Optimal. Leaf size=19 \[ \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \]
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Rubi [A] time = 0.06, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3198, 261} \[ \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 261
Rule 3198
Rubi steps
\begin {align*} \int \cosh (x) \sinh (x) \sqrt {a+b \sinh ^2(x)} \, dx &=\operatorname {Subst}\left (\int x \sqrt {a+b x^2} \, dx,x,\sinh (x)\right )\\ &=\frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 154, normalized size = 8.11 \[ \frac {\sqrt {2} {\left (b \cosh \relax (x)^{4} + 4 \, b \cosh \relax (x) \sinh \relax (x)^{3} + b \sinh \relax (x)^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} + 2 \, a - b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} + {\left (2 \, a - b\right )} \cosh \relax (x)\right )} \sinh \relax (x) + b\right )} \sqrt {\frac {b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a - b}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{24 \, {\left (b \cosh \relax (x)^{3} + 3 \, b \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, b \cosh \relax (x) \sinh \relax (x)^{2} + b \sinh \relax (x)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sinh \relax (x)^{2} + a} \cosh \relax (x) \sinh \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 16, normalized size = 0.84 \[ \frac {\left (a +b \left (\sinh ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 15, normalized size = 0.79 \[ \frac {{\left (b \sinh \relax (x)^{2} + a\right )}^{\frac {3}{2}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.87, size = 15, normalized size = 0.79 \[ \frac {{\left (b\,{\mathrm {sinh}\relax (x)}^2+a\right )}^{3/2}}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.32, size = 46, normalized size = 2.42 \[ \begin {cases} \frac {a \sqrt {a + b \sinh ^{2}{\relax (x )}}}{3 b} + \frac {\sqrt {a + b \sinh ^{2}{\relax (x )}} \sinh ^{2}{\relax (x )}}{3} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} \cosh ^{2}{\relax (x )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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