Optimal. Leaf size=150 \[ \frac {\text {Li}_2\left (-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b}-\frac {\text {Li}_2\left (-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b}+\frac {1}{2} x \log \left (\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}+1\right )-\frac {1}{2} x \log \left (\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}+1\right )+x \tanh ^{-1}(d \tanh (a+b x)+c) \]
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Rubi [A] time = 0.22, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6235, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b}-\frac {\text {PolyLog}\left (2,-\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b}+\frac {1}{2} x \log \left (\frac {(-c-d+1) e^{2 a+2 b x}}{-c+d+1}+1\right )-\frac {1}{2} x \log \left (\frac {(c+d+1) e^{2 a+2 b x}}{c-d+1}+1\right )+x \tanh ^{-1}(d \tanh (a+b x)+c) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 6235
Rubi steps
\begin {align*} \int \tanh ^{-1}(c+d \tanh (a+b x)) \, dx &=x \tanh ^{-1}(c+d \tanh (a+b x))+(b (1-c-d)) \int \frac {e^{2 a+2 b x} x}{1-c+d+(1-c-d) e^{2 a+2 b x}} \, dx-(b (1+c+d)) \int \frac {e^{2 a+2 b x} x}{1+c-d+(1+c+d) e^{2 a+2 b x}} \, dx\\ &=x \tanh ^{-1}(c+d \tanh (a+b x))+\frac {1}{2} x \log \left (1+\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac {1}{2} x \log \left (1+\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )-\frac {1}{2} \int \log \left (1+\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right ) \, dx+\frac {1}{2} \int \log \left (1+\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right ) \, dx\\ &=x \tanh ^{-1}(c+d \tanh (a+b x))+\frac {1}{2} x \log \left (1+\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac {1}{2} x \log \left (1+\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {(1-c-d) x}{1-c+d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {(1+c+d) x}{1+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=x \tanh ^{-1}(c+d \tanh (a+b x))+\frac {1}{2} x \log \left (1+\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac {1}{2} x \log \left (1+\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac {\text {Li}_2\left (-\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac {\text {Li}_2\left (-\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 6.81, size = 131, normalized size = 0.87 \[ \frac {\text {Li}_2\left (-\frac {(c+d-1) e^{2 (a+b x)}}{c-d-1}\right )-\text {Li}_2\left (-\frac {(c+d+1) e^{2 (a+b x)}}{c-d+1}\right )+2 b x \left (\log \left (\frac {(c+d-1) e^{2 (a+b x)}}{c-d-1}+1\right )-\log \left (\frac {(c+d+1) e^{2 (a+b x)}}{c-d+1}+1\right )\right )}{4 b}+x \tanh ^{-1}(d \tanh (a+b x)+c) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 552, normalized size = 3.68 \[ \frac {b x \log \left (-\frac {{\left (c + 1\right )} \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}{{\left (c - 1\right )} \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}\right ) + a \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d + 1\right )} \sqrt {-\frac {c + d + 1}{c - d + 1}}\right ) + a \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d + 1\right )} \sqrt {-\frac {c + d + 1}{c - d + 1}}\right ) - a \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d - 1\right )} \sqrt {-\frac {c + d - 1}{c - d - 1}}\right ) - a \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d - 1\right )} \sqrt {-\frac {c + d - 1}{c - d - 1}}\right ) - {\left (b x + a\right )} \log \left (\sqrt {-\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\left (b x + a\right )} \log \left (-\sqrt {-\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b x + a\right )} \log \left (\sqrt {-\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b x + a\right )} \log \left (-\sqrt {-\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\rm Li}_2\left (\sqrt {-\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - {\rm Li}_2\left (-\sqrt {-\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\rm Li}_2\left (\sqrt {-\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\rm Li}_2\left (-\sqrt {-\frac {c + d - 1}{c - d - 1}} {\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 \[ \int \operatorname {artanh}\left (d \tanh \left (b x + a\right ) + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 306, normalized size = 2.04 \[ -\frac {\arctanh \left (c +d \tanh \left (b x +a \right )\right ) \ln \left (d \tanh \left (b x +a \right )-d \right )}{2 b}+\frac {\arctanh \left (c +d \tanh \left (b x +a \right )\right ) \ln \left (d \tanh \left (b x +a \right )+d \right )}{2 b}+\frac {\dilog \left (\frac {d \tanh \left (b x +a \right )+c -1}{c -d -1}\right )}{4 b}+\frac {\ln \left (d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {d \tanh \left (b x +a \right )+c -1}{c -d -1}\right )}{4 b}-\frac {\dilog \left (\frac {d \tanh \left (b x +a \right )+c +1}{1+c -d}\right )}{4 b}-\frac {\ln \left (d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {d \tanh \left (b x +a \right )+c +1}{1+c -d}\right )}{4 b}+\frac {\dilog \left (\frac {d \tanh \left (b x +a \right )+c +1}{1+c +d}\right )}{4 b}+\frac {\ln \left (d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {d \tanh \left (b x +a \right )+c +1}{1+c +d}\right )}{4 b}-\frac {\dilog \left (\frac {d \tanh \left (b x +a \right )+c -1}{c +d -1}\right )}{4 b}-\frac {\ln \left (d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {d \tanh \left (b x +a \right )+c -1}{c +d -1}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 142, normalized size = 0.95 \[ -\frac {1}{4} \, b d {\left (\frac {2 \, b x \log \left (\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}\right )}{b^{2} d} - \frac {2 \, b x \log \left (\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}\right )}{b^{2} d}\right )} + x \operatorname {artanh}\left (d \tanh \left (b x + a\right ) + c\right ) \]
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 \[ \int \mathrm {atanh}\left (c+d\,\mathrm {tanh}\left (a+b\,x\right )\right ) \,d x \]
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 \[ \int \operatorname {atanh}{\left (c + d \tanh {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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