Optimal. Leaf size=108 \[ \frac {2 \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3127, 2661} \[ \frac {2 \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \]
Antiderivative was successfully verified.
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Rule 2661
Rule 3127
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \, dx &=\frac {\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}} \, dx}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}\\ &=\frac {2 F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}\\ \end {align*}
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Mathematica [C] time = 0.56, size = 285, normalized size = 2.64 \[ \frac {2 \sec \left (\tan ^{-1}\left (\frac {b}{c}\right )+d+e x\right ) \sqrt {-\frac {c \sqrt {\frac {b^2}{c^2}+1} \left (\sin \left (\tan ^{-1}\left (\frac {b}{c}\right )+d+e x\right )-1\right )}{a+c \sqrt {\frac {b^2}{c^2}+1}}} \sqrt {\frac {c \sqrt {\frac {b^2}{c^2}+1} \left (\sin \left (\tan ^{-1}\left (\frac {b}{c}\right )+d+e x\right )+1\right )}{c \sqrt {\frac {b^2}{c^2}+1}-a}} \sqrt {a+c \sqrt {\frac {b^2}{c^2}+1} \sin \left (\tan ^{-1}\left (\frac {b}{c}\right )+d+e x\right )} F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};\frac {a+\sqrt {\frac {b^2}{c^2}+1} c \sin \left (d+e x+\tan ^{-1}\left (\frac {b}{c}\right )\right )}{a-\sqrt {\frac {b^2}{c^2}+1} c},\frac {a+\sqrt {\frac {b^2}{c^2}+1} c \sin \left (d+e x+\tan ^{-1}\left (\frac {b}{c}\right )\right )}{a+\sqrt {\frac {b^2}{c^2}+1} c}\right )}{c e \sqrt {\frac {b^2}{c^2}+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 3.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 303, normalized size = 2.81 \[ -\frac {2 \left (-a +\sqrt {b^{2}+c^{2}}\right ) \sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a}{-a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {-\frac {\left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {\frac {\left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right ) \sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\, \EllipticF \left (\sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a}{-a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {-a +\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right )}{\sqrt {b^{2}+c^{2}}\, \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a \sqrt {b^{2}+c^{2}}}{\sqrt {b^{2}+c^{2}}}}\, e} \]
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 \[ \int \frac {1}{\sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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