Optimal. Leaf size=31 \[ \frac {x^2}{4}+\frac {\cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5429, 2715, 8}
\begin {gather*} \frac {\sinh \left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{4 b}+\frac {x^2}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 5429
Rubi steps
\begin {align*} \int x \cosh ^2\left (a+b x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \cosh ^2(a+b x) \, dx,x,x^2\right )\\ &=\frac {\cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}+\frac {1}{4} \text {Subst}\left (\int 1 \, dx,x,x^2\right )\\ &=\frac {x^2}{4}+\frac {\cosh \left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.87 \begin {gather*} \frac {2 \left (a+b x^2\right )+\sinh \left (2 \left (a+b x^2\right )\right )}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.65, size = 34, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {\cosh \left (x^{2} b +a \right ) \sinh \left (x^{2} b +a \right )}{2}+\frac {x^{2} b}{2}+\frac {a}{2}}{2 b}\) | \(34\) |
default | \(\frac {\frac {\cosh \left (x^{2} b +a \right ) \sinh \left (x^{2} b +a \right )}{2}+\frac {x^{2} b}{2}+\frac {a}{2}}{2 b}\) | \(34\) |
risch | \(\frac {x^{2}}{4}+\frac {{\mathrm e}^{2 x^{2} b +2 a}}{16 b}-\frac {{\mathrm e}^{-2 x^{2} b -2 a}}{16 b}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 38, normalized size = 1.23 \begin {gather*} \frac {1}{4} \, x^{2} + \frac {e^{\left (2 \, b x^{2} + 2 \, a\right )}}{16 \, b} - \frac {e^{\left (-2 \, b x^{2} - 2 \, a\right )}}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 28, normalized size = 0.90 \begin {gather*} \frac {b x^{2} + \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )}{4 \, b} \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. 60 vs.
\(2 (24) = 48\).
time = 0.13, size = 60, normalized size = 1.94 \begin {gather*} \begin {cases} - \frac {x^{2} \sinh ^{2}{\left (a + b x^{2} \right )}}{4} + \frac {x^{2} \cosh ^{2}{\left (a + b x^{2} \right )}}{4} + \frac {\sinh {\left (a + b x^{2} \right )} \cosh {\left (a + b x^{2} \right )}}{4 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \cosh ^{2}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 54, normalized size = 1.74 \begin {gather*} \frac {4 \, b x^{2} - {\left (2 \, e^{\left (2 \, b x^{2} + 2 \, a\right )} + 1\right )} e^{\left (-2 \, b x^{2} - 2 \, a\right )} + 4 \, a + e^{\left (2 \, b x^{2} + 2 \, a\right )}}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 22, normalized size = 0.71 \begin {gather*} \frac {\mathrm {sinh}\left (2\,b\,x^2+2\,a\right )}{8\,b}+\frac {x^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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