Optimal. Leaf size=66 \[ \frac{5 \sinh ^3(a+b x)}{6 b}-\frac{5 \sinh (a+b x)}{2 b}-\frac{\sinh ^3(a+b x) \tanh ^2(a+b x)}{2 b}+\frac{5 \tan ^{-1}(\sinh (a+b x))}{2 b} \]
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Rubi [A] time = 0.0424965, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2592, 288, 302, 203} \[ \frac{5 \sinh ^3(a+b x)}{6 b}-\frac{5 \sinh (a+b x)}{2 b}-\frac{\sinh ^3(a+b x) \tanh ^2(a+b x)}{2 b}+\frac{5 \tan ^{-1}(\sinh (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 2592
Rule 288
Rule 302
Rule 203
Rubi steps
\begin{align*} \int \sinh ^3(a+b x) \tanh ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^2} \, dx,x,\sinh (a+b x)\right )}{b}\\ &=-\frac{\sinh ^3(a+b x) \tanh ^2(a+b x)}{2 b}+\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{2 b}\\ &=-\frac{\sinh ^3(a+b x) \tanh ^2(a+b x)}{2 b}+\frac{5 \operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,\sinh (a+b x)\right )}{2 b}\\ &=-\frac{5 \sinh (a+b x)}{2 b}+\frac{5 \sinh ^3(a+b x)}{6 b}-\frac{\sinh ^3(a+b x) \tanh ^2(a+b x)}{2 b}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (a+b x)\right )}{2 b}\\ &=\frac{5 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{5 \sinh (a+b x)}{2 b}+\frac{5 \sinh ^3(a+b x)}{6 b}-\frac{\sinh ^3(a+b x) \tanh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.103281, size = 65, normalized size = 0.98 \[ \frac{2 \sinh ^3(a+b x) \tanh ^2(a+b x)+15 \tan ^{-1}(\sinh (a+b x))-10 \sinh (a+b x) \tanh ^2(a+b x)-15 \tanh (a+b x) \text{sech}(a+b x)}{6 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 92, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{5}}{3\,b \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}-{\frac{5\, \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{3\,b \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}-5\,{\frac{\sinh \left ( bx+a \right ) }{b \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}+{\frac{5\,{\rm sech} \left (bx+a\right )\tanh \left ( bx+a \right ) }{2\,b}}+5\,{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63116, size = 157, normalized size = 2.38 \begin{align*} \frac{27 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} - \frac{5 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac{25 \, e^{\left (-2 \, b x - 2 \, a\right )} + 77 \, e^{\left (-4 \, b x - 4 \, a\right )} + 3 \, e^{\left (-6 \, b x - 6 \, a\right )} - 1}{24 \, b{\left (e^{\left (-3 \, b x - 3 \, a\right )} + 2 \, e^{\left (-5 \, b x - 5 \, a\right )} + e^{\left (-7 \, b x - 7 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94062, size = 2390, normalized size = 36.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh ^{3}{\left (a + b x \right )} \tanh ^{3}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37744, size = 130, normalized size = 1.97 \begin{align*} \frac{{\left (27 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} +{\left (e^{\left (3 \, b x + 30 \, a\right )} - 27 \, e^{\left (b x + 28 \, a\right )}\right )} e^{\left (-27 \, a\right )} - \frac{24 \,{\left (e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} + 120 \, \arctan \left (e^{\left (b x + a\right )}\right )}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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