Optimal. Leaf size=69 \[ -\frac {\text {Li}_2\left (-\left ((d+1) e^{2 a+2 b x}\right )\right )}{4 b}-\frac {1}{2} x \log \left ((d+1) e^{2 a+2 b x}+1\right )+x \tanh ^{-1}(d \tanh (a+b x)+d+1)+\frac {b x^2}{2} \]
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Rubi [A] time = 0.14, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6231, 2184, 2190, 2279, 2391} \[ -\frac {\text {PolyLog}\left (2,-(d+1) e^{2 a+2 b x}\right )}{4 b}-\frac {1}{2} x \log \left ((d+1) e^{2 a+2 b x}+1\right )+x \tanh ^{-1}(d \tanh (a+b x)+d+1)+\frac {b x^2}{2} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2279
Rule 2391
Rule 6231
Rubi steps
\begin {align*} \int \tanh ^{-1}(1+d+d \tanh (a+b x)) \, dx &=x \tanh ^{-1}(1+d+d \tanh (a+b x))+b \int \frac {x}{1+(1+d) e^{2 a+2 b x}} \, dx\\ &=\frac {b x^2}{2}+x \tanh ^{-1}(1+d+d \tanh (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 \tanh ^{-1}(1+d+d \tanh (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 \tanh ^{-1}(1+d+d \tanh (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 \tanh ^{-1}(1+d+d \tanh (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*}
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Mathematica [B] time = 4.76, size = 201, normalized size = 2.91 \[ \frac {-2 \text {Li}_2\left (-\sqrt {-d-1} e^{a+b x}\right )-2 \text {Li}_2\left (\sqrt {-d-1} e^{a+b x}\right )-2 \log \left (e^{a+b x}\right ) \log \left (1-\sqrt {-d-1} e^{a+b x}\right )-2 \log \left (e^{a+b x}\right ) \log \left (\sqrt {-d-1} e^{a+b x}+1\right )+2 \log \left (e^{a+b x}\right ) \log \left ((d+1) e^{a+b x}+e^{-a-b x}\right )-2 b x \log (d \sinh (a+b x)+(d+2) \cosh (a+b x))+\log ^2\left (e^{a+b x}\right )+b^2 x^2}{4 b}+x \tanh ^{-1}(d \tanh (a+b x)+d+1) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.58, size = 239, normalized size = 3.46 \[ \frac {b^{2} x^{2} + b x \log \left (-\frac {{\left (d + 2\right )} \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}{d \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}\right ) + a \log \left (2 \, {\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d + 1\right )} \sinh \left (b x + a\right ) + \sqrt {-4 \, d - 4}\right ) + a \log \left (2 \, {\left (d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (d + 1\right )} \sinh \left (b x + a\right ) - \sqrt {-4 \, d - 4}\right ) - {\left (b x + a\right )} \log \left (\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\left (b x + a\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 \, d - 4} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {artanh}\left (d \tanh \left (b x + a\right ) + d + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 247, normalized size = 3.58 \[ -\frac {\arctanh \left (1+d +d \tanh \left (b x +a \right )\right ) \ln \left (d \tanh \left (b x +a \right )-d \right )}{2 b}+\frac {\arctanh \left (1+d +d \tanh \left (b x +a \right )\right ) \ln \left (d \tanh \left (b x +a \right )+d \right )}{2 b}+\frac {\ln \left (d \tanh \left (b x +a \right )+d \right )^{2}}{8 b}-\frac {\dilog \left (1+\frac {d \tanh \left (b x +a \right )}{2}+\frac {d}{2}\right )}{4 b}-\frac {\ln \left (d \tanh \left (b x +a \right )+d \right ) \ln \left (1+\frac {d \tanh \left (b x +a \right )}{2}+\frac {d}{2}\right )}{4 b}-\frac {\dilog \left (\frac {d \tanh \left (b x +a \right )+d}{2 d}\right )}{4 b}-\frac {\ln \left (d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {d \tanh \left (b x +a \right )+d}{2 d}\right )}{4 b}+\frac {\dilog \left (\frac {d \tanh \left (b x +a \right )+d +2}{2 d +2}\right )}{4 b}+\frac {\ln \left (d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {d \tanh \left (b x +a \right )+d +2}{2 d +2}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 72, normalized size = 1.04 \[ \frac {1}{4} \, b d {\left (\frac {2 \, x^{2}}{d} - \frac {2 \, b x \log \left ({\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-{\left (d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )}{b^{2} d}\right )} + x \operatorname {artanh}\left (d \tanh \left (b x + a\right ) + d + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {atanh}\left (d+d\,\mathrm {tanh}\left (a+b\,x\right )+1\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {atanh}{\left (d \tanh {\left (a + b x \right )} + d + 1 \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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