Optimal. Leaf size=42 \[ \frac {\sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{\sqrt {1+\frac {b \cos ^2(x)}{a}}} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, 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 = {3257, 3256}
\begin {gather*} \frac {\sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cos ^2(x)}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3256
Rule 3257
Rubi steps
\begin {align*} \int \sqrt {a+b \cos ^2(x)} \, dx &=\frac {\sqrt {a+b \cos ^2(x)} \int \sqrt {1+\frac {b \cos ^2(x)}{a}} \, dx}{\sqrt {1+\frac {b \cos ^2(x)}{a}}}\\ &=\frac {\sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{\sqrt {1+\frac {b \cos ^2(x)}{a}}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 46, normalized size = 1.10 \begin {gather*} \frac {\sqrt {2 a+b+b \cos (2 x)} E\left (x\left |\frac {b}{a+b}\right .\right )}{\sqrt {\frac {2 a+b+b \cos (2 x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 49, normalized size = 1.17
method | result | size |
default | \(-\frac {a \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \left (\cos ^{2}\left (x \right )\right )}{a}}\, \EllipticE \left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right )}{\sin \left (x \right ) \sqrt {a +b \left (\cos ^{2}\left (x \right )\right )}}\) | \(49\) |
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]
time = 0.10, size = 12, normalized size = 0.29 \begin {gather*} {\rm integral}\left (\sqrt {b \cos \left (x\right )^{2} + a}, x\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 \sqrt {a + b \cos ^{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 = -0.02 \begin {gather*} \int \sqrt {b\,{\cos \left (x\right )}^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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