Optimal. Leaf size=11 \[ \frac {\sinh (x)}{b+a \cosh (x)} \]
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Rubi [A]
time = 0.06, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3370, 2833, 8}
\begin {gather*} \frac {\sinh (x)}{a \cosh (x)+b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2833
Rule 3370
Rubi steps
\begin {align*} \int \frac {a+b \cosh (x)}{b^2+2 a b \cosh (x)+a^2 \cosh ^2(x)} \, dx &=\left (4 a^2\right ) \int \frac {a+b \cosh (x)}{\left (2 a b+2 a^2 \cosh (x)\right )^2} \, dx\\ &=\frac {\sinh (x)}{b+a \cosh (x)}+\frac {\int 0 \, dx}{a^2-b^2}\\ &=\frac {\sinh (x)}{b+a \cosh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 11, normalized size = 1.00 \begin {gather*} \frac {\sinh (x)}{b+a \cosh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs.
\(2(11)=22\).
time = 0.65, size = 29, normalized size = 2.64
method | result | size |
risch | \(-\frac {2 \left ({\mathrm e}^{x} b +a \right )}{a \left ({\mathrm e}^{2 x} a +2 \,{\mathrm e}^{x} b +a \right )}\) | \(27\) |
default | \(\frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+a +b}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 54 vs.
\(2 (11) = 22\).
time = 0.40, size = 54, normalized size = 4.91 \begin {gather*} -\frac {2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 26 vs.
\(2 (11) = 22\).
time = 0.41, size = 26, normalized size = 2.36 \begin {gather*} -\frac {2 \, {\left (b e^{x} + a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} + a\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.93, size = 51, normalized size = 4.64 \begin {gather*} -\frac {\frac {2\,{\mathrm {e}}^x\,\left (a\,b^3-a^3\,b\right )}{a\,\left (a\,b^2-a^3\right )}+2}{a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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