Optimal. Leaf size=46 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a \cosh (c+d x)+a}}\right )}{\sqrt {a} d} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2649, 206} \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a \cosh (c+d x)+a}}\right )}{\sqrt {a} d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+a \cosh (c+d x)}} \, dx &=\frac {(2 i) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {i a \sinh (c+d x)}{\sqrt {a+a \cosh (c+d x)}}\right )}{d}\\ &=\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a+a \cosh (c+d x)}}\right )}{\sqrt {a} d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.87 \[ \frac {2 \cosh \left (\frac {1}{2} (c+d x)\right ) \tan ^{-1}\left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{d \sqrt {a (\cosh (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 149, normalized size = 3.24 \[ \left [\frac {\sqrt {2} \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} \sqrt {-\frac {1}{a}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + \cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1}\right )}{d}, \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{\sqrt {a}}\right )}{\sqrt {a} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 21, normalized size = 0.46 \[ \frac {2 \, \sqrt {2} \arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{\sqrt {a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 103, normalized size = 2.24 \[ -\frac {\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \ln \left (\frac {2 \sqrt {a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {-a}-2 a}{\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) \sqrt {2}}{\sqrt {-a}\, \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 86, normalized size = 1.87 \[ 2 \, \sqrt {2} {\left (\frac {\arctan \left (e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )}{\sqrt {a} d} + \frac {e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{{\left (\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a}\right )} d}\right )} - \frac {2 \, \sqrt {2} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{\sqrt {a} d e^{\left (d x + c\right )} + \sqrt {a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \cosh {\left (c + d x \right )} + a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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