Optimal. Leaf size=15 \[ \frac {\cosh ^3(a+b x)}{3 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2645, 30}
\begin {gather*} \frac {\cosh ^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2645
Rubi steps
\begin {align*} \int \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac {\text {Subst}\left (\int x^2 \, dx,x,\cosh (a+b x)\right )}{b}\\ &=\frac {\cosh ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\cosh ^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 14, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {\cosh ^{3}\left (b x +a \right )}{3 b}\) | \(14\) |
default | \(\frac {\cosh ^{3}\left (b x +a \right )}{3 b}\) | \(14\) |
risch | \(\frac {{\mathrm e}^{3 b x +3 a}}{24 b}+\frac {{\mathrm e}^{b x +a}}{8 b}+\frac {{\mathrm e}^{-b x -a}}{8 b}+\frac {{\mathrm e}^{-3 b x -3 a}}{24 b}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 13, normalized size = 0.87 \begin {gather*} \frac {\cosh \left (b x + a\right )^{3}}{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. 38 vs.
\(2 (13) = 26\).
time = 0.37, size = 38, normalized size = 2.53 \begin {gather*} \frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, \cosh \left (b x + a\right )}{12 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 20, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {\cosh ^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sinh {\left (a \right )} \cosh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (13) = 26\).
time = 0.39, size = 54, normalized size = 3.60 \begin {gather*} \frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} + \frac {e^{\left (b x + a\right )}}{8 \, b} + \frac {e^{\left (-b x - a\right )}}{8 \, b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.46, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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