Optimal. Leaf size=26 \[ 2 a \sqrt {x}-\frac {2 b \tanh ^{-1}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d} \]
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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {14, 5545, 3855}
\begin {gather*} 2 a \sqrt {x}-\frac {2 b \tanh ^{-1}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 3855
Rule 5545
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx &=\int \left (\frac {a}{\sqrt {x}}+\frac {b \text {csch}\left (c+d \sqrt {x}\right )}{\sqrt {x}}\right ) \, dx\\ &=2 a \sqrt {x}+b \int \frac {\text {csch}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx\\ &=2 a \sqrt {x}+(2 b) \text {Subst}\left (\int \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )\\ &=2 a \sqrt {x}-\frac {2 b \tanh ^{-1}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 34, normalized size = 1.31 \begin {gather*} \frac {2 \left (a \left (c+d \sqrt {x}\right )+b \log \left (\tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.09, size = 26, normalized size = 1.00
method | result | size |
derivativedivides | \(2 a \sqrt {x}+\frac {2 b \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{d}\) | \(26\) |
default | \(2 a \sqrt {x}+\frac {2 b \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{d}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 25, normalized size = 0.96 \begin {gather*} 2 \, a \sqrt {x} + \frac {2 \, b \log \left (\tanh \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )\right )}{d} \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. 55 vs.
\(2 (22) = 44\).
time = 0.41, size = 55, normalized size = 2.12 \begin {gather*} \frac {2 \, {\left (a d \sqrt {x} - b \log \left (\cosh \left (d \sqrt {x} + c\right ) + \sinh \left (d \sqrt {x} + c\right ) + 1\right ) + b \log \left (\cosh \left (d \sqrt {x} + c\right ) + \sinh \left (d \sqrt {x} + c\right ) - 1\right )\right )}}{d} \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 {a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}}{\sqrt {x}}\, dx \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. 49 vs.
\(2 (22) = 44\).
time = 0.39, size = 49, normalized size = 1.88 \begin {gather*} \frac {2 \, {\left (d \sqrt {x} + c\right )} a}{d} - \frac {2 \, b \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right )}{d} + \frac {2 \, b \log \left ({\left | e^{\left (d \sqrt {x} + c\right )} - 1 \right |}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.44, size = 47, normalized size = 1.81 \begin {gather*} 2\,a\,\sqrt {x}-\frac {4\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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