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 = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 203
Rule 288
Rule 302
Rule 2592
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*}
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Mathematica [A] time = 0.11, 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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fricas [B] time = 0.45, size = 851, normalized size = 12.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 96, normalized size = 1.45 \[ \frac {{\left (27 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} + {\left (e^{\left (3 \, b x + 30 \, a\right )} - 27 \, e^{\left (b x + 28 \, a\right )}\right )} e^{\left (-27 \, a\right )} - \frac {24 \, {\left (e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} + 120 \, \arctan \left (e^{\left (b x + a\right )}\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 92, normalized size = 1.39 \[ \frac {\sinh ^{5}\left (b x +a \right )}{3 b \cosh \left (b x +a \right )^{2}}-\frac {5 \left (\sinh ^{3}\left (b x +a \right )\right )}{3 b \cosh \left (b x +a \right )^{2}}-\frac {5 \sinh \left (b x +a \right )}{b \cosh \left (b x +a \right )^{2}}+\frac {5 \,\mathrm {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2 b}+\frac {5 \arctan \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 116, normalized size = 1.76 \[ \frac {27 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} - \frac {5 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac {25 \, e^{\left (-2 \, b x - 2 \, a\right )} + 77 \, e^{\left (-4 \, b x - 4 \, a\right )} + 3 \, e^{\left (-6 \, b x - 6 \, a\right )} - 1}{24 \, b {\left (e^{\left (-3 \, b x - 3 \, a\right )} + 2 \, e^{\left (-5 \, b x - 5 \, a\right )} + e^{\left (-7 \, b x - 7 \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 136, normalized size = 2.06 \[ \frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {9\,{\mathrm {e}}^{a+b\,x}}{8\,b}+\frac {9\,{\mathrm {e}}^{-a-b\,x}}{8\,b}-\frac {{\mathrm {e}}^{-3\,a-3\,b\,x}}{24\,b}+\frac {{\mathrm {e}}^{3\,a+3\,b\,x}}{24\,b}+\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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 \[ \int \sinh ^{3}{\left (a + b x \right )} \tanh ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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