Optimal. Leaf size=57 \[ \frac {2 \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, 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 = {2734, 2732}
\begin {gather*} \frac {2 \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rubi steps
\begin {align*} \int \sqrt {a+b \cos (c+d x)} \, dx &=\frac {\sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\\ &=\frac {2 \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 57, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \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. \(169\) vs.
\(2(82)=164\).
time = 0.26, size = 170, normalized size = 2.98
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) \left (a -b \right )}{\sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b}\, d}\) | \(170\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right ) {\mathrm e}^{-i \left (d x +c \right )}}}{d}-\frac {i \left (\frac {2 a \left (a +\sqrt {a^{2}-b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}}\, \sqrt {-\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{a +\sqrt {a^{2}-b^{2}}}}\, \EllipticF \left (\sqrt {\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )}{b \sqrt {b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 a \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )} b}}+b \left (-\frac {2 \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b \sqrt {\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right ) {\mathrm e}^{i \left (d x +c \right )}}}+\frac {2 \left (a +\sqrt {a^{2}-b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}}\, \sqrt {-\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{a +\sqrt {a^{2}-b^{2}}}}\, \left (\left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right ) \EllipticE \left (\sqrt {\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )+\frac {\left (-a +\sqrt {a^{2}-b^{2}}\right ) \EllipticF \left (\sqrt {\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )}{b}\right )}{b \sqrt {b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 a \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )} b}}\right )\right ) \sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right ) {\mathrm e}^{-i \left (d x +c \right )}}\, \sqrt {\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right ) {\mathrm e}^{i \left (d x +c \right )}}}{d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}\) | \(1046\) |
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 = 0.13, size = 355, normalized size = 6.23 \begin {gather*} \frac {-i \, \sqrt {2} a \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + i \, \sqrt {2} a \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + 3 i \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 i \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right )}{3 \, b d} \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 {\left (c + d 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 {a+b\,\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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