Optimal. Leaf size=19 \[ \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \]
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Rubi [A]
time = 0.04, 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 = {3277, 267}
\begin {gather*} \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 3277
Rubi steps
\begin {align*} \int \cosh (x) \sinh (x) \sqrt {a+b \sinh ^2(x)} \, dx &=\text {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 \begin {gather*} \frac {\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 16, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\left (a +b \left (\sinh ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}{3 b}\) | \(16\) |
default | \(\frac {\left (a +b \left (\sinh ^{2}\left (x \right )\right )\right )^{\frac {3}{2}}}{3 b}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs.
\(2 (15) = 30\).
time = 0.34, size = 154, normalized size = 8.11 \begin {gather*} \frac {\sqrt {2} {\left (b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a - b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b\right )} \sqrt {\frac {b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a - b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{24 \, {\left (b \cosh \left (x\right )^{3} + 3 \, b \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, b \cosh \left (x\right ) \sinh \left (x\right )^{2} + b \sinh \left (x\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (14) = 28\).
time = 0.27, size = 46, normalized size = 2.42 \begin {gather*} \begin {cases} \frac {a \sqrt {a + b \sinh ^{2}{\left (x \right )}}}{3 b} + \frac {\sqrt {a + b \sinh ^{2}{\left (x \right )}} \sinh ^{2}{\left (x \right )}}{3} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} \cosh ^{2}{\left (x \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {{\left (b\,{\mathrm {sinh}\left (x\right )}^2+a\right )}^{3/2}}{3\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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