Optimal. Leaf size=38 \[ \frac {2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {\cos (a+b x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, 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 = {2721, 2719}
\begin {gather*} \frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{b \sqrt {\cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rubi steps
\begin {align*} \int \sqrt {c \cos (a+b x)} \, dx &=\frac {\sqrt {c \cos (a+b x)} \int \sqrt {\cos (a+b x)} \, dx}{\sqrt {\cos (a+b x)}}\\ &=\frac {2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {\cos (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {\cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs.
\(2(60)=120\).
time = 0.07, size = 142, normalized size = 3.74
method | result | size |
default | \(\frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, c \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(142\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {c \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{-i \left (b x +a \right )}}}{b}-\frac {i \left (-\frac {2 \left (c \,{\mathrm e}^{2 i \left (b x +a \right )}+c \right )}{c \sqrt {{\mathrm e}^{i \left (b x +a \right )} \left (c \,{\mathrm e}^{2 i \left (b x +a \right )}+c \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {c \,{\mathrm e}^{3 i \left (b x +a \right )}+c \,{\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {c \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{-i \left (b x +a \right )}}\, \sqrt {c \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{i \left (b x +a \right )}}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}\) | \(299\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 63, normalized size = 1.66 \begin {gather*} \frac {i \, \sqrt {2} \sqrt {c} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - i \, \sqrt {2} \sqrt {c} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \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 {c \cos {\left (a + b 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 = -0.03 \begin {gather*} \int \sqrt {c\,\cos \left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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