Optimal. Leaf size=49 \[ -\frac {i \sqrt {a+b \cosh ^2(x)} E\left (i x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cosh ^2(x)}{a}+1}} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3178, 3177} \[ -\frac {i \sqrt {a+b \cosh ^2(x)} E\left (i x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cosh ^2(x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3177
Rule 3178
Rubi steps
\begin {align*} \int \sqrt {a+b \cosh ^2(x)} \, dx &=\frac {\sqrt {a+b \cosh ^2(x)} \int \sqrt {1+\frac {b \cosh ^2(x)}{a}} \, dx}{\sqrt {1+\frac {b \cosh ^2(x)}{a}}}\\ &=-\frac {i \sqrt {a+b \cosh ^2(x)} E\left (\frac {\pi }{2}+i x|-\frac {b}{a}\right )}{\sqrt {1+\frac {b \cosh ^2(x)}{a}}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 53, normalized size = 1.08 \[ -\frac {i \sqrt {2 a+b \cosh (2 x)+b} E\left (i x\left |\frac {b}{a+b}\right .\right )}{\sqrt {\frac {2 a+b \cosh (2 x)+b}{a+b}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \cosh \relax (x)^{2} + a}, x\right ) \]
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 \[ \int \sqrt {b \cosh \relax (x)^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 114, normalized size = 2.33 \[ \frac {\sqrt {\frac {a +b \left (\cosh ^{2}\relax (x )\right )}{a}}\, \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \left (a \EllipticF \left (\cosh \relax (x ) \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a}{b}}\right )+b \EllipticF \left (\cosh \relax (x ) \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a}{b}}\right )-b \EllipticE \left (\cosh \relax (x ) \sqrt {-\frac {b}{a}}, \sqrt {-\frac {a}{b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sinh \relax (x ) \sqrt {a +b \left (\cosh ^{2}\relax (x )\right )}} \]
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 \[ \int \sqrt {b \cosh \relax (x)^{2} + a}\,{d x} \]
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 \[ \int \sqrt {b\,{\mathrm {cosh}\relax (x)}^2+a} \,d x \]
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 \[ \int \sqrt {a + b \cosh ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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