Optimal. Leaf size=83 \[ \cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right )+\sqrt {a+a \cosh (c+d x)} \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3400, 3384,
3379, 3382} \begin {gather*} \cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}+\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3400
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx &=\left (\sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \frac {\sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )}{x} \, dx\\ &=\left (\cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \frac {\cosh \left (\frac {d x}{2}\right )}{x} \, dx+\left (\sqrt {a+a \cosh (c+d x)} \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right ) \sinh \left (\frac {c}{2}\right )\right ) \int \frac {\sinh \left (\frac {d x}{2}\right )}{x} \, dx\\ &=\cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right )+\sqrt {a+a \cosh (c+d x)} \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 54, normalized size = 0.65 \begin {gather*} \sqrt {a (1+\cosh (c+d x))} \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (\cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right )+\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.77, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +a \cosh \left (d x +c \right )}}{x}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 32, normalized size = 0.39 \begin {gather*} \frac {1}{2} \, \sqrt {2} {\left (\sqrt {a} {\rm Ei}\left (\frac {1}{2} \, d x\right ) e^{\left (\frac {1}{2} \, c\right )} + \sqrt {a} {\rm Ei}\left (-\frac {1}{2} \, d x\right ) e^{\left (-\frac {1}{2} \, c\right )}\right )} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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