1}(\tanh (a+b x))}-\frac {\log (x)}{b x-\coth ^{-1}(\tanh (a+b x))}}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b x-\coth ^{-1}(\tanh (a+b x))}\) |
1/(x*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]) + (2*b*(-(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] ])
3.2.73.3.1 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]
Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simp lify[D[v, x]]}, Simp[(-u^(m + 1))*(v^(n + 1)/((m + 1)*(b*u - a*v))), x] + S imp[b*((m + n + 2)/((m + 1)*(b*u - a*v))) Int[u^(m + 1)*v^n, x], x] /; Ne Q[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m , -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.35 (sec) , antiderivative size = 5357, normalized size of antiderivative = 52.52
\[\text {output too large to display}\]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {4 \, {\left (8 \, a b x - \pi ^{2} + 4 i \, \pi {\left (b x + a\right )} + 4 \, a^{2} - 4 \, {\left (2 \, b^{2} x^{2} + i \, \pi b x + 2 \, a b x\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right ) + 4 \, {\left (2 \, b^{2} x^{2} + i \, \pi b x + 2 \, a b x\right )} \log \left (x\right )\right )}}{16 \, a^{3} b x^{2} + \pi ^{4} x + 16 \, a^{4} x - 2 i \, \pi ^{3} {\left (b x^{2} + 4 \, a x\right )} - 12 \, \pi ^{2} {\left (a b x^{2} + 2 \, a^{2} x\right )} + 8 i \, \pi {\left (3 \, a^{2} b x^{2} + 4 \, a^{3} x\right )}} \]
-4*(8*a*b*x - pi^2 + 4*I*pi*(b*x + a) + 4*a^2 - 4*(2*b^2*x^2 + I*pi*b*x + 2*a*b*x)*log(I*pi + 2*b*x + 2*a) + 4*(2*b^2*x^2 + I*pi*b*x + 2*a*b*x)*log( x))/(16*a^3*b*x^2 + pi^4*x + 16*a^4*x - 2*I*pi^3*(b*x^2 + 4*a*x) - 12*pi^2 *(a*b*x^2 + 2*a^2*x) + 8*I*pi*(3*a^2*b*x^2 + 4*a^3*x))
\[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx=\int \frac {1}{x^{2} \operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx=-\frac {16 \, b \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} + \frac {16 \, b \log \left (x\right )}{-i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3}} - \frac {4 \, {\left (i \, \pi - 4 \, b x - 2 \, a\right )}}{2 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x^{2} - {\left (i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3}\right )} x} \]
-16*b*log(-I*pi + 2*b*x + 2*a)/(-I*pi^3 + 6*pi^2*a + 12*I*pi*a^2 - 8*a^3) + 16*b*log(x)/(-I*pi^3 + 6*pi^2*a + 12*I*pi*a^2 - 8*a^3) - 4*(I*pi - 4*b*x - 2*a)/(2*(pi^2*b + 4*I*pi*a*b - 4*a^2*b)*x^2 - (I*pi^3 - 6*pi^2*a - 12*I *pi*a^2 + 8*a^3)*x)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {16 i \, b \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{3} - 6 i \, \pi ^{2} a - 12 \, \pi a^{2} + 8 i \, a^{3}} - \frac {16 i \, b \log \left (x\right )}{\pi ^{3} - 6 i \, \pi ^{2} a - 12 \, \pi a^{2} + 8 i \, a^{3}} + \frac {8 \, b}{2 \, \pi ^{2} b x - 8 i \, \pi a b x - 8 \, a^{2} b x + i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}} + \frac {4}{\pi ^{2} x - 4 i \, \pi a x - 4 \, a^{2} x} \]
16*I*b*log(I*pi + 2*b*x + 2*a)/(pi^3 - 6*I*pi^2*a - 12*pi*a^2 + 8*I*a^3) - 16*I*b*log(x)/(pi^3 - 6*I*pi^2*a - 12*pi*a^2 + 8*I*a^3) + 8*b/(2*pi^2*b*x - 8*I*pi*a*b*x - 8*a^2*b*x + I*pi^3 + 6*pi^2*a - 12*I*pi*a^2 - 8*a^3) + 4 /(pi^2*x - 4*I*pi*a*x - 4*a^2*x)
Time = 6.68 (sec) , antiderivative size = 453, normalized size of antiderivative = 4.44 \[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx=\frac {4\,{\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\left (8\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+b\,x\,\mathrm {atan}\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}{\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 )\,32{}\mathrm {i}\right )+4\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2-16\,b^2\,x^2+b\,x\,\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\mathrm {atan}\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}-\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}{\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 )\,32{}\mathrm {i}}{x\,\left (\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\right )\,{\left (\ln \left (-\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3} \]
(4*log(-1/(exp(2*a)*exp(2*b*x) - 1))^2 - log((exp(2*a)*exp(2*b*x))/(exp(2* a)*exp(2*b*x) - 1))*(8*log(-1/(exp(2*a)*exp(2*b*x) - 1)) + b*x*atan((log(( 2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1))*1i - log(-2/(exp(2*a)*ex p(2*b*x) - 1))*1i + b*x*2i)/(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*ex p(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x))*32i) + 4*log((exp( 2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1))^2 - 16*b^2*x^2 + b*x*log(-1/(e xp(2*a)*exp(2*b*x) - 1))*atan((log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2 *b*x) - 1))*1i - log(-2/(exp(2*a)*exp(2*b*x) - 1))*1i + b*x*2i)/(log(-2/(e xp(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))*32i)/(x*(log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) - log(-1/(exp(2*a)*exp(2*b*x) - 1)))*(log(-1/(exp(2*a)*exp(2*b*x) - 1)) - log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^3)