Optimal. Leaf size=101 \[ \frac {e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {8 e^{3 a+3 b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}+\frac {\tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2282, 12, 288, 199, 206} \[ \frac {e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {8 e^{3 a+3 b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}+\frac {\tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 199
Rule 206
Rule 288
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \text {csch}^4(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {16 x^4}{\left (1-x^2\right )^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {16 \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {8 e^{3 a+3 b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {8 \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {8 e^{3 a+3 b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {8 e^{3 a+3 b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {8 e^{3 a+3 b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}+\frac {\tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 75, normalized size = 0.74 \[ \frac {3 e^{a+b x}-8 e^{3 (a+b x)}-3 e^{5 (a+b x)}+3 \left (e^{2 (a+b x)}-1\right )^3 \tanh ^{-1}\left (e^{a+b x}\right )}{3 b \left (e^{2 (a+b x)}-1\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.05, size = 705, normalized size = 6.98 \[ -\frac {6 \, \cosh \left (b x + a\right )^{5} + 30 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 6 \, \sinh \left (b x + a\right )^{5} + 4 \, {\left (15 \, \cosh \left (b x + a\right )^{2} + 4\right )} \sinh \left (b x + a\right )^{3} + 16 \, \cosh \left (b x + a\right )^{3} + 12 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - 3 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 3 \, \cosh \left (b x + a\right )^{2} + 6 \, {\left (\cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - 3 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 3 \, \cosh \left (b x + a\right )^{2} + 6 \, {\left (\cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 8 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - 6 \, \cosh \left (b x + a\right )}{6 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} - 3 \, b \cosh \left (b x + a\right )^{4} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} - 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{4} - 6 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 6 \, {\left (b \cosh \left (b x + a\right )^{5} - 2 \, b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 75, normalized size = 0.74 \[ -\frac {\frac {2 \, {\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} + 8 \, e^{\left (3 \, b x + 3 \, a\right )} - 3 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} - 3 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 0.37 \[ \frac {-\frac {\mathrm {csch}\left (b x +a \right ) \coth \left (b x +a \right )}{2}+\arctanh \left ({\mathrm e}^{b x +a}\right )-\frac {1}{3 \sinh \left (b x +a \right )^{3}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 100, normalized size = 0.99 \[ \frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{2 \, b} - \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{2 \, b} - \frac {3 \, e^{\left (5 \, b x + 5 \, a\right )} + 8 \, e^{\left (3 \, b x + 3 \, a\right )} - 3 \, e^{\left (b x + a\right )}}{3 \, b {\left (e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 135, normalized size = 1.34 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{3\,a+3\,b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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 \[ e^{a} \int e^{b x} \operatorname {csch}^{4}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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