Optimal. Leaf size=31 \[ \frac {\sinh ^5(a+b x)}{5 b}+\frac {\sinh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2564, 14} \[ \frac {\sinh ^5(a+b x)}{5 b}+\frac {\sinh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rubi steps
\begin {align*} \int \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\frac {i \operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,i \sinh (a+b x)\right )}{b}\\ &=\frac {i \operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,i \sinh (a+b x)\right )}{b}\\ &=\frac {\sinh ^3(a+b x)}{3 b}+\frac {\sinh ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 27, normalized size = 0.87 \[ \frac {\sinh ^3(a+b x) (3 \cosh (2 (a+b x))+7)}{30 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 64, normalized size = 2.06 \[ \frac {3 \, \sinh \left (b x + a\right )^{5} + 5 \, {\left (6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{3} + 15 \, {\left (\cosh \left (b x + a\right )^{4} + \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )}{240 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 82, normalized size = 2.65 \[ \frac {e^{\left (5 \, b x + 5 \, a\right )}}{160 \, b} + \frac {e^{\left (3 \, b x + 3 \, a\right )}}{96 \, b} - \frac {e^{\left (b x + a\right )}}{16 \, b} + \frac {e^{\left (-b x - a\right )}}{16 \, b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} - \frac {e^{\left (-5 \, b x - 5 \, a\right )}}{160 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 42, normalized size = 1.35 \[ \frac {\frac {\sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{5}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 78, normalized size = 2.52 \[ \frac {{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 30 \, e^{\left (-4 \, b x - 4 \, a\right )} + 3\right )} e^{\left (5 \, b x + 5 \, a\right )}}{480 \, b} + \frac {30 \, e^{\left (-b x - a\right )} - 5 \, e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, e^{\left (-5 \, b x - 5 \, a\right )}}{480 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 26, normalized size = 0.84 \[ \frac {3\,{\mathrm {sinh}\left (a+b\,x\right )}^5+5\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{15\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.46, size = 44, normalized size = 1.42 \[ \begin {cases} - \frac {2 \sinh ^{5}{\left (a + b x \right )}}{15 b} + \frac {\sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sinh ^{2}{\relax (a )} \cosh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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