Optimal. Leaf size=26 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A]
time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3273, 65, 214}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3273
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^2(x)}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\cosh ^2(x)\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^2(x)}\right )}{b}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 31, normalized size = 1.19
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\cosh ^{2}\left (x \right )\right )}}{\cosh \left (x \right )}\right )}{\sqrt {a}}\) | \(31\) |
default | \(-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\cosh ^{2}\left (x \right )\right )}}{\cosh \left (x \right )}\right )}{\sqrt {a}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 68 vs.
\(2 (20) = 40\).
time = 0.44, size = 248, normalized size = 9.54 \begin {gather*} \left [\frac {\log \left (\frac {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 (4 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + 4 \, a + b\right )} \sinh \left (x\right )^{2} - 4 \, \sqrt {2} \sqrt {a} \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}}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 4 \, {\left (b \cosh \left (x\right )^{3} + {\left (4 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \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}}}}{2 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}\right )}{a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \cosh ^{2}{\left (x \right )}}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {b\,{\mathrm {cosh}\left (x\right )}^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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