Optimal. Leaf size=49 \[ -\frac {3 \text {ArcTan}(\sinh (a+b x))}{2 b}+\frac {3 \sinh (a+b x)}{2 b}-\frac {\sinh (a+b x) \tanh ^2(a+b x)}{2 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2672, 294, 327,
209} \begin {gather*} -\frac {3 \text {ArcTan}(\sinh (a+b x))}{2 b}+\frac {3 \sinh (a+b x)}{2 b}-\frac {\sinh (a+b x) \tanh ^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 294
Rule 327
Rule 2672
Rubi steps
\begin {align*} \int \sinh (a+b x) \tanh ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=-\frac {\sinh (a+b x) \tanh ^2(a+b x)}{2 b}+\frac {3 \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{2 b}\\ &=\frac {3 \sinh (a+b x)}{2 b}-\frac {\sinh (a+b x) \tanh ^2(a+b x)}{2 b}-\frac {3 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{2 b}\\ &=-\frac {3 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac {3 \sinh (a+b x)}{2 b}-\frac {\sinh (a+b x) \tanh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 48, normalized size = 0.98 \begin {gather*} \frac {\sinh (a+b x) \tanh ^2(a+b x)}{b}-\frac {3 (\text {ArcTan}(\sinh (a+b x))-\text {sech}(a+b x) \tanh (a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.42, size = 93, normalized size = 1.90
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}-\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i\right )}{2 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i\right )}{2 b}\) | \(93\) |
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. 91 vs.
\(2 (43) = 86\).
time = 0.48, size = 91, normalized size = 1.86 \begin {gather*} \frac {3 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} + \frac {4 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + 1}{2 \, b {\left (e^{\left (-b x - a\right )} + 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + e^{\left (-5 \, b x - 5 \, a\right )}\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. 463 vs.
\(2 (43) = 86\).
time = 0.36, size = 463, normalized size = 9.45 \begin {gather*} \frac {\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} - 6 \, {\left (\cosh \left (b x + a\right )^{5} + 5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )^{3} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + {\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 ) + \cosh \left (b x + a\right )\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 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}{2 \, {\left (b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + b \sinh \left (b x + a\right )^{5} + 2 \, b \cosh \left (b x + a\right )^{3} + 2 \, {\left (5 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{3} + 2 \, {\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 )^{2} + b \cosh \left (b x + a\right ) + {\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 )\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*} \int \sinh {\left (a + b x \right )} \tanh ^{3}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (43) = 86\).
time = 0.40, size = 97, normalized size = 1.98 \begin {gather*} -\frac {3 \, \pi - \frac {4 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4} + 6 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right ) - 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.46, size = 107, normalized size = 2.18 \begin {gather*} \frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}+\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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