Optimal. Leaf size=57 \[ \frac {(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x)^2}{2 d}+\frac {d \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2} \]
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Rubi [A] time = 0.09, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3718, 2190, 2279, 2391} \[ \frac {d \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {(c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rubi steps
\begin {align*} \int (c+d x) \tanh (e+f x) \, dx &=-\frac {(c+d x)^2}{2 d}+2 \int \frac {e^{2 (e+f x)} (c+d x)}{1+e^{2 (e+f x)}} \, dx\\ &=-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {d \int \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}-\frac {d \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^2}\\ &=-\frac {(c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d \text {Li}_2\left (-e^{2 (e+f x)}\right )}{2 f^2}\\ \end {align*}
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Mathematica [C] time = 4.42, size = 210, normalized size = 3.68 \[ \frac {c \log (\cosh (e+f x))}{f}+\frac {d \text {csch}(e) \text {sech}(e) \left (f^2 x^2 e^{-\tanh ^{-1}(\coth (e))}-\frac {i \coth (e) \left (i \text {Li}_2\left (e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )-f x \left (-\pi +2 i \tanh ^{-1}(\coth (e))\right )-2 \left (i \tanh ^{-1}(\coth (e))+i f x\right ) \log \left (1-e^{2 i \left (i \tanh ^{-1}(\coth (e))+i f x\right )}\right )+2 i \tanh ^{-1}(\coth (e)) \log \left (i \sinh \left (\tanh ^{-1}(\coth (e))+f x\right )\right )-\pi \log \left (e^{2 f x}+1\right )+\pi \log (\cosh (f x))\right )}{\sqrt {1-\coth ^2(e)}}\right )}{2 f^2 \sqrt {\text {csch}^2(e) \left (\sinh ^2(e)-\cosh ^2(e)\right )}}+\frac {1}{2} d x^2 \tanh (e) \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.46, size = 171, normalized size = 3.00 \[ -\frac {d f^{2} x^{2} + 2 \, c f^{2} x - 2 \, d {\rm Li}_2\left (i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right )\right ) - 2 \, d {\rm Li}_2\left (-i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right )\right ) + 2 \, {\left (d e - c f\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + i\right ) + 2 \, {\left (d e - c f\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - i\right ) - 2 \, {\left (d f x + d e\right )} \log \left (i \, \cosh \left (f x + e\right ) + i \, \sinh \left (f x + e\right ) + 1\right ) - 2 \, {\left (d f x + d e\right )} \log \left (-i \, \cosh \left (f x + e\right ) - i \, \sinh \left (f x + e\right ) + 1\right )}{2 \, f^{2}} \]
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 \[ \int {\left (d x + c\right )} \tanh \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 109, normalized size = 1.91 \[ -\frac {d \,x^{2}}{2}+c x +\frac {c \ln \left ({\mathrm e}^{2 f x +2 e}+1\right )}{f}-\frac {2 c \ln \left ({\mathrm e}^{f x +e}\right )}{f}-\frac {2 d e x}{f}-\frac {d \,e^{2}}{f^{2}}+\frac {d \ln \left ({\mathrm e}^{2 f x +2 e}+1\right ) x}{f}+\frac {d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{2}}+\frac {2 d e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 78, normalized size = 1.37 \[ -\frac {1}{2} \, d x^{2} + \frac {c \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{2 \, f} + \frac {c \log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{2 \, f} + \frac {{\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )} d}{2 \, f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {tanh}\left (e+f\,x\right )\,\left (c+d\,x\right ) \,d x \]
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 \[ \int \left (c + d x\right ) \tanh {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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