Optimal. Leaf size=117 \[ \frac {e^{c (a+b x)}}{b c}-\frac {4 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};1+\frac {b c}{2 e};-e^{2 (d+e x)}\right )}{b c}+\frac {4 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};1+\frac {b c}{2 e};-e^{2 (d+e x)}\right )}{b c} \]
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Rubi [A]
time = 0.09, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5592, 2225,
2283} \begin {gather*} -\frac {4 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac {4 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac {e^{c (a+b x)}}{b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2283
Rule 5592
Rubi steps
\begin {align*} \int e^{c (a+b x)} \tanh ^2(d+e x) \, dx &=\int \left (e^{c (a+b x)}+\frac {4 e^{c (a+b x)}}{\left (1+e^{2 (d+e x)}\right )^2}-\frac {4 e^{c (a+b x)}}{1+e^{2 (d+e x)}}\right ) \, dx\\ &=4 \int \frac {e^{c (a+b x)}}{\left (1+e^{2 (d+e x)}\right )^2} \, dx-4 \int \frac {e^{c (a+b x)}}{1+e^{2 (d+e x)}} \, dx+\int e^{c (a+b x)} \, dx\\ &=\frac {e^{c (a+b x)}}{b c}-\frac {4 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};1+\frac {b c}{2 e};-e^{2 (d+e x)}\right )}{b c}+\frac {4 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};1+\frac {b c}{2 e};-e^{2 (d+e x)}\right )}{b c}\\ \end {align*}
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Mathematica [A]
time = 2.31, size = 169, normalized size = 1.44 \begin {gather*} \frac {e^{c (a+b x)} \left (2 b^2 c^2 e^{2 (d+e x)} \, _2F_1\left (1,1+\frac {b c}{2 e};2+\frac {b c}{2 e};-e^{2 (d+e x)}\right )-(b c+2 e) \left (2 b c e^{2 d} \, _2F_1\left (1,\frac {b c}{2 e};1+\frac {b c}{2 e};-e^{2 (d+e x)}\right )-\left (1+e^{2 d}\right ) (e-b c \text {sech}(d) \text {sech}(d+e x) \sinh (e x))\right )\right )}{b c e (b c+2 e) \left (1+e^{2 d}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.30, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{c \left (b x +a \right )} \left (\tanh ^{2}\left (e x +d \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} e^{a c} \int e^{b c x} \tanh ^{2}{\left (d + e x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\mathrm {tanh}\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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