Optimal. Leaf size=76 \[ -\frac {\text {Li}_2\left ((1-d) e^{2 a+2 b x}\right )}{4 b}-\frac {1}{2} x \log \left (1-(1-d) e^{2 a+2 b x}\right )+x \coth ^{-1}(d (-\coth (a+b x))-d+1)+\frac {b x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6234, 2184, 2190, 2279, 2391} \[ -\frac {\text {PolyLog}\left (2,(1-d) e^{2 a+2 b x}\right )}{4 b}-\frac {1}{2} x \log \left (1-(1-d) e^{2 a+2 b x}\right )+x \coth ^{-1}(d (-\coth (a+b x))-d+1)+\frac {b x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2184
Rule 2190
Rule 2279
Rule 2391
Rule 6234
Rubi steps
\begin {align*} \int \coth ^{-1}(1-d-d \coth (a+b x)) \, dx &=x \coth ^{-1}(1-d-d \coth (a+b x))+b \int \frac {x}{1+(-1+d) e^{2 a+2 b x}} \, dx\\ &=\frac {b x^2}{2}+x \coth ^{-1}(1-d-d \coth (a+b x))+(b (1-d)) \int \frac {e^{2 a+2 b x} x}{1+(-1+d) e^{2 a+2 b x}} \, dx\\ &=\frac {b x^2}{2}+x \coth ^{-1}(1-d-d \coth (a+b x))-\frac {1}{2} x \log \left (1-(1-d) e^{2 a+2 b x}\right )+\frac {1}{2} \int \log \left (1+(-1+d) e^{2 a+2 b x}\right ) \, dx\\ &=\frac {b x^2}{2}+x \coth ^{-1}(1-d-d \coth (a+b x))-\frac {1}{2} x \log \left (1-(1-d) e^{2 a+2 b x}\right )+\frac {\operatorname {Subst}\left (\int \frac {\log (1+(-1+d) x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=\frac {b x^2}{2}+x \coth ^{-1}(1-d-d \coth (a+b x))-\frac {1}{2} x \log \left (1-(1-d) e^{2 a+2 b x}\right )-\frac {\text {Li}_2\left ((1-d) e^{2 a+2 b x}\right )}{4 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.79, size = 208, normalized size = 2.74 \[ \frac {-2 \text {Li}_2\left (-\sqrt {1-d} e^{a+b x}\right )-2 \text {Li}_2\left (\sqrt {1-d} e^{a+b x}\right )-2 \log \left (e^{a+b x}\right ) \log \left (1-\sqrt {1-d} e^{a+b x}\right )-2 \log \left (e^{a+b x}\right ) \log \left (\sqrt {1-d} e^{a+b x}+1\right )+2 \log \left (e^{a+b x}\right ) \log \left (e^{-a-b x} \left ((d-1) e^{2 (a+b x)}+1\right )\right )-2 b x \log ((d-2) \sinh (a+b x)+d \cosh (a+b x))+\log ^2\left (e^{a+b x}\right )+b^2 x^2}{4 b}+x \coth ^{-1}(d (-\coth (a+b x))-d+1) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.84, size = 239, normalized size = 3.14 \[ \frac {b^{2} x^{2} - b x \log \left (\frac {d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + {\left (d - 2\right )} \sinh \left (b x + a\right )}\right ) + a \log \left (2 \, {\left (d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d - 1\right )} \sinh \left (b x + a\right ) + \sqrt {-4 \, d + 4}\right ) + a \log \left (2 \, {\left (d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d - 1\right )} \sinh \left (b x + a\right ) - \sqrt {-4 \, d + 4}\right ) - {\left (b x + a\right )} \log \left (\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\left (b x + a\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 \, d + 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arcoth}\left (-d \coth \left (b x + a\right ) - d + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.52, size = 271, normalized size = 3.57 \[ -\frac {\mathrm {arccoth}\left (1-d -d \coth \left (b x +a \right )\right ) \ln \left (-d \coth \left (b x +a \right )+d \right )}{2 b}+\frac {\mathrm {arccoth}\left (1-d -d \coth \left (b x +a \right )\right ) \ln \left (-d \coth \left (b x +a \right )-d \right )}{2 b}+\frac {\ln \left (-d \coth \left (b x +a \right )-d \right )^{2}}{8 b}-\frac {\dilog \left (1-\frac {d \coth \left (b x +a \right )}{2}-\frac {d}{2}\right )}{4 b}-\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (1-\frac {d \coth \left (b x +a \right )}{2}-\frac {d}{2}\right )}{4 b}-\frac {\dilog \left (-\frac {-d \coth \left (b x +a \right )-d}{2 d}\right )}{4 b}-\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (-\frac {-d \coth \left (b x +a \right )-d}{2 d}\right )}{4 b}+\frac {\dilog \left (\frac {-d \coth \left (b x +a \right )-d +2}{-2 d +2}\right )}{4 b}+\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-d +2}{-2 d +2}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.11, size = 73, normalized size = 0.96 \[ \frac {1}{4} \, b d {\left (\frac {2 \, x^{2}}{d} - \frac {2 \, b x \log \left ({\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-{\left (d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )}{b^{2} d}\right )} - x \operatorname {arcoth}\left (d \coth \left (b x + a\right ) + d - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\mathrm {acoth}\left (d+d\,\mathrm {coth}\left (a+b\,x\right )-1\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acoth}{\left (- d \coth {\left (a + b x \right )} - d + 1 \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________