Optimal. Leaf size=167 \[ -\frac {6 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac {12 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}-\frac {8 e^{c (a+b x)} \, _2F_1\left (3,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac {e^{c (a+b x)}}{b c} \]
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Rubi [A] time = 0.19, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5484, 2194, 2251} \[ -\frac {6 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac {12 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}-\frac {8 e^{c (a+b x)} \, _2F_1\left (3,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac {e^{c (a+b x)}}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 5484
Rubi steps
\begin {align*} \int e^{c (a+b x)} \tanh ^3(d+e x) \, dx &=\int \left (e^{c (a+b x)}-\frac {8 e^{c (a+b x)}}{\left (1+e^{2 (d+e x)}\right )^3}+\frac {12 e^{c (a+b x)}}{\left (1+e^{2 (d+e x)}\right )^2}-\frac {6 e^{c (a+b x)}}{1+e^{2 (d+e x)}}\right ) \, dx\\ &=-\left (6 \int \frac {e^{c (a+b x)}}{1+e^{2 (d+e x)}} \, dx\right )-8 \int \frac {e^{c (a+b x)}}{\left (1+e^{2 (d+e x)}\right )^3} \, dx+12 \int \frac {e^{c (a+b x)}}{\left (1+e^{2 (d+e x)}\right )^2} \, dx+\int e^{c (a+b x)} \, dx\\ &=\frac {e^{c (a+b x)}}{b c}-\frac {6 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 {12 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}-\frac {8 e^{c (a+b x)} \, _2F_1\left (3,\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 = 4.45, size = 205, normalized size = 1.23 \[ \frac {1}{2} e^{a c} \left (\frac {2 e^{2 d} \left (b^2 c^2+2 e^2\right ) \left (\frac {e^{x (b c+2 e)} \, _2F_1\left (1,\frac {b c}{2 e}+1;\frac {b c}{2 e}+2;-e^{2 (d+e x)}\right )}{b c+2 e}-\frac {e^{b c x} \, _2F_1\left (1,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}\right )}{\left (e^{2 d}+1\right ) e^2}-\frac {b c \text {sech}(d) e^{b c x} \sinh (e x) \text {sech}(d+e x)}{e^2}+\frac {e^{b c x} \text {sech}^2(d+e x)}{e}+\frac {2 \tanh (d) e^{b c x}}{b c}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (b c x + a c\right )} \tanh \left (e x + d\right )^{3}, x\right ) \]
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 \[ \int e^{\left ({\left (b x + a\right )} c\right )} \tanh \left (e x + d\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{c \left (b x +a \right )} \left (\tanh ^{3}\left (e x +d \right )\right )\, dx \]
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 \[ 48 \, {\left (b^{2} c^{2} e e^{\left (a c\right )} + 2 \, e^{3} e^{\left (a c\right )}\right )} \int \frac {e^{\left (b c x\right )}}{b^{3} c^{3} - 12 \, b^{2} c^{2} e + 44 \, b c e^{2} - 48 \, e^{3} + {\left (b^{3} c^{3} e^{\left (8 \, d\right )} - 12 \, b^{2} c^{2} e e^{\left (8 \, d\right )} + 44 \, b c e^{2} e^{\left (8 \, d\right )} - 48 \, e^{3} e^{\left (8 \, d\right )}\right )} e^{\left (8 \, e x\right )} + 4 \, {\left (b^{3} c^{3} e^{\left (6 \, d\right )} - 12 \, b^{2} c^{2} e e^{\left (6 \, d\right )} + 44 \, b c e^{2} e^{\left (6 \, d\right )} - 48 \, e^{3} e^{\left (6 \, d\right )}\right )} e^{\left (6 \, e x\right )} + 6 \, {\left (b^{3} c^{3} e^{\left (4 \, d\right )} - 12 \, b^{2} c^{2} e e^{\left (4 \, d\right )} + 44 \, b c e^{2} e^{\left (4 \, d\right )} - 48 \, e^{3} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 4 \, {\left (b^{3} c^{3} e^{\left (2 \, d\right )} - 12 \, b^{2} c^{2} e e^{\left (2 \, d\right )} + 44 \, b c e^{2} e^{\left (2 \, d\right )} - 48 \, e^{3} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}}\,{d x} - \frac {{\left (b^{3} c^{3} e^{\left (a c\right )} + 36 \, b^{2} c^{2} e e^{\left (a c\right )} + 44 \, b c e^{2} e^{\left (a c\right )} + 48 \, e^{3} e^{\left (a c\right )} - {\left (b^{3} c^{3} e^{\left (a c + 6 \, d\right )} - 12 \, b^{2} c^{2} e e^{\left (a c + 6 \, d\right )} + 44 \, b c e^{2} e^{\left (a c + 6 \, d\right )} - 48 \, e^{3} e^{\left (a c + 6 \, d\right )}\right )} e^{\left (6 \, e x\right )} + 3 \, {\left (b^{3} c^{3} e^{\left (a c + 4 \, d\right )} - 8 \, b^{2} c^{2} e e^{\left (a c + 4 \, d\right )} + 4 \, b c e^{2} e^{\left (a c + 4 \, d\right )} + 48 \, e^{3} e^{\left (a c + 4 \, d\right )}\right )} e^{\left (4 \, e x\right )} - 3 \, {\left (b^{3} c^{3} e^{\left (a c + 2 \, d\right )} - 28 \, b c e^{2} e^{\left (a c + 2 \, d\right )} - 48 \, e^{3} e^{\left (a c + 2 \, d\right )}\right )} e^{\left (2 \, e x\right )}\right )} e^{\left (b c x\right )}}{b^{4} c^{4} - 12 \, b^{3} c^{3} e + 44 \, b^{2} c^{2} e^{2} - 48 \, b c e^{3} + {\left (b^{4} c^{4} e^{\left (6 \, d\right )} - 12 \, b^{3} c^{3} e e^{\left (6 \, d\right )} + 44 \, b^{2} c^{2} e^{2} e^{\left (6 \, d\right )} - 48 \, b c e^{3} e^{\left (6 \, d\right )}\right )} e^{\left (6 \, e x\right )} + 3 \, {\left (b^{4} c^{4} e^{\left (4 \, d\right )} - 12 \, b^{3} c^{3} e e^{\left (4 \, d\right )} + 44 \, b^{2} c^{2} e^{2} e^{\left (4 \, d\right )} - 48 \, b c e^{3} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 3 \, {\left (b^{4} c^{4} e^{\left (2 \, d\right )} - 12 \, b^{3} c^{3} e e^{\left (2 \, d\right )} + 44 \, b^{2} c^{2} e^{2} e^{\left (2 \, d\right )} - 48 \, b c e^{3} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\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 \[ \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\mathrm {tanh}\left (d+e\,x\right )}^3 \,d x \]
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 \[ e^{a c} \int e^{b c x} \tanh ^{3}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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