Optimal. Leaf size=31 \[ -\frac {2 e^{4 a+4 b x}}{b \left (1-e^{2 a+2 b x}\right )^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2320, 12, 270}
\begin {gather*} -\frac {2 e^{4 a+4 b x}}{b \left (1-e^{2 a+2 b x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \text {csch}^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {8 x^3}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {8 \text {Subst}\left (\int \frac {x^3}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac {2 e^{4 a+4 b x}}{b \left (1-e^{2 a+2 b x}\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.94 \begin {gather*} -\frac {2 e^{4 a+4 b x}}{b \left (-1+e^{2 a+2 b x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.13, size = 32, normalized size = 1.03
method | result | size |
risch | \(-\frac {2 \left (2 \,{\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (27) = 54\).
time = 0.27, size = 68, normalized size = 2.19 \begin {gather*} -\frac {4 \, e^{\left (2 \, b x + 2 \, a\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} + \frac {2}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs.
\(2 (27) = 54\).
time = 0.45, size = 88, normalized size = 2.84 \begin {gather*} -\frac {2 \, {\left (\cosh \left (b x + a\right ) + 3 \, \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right ) + 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )} \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} \int e^{b x} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 31, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 31, normalized size = 1.00 \begin {gather*} -\frac {2\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}{b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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