Optimal. Leaf size=38 \[ \frac {\sinh ^3(a+b x)}{3 b}-\frac {\sinh (a+b x)}{b}+\frac {\tan ^{-1}(\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2592, 302, 203} \[ \frac {\sinh ^3(a+b x)}{3 b}-\frac {\sinh (a+b x)}{b}+\frac {\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 203
Rule 302
Rule 2592
Rubi steps
\begin {align*} \int \sinh ^3(a+b x) \tanh (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\sinh (a+b x)\right )}{b}\\ &=-\frac {\sinh (a+b x)}{b}+\frac {\sinh ^3(a+b x)}{3 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=\frac {\tan ^{-1}(\sinh (a+b x))}{b}-\frac {\sinh (a+b x)}{b}+\frac {\sinh ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.00 \[ \frac {\sinh ^3(a+b x)}{3 b}-\frac {\sinh (a+b x)}{b}+\frac {\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 290, normalized size = 7.63 \[ \frac {\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 15 \, {\left (\cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - 15 \, \cosh \left (b x + a\right )^{4} + 20 \, {\left (\cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 15 \, {\left (\cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 48 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3}\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 15 \, \cosh \left (b x + a\right )^{2} + 6 \, {\left (\cosh \left (b x + a\right )^{5} - 10 \, \cosh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1}{24 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 63, normalized size = 1.66 \[ \frac {{\left (15 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} + {\left (e^{\left (3 \, b x + 18 \, a\right )} - 15 \, e^{\left (b x + 16 \, a\right )}\right )} e^{\left (-15 \, a\right )} + 48 \, \arctan \left (e^{\left (b x + a\right )}\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 38, normalized size = 1.00 \[ \frac {\sinh ^{3}\left (b x +a \right )}{3 b}-\frac {\sinh \left (b x +a \right )}{b}+\frac {2 \arctan \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 71, normalized size = 1.87 \[ -\frac {{\left (15 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} + \frac {15 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} - \frac {2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 77, normalized size = 2.03 \[ \frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {5\,{\mathrm {e}}^{a+b\,x}}{8\,b}+\frac {5\,{\mathrm {e}}^{-a-b\,x}}{8\,b}-\frac {{\mathrm {e}}^{-3\,a-3\,b\,x}}{24\,b}+\frac {{\mathrm {e}}^{3\,a+3\,b\,x}}{24\,b} \]
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 \[ \int \sinh ^{3}{\left (a + b x \right )} \tanh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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