Optimal. Leaf size=75 \[ \frac {2 \sqrt {a \cos (c+d x)+b \sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{d \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}} \]
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Rubi [A] time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3078, 2639} \[ \frac {2 \sqrt {a \cos (c+d x)+b \sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{d \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3078
Rubi steps
\begin {align*} \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx &=\frac {\sqrt {a \cos (c+d x)+b \sin (c+d x)} \int \sqrt {\cos \left (c+d x-\tan ^{-1}(a,b)\right )} \, dx}{\sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}\\ &=\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right ) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{d \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}\\ \end {align*}
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Mathematica [C] time = 1.10, size = 268, normalized size = 3.57 \[ \frac {\cos \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right ) \left (\sqrt {\sin ^2\left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )} \left (b \left (a^2+b^2\right ) \sin \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )-2 a \left (a^2+b^2\right ) \cos \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )+2 a^2 \sqrt {\frac {b^2}{a^2}+1} \sqrt {a \cos (c+d x)+b \sin (c+d x)} \sqrt {a \sqrt {\frac {b^2}{a^2}+1} \cos \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )}\right )-b \left (a^2+b^2\right ) \sin \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2\left (c+d x-\tan ^{-1}\left (\frac {b}{a}\right )\right )\right )\right )}{b d \sqrt {\sin ^2\left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )} \left (a \sqrt {\frac {b^2}{a^2}+1} \cos \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 159, normalized size = 2.12 \[ -\frac {\sqrt {a^{2}+b^{2}}\, \sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \sqrt {-2 \sin \left (d x +c -\arctan \left (-a , b\right )\right )+2}\, \sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \left (2 \EllipticE \left (\sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {\sin \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {a^{2}+b^{2}}}\, d} \]
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 \[ \int \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a\,\cos \left (c+d\,x\right )+b\,\sin \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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