Optimal. Leaf size=180 \[ \frac {2 (d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} F\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 e \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}} \]
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Rubi [A] time = 0.19, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3149, 3119, 2653, 3127, 2661} \[ \frac {2 (d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} F\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 e \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 3119
Rule 3127
Rule 3149
Rubi steps
\begin {align*} \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx &=e \int \sqrt {a+b \cos (x)+c \sin (x)} \, dx+(d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx\\ &=\frac {\left (e \sqrt {a+b \cos (x)+c \sin (x)}\right ) \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}} \, dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+\frac {\left ((d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}} \, dx}{\sqrt {a+b \cos (x)+c \sin (x)}}\\ &=\frac {2 e E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+\frac {2 (d-a e) F\left (\frac {1}{2} \left (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 (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{\sqrt {a+b \cos (x)+c \sin (x)}}\\ \end {align*}
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Mathematica [C] time = 6.31, size = 1319, normalized size = 7.33 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b e \cos \relax (x) + c e \sin \relax (x) + d}{\sqrt {b \cos \relax (x) + c \sin \relax (x) + a}}, x\right ) \]
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 \[ \int \frac {b e \cos \relax (x) + c e \sin \relax (x) + d}{\sqrt {b \cos \relax (x) + c \sin \relax (x) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.65, size = 777, normalized size = 4.32 \[ \frac {\sqrt {-\frac {\left (-b^{2} \sin \left (x -\arctan \left (-b , c\right )\right )-c^{2} \sin \left (x -\arctan \left (-b , c\right )\right )-a \sqrt {b^{2}+c^{2}}\right ) \left (\cos ^{2}\left (x -\arctan \left (-b , c\right )\right )\right )}{\sqrt {b^{2}+c^{2}}}}\, \left (\frac {2 \left (b^{2} e +c^{2} e \right ) \left (\frac {a}{\sqrt {b^{2}+c^{2}}}-1\right ) \sqrt {\frac {-\sqrt {b^{2}+c^{2}}\, \sin \left (x -\arctan \left (-b , c\right )\right )-a}{-a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {\frac {\left (-\sin \left (x -\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 (x -\arctan \left (-b , c\right )\right )\right ) \sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\, \left (\left (-\frac {a}{\sqrt {b^{2}+c^{2}}}-1\right ) \EllipticE \left (\sqrt {\frac {-\sqrt {b^{2}+c^{2}}\, \sin \left (x -\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 )+\EllipticF \left (\sqrt {\frac {-\sqrt {b^{2}+c^{2}}\, \sin \left (x -\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 )\right )}{\sqrt {\left (\cos ^{2}\left (x -\arctan \left (-b , c\right )\right )\right ) \left (\sqrt {b^{2}+c^{2}}\, \sin \left (x -\arctan \left (-b , c\right )\right )+a \right )}}+\frac {2 d \sqrt {b^{2}+c^{2}}\, \left (\frac {a}{\sqrt {b^{2}+c^{2}}}-1\right ) \sqrt {\frac {-\sqrt {b^{2}+c^{2}}\, \sin \left (x -\arctan \left (-b , c\right )\right )-a}{-a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {\frac {\left (-\sin \left (x -\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 (x -\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 (x -\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 {-\frac {\left (-b^{2} \sin \left (x -\arctan \left (-b , c\right )\right )-c^{2} \sin \left (x -\arctan \left (-b , c\right )\right )-a \sqrt {b^{2}+c^{2}}\right ) \left (\cos ^{2}\left (x -\arctan \left (-b , c\right )\right )\right )}{\sqrt {b^{2}+c^{2}}}}}\right )}{\sqrt {b^{2}+c^{2}}\, \cos \left (x -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (x -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (x -\arctan \left (-b , c\right )\right )+a \sqrt {b^{2}+c^{2}}}{\sqrt {b^{2}+c^{2}}}}} \]
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 \[ \int \frac {b e \cos \relax (x) + c e \sin \relax (x) + d}{\sqrt {b \cos \relax (x) + c \sin \relax (x) + a}}\,{d x} \]
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 \[ \int \frac {d+b\,e\,\cos \relax (x)+c\,e\,\sin \relax (x)}{\sqrt {a+b\,\cos \relax (x)+c\,\sin \relax (x)}} \,d x \]
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 \[ \int \frac {b e \cos {\relax (x )} + c e \sin {\relax (x )} + d}{\sqrt {a + b \cos {\relax (x )} + c \sin {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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