Optimal. Leaf size=57 \[ -\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {3112} \[ -\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]
Antiderivative was successfully verified.
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Rule 3112
Rubi steps
\begin {align*} \int \sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx &=-\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt {-\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}\\ \end {align*}
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Mathematica [C] time = 21.95, size = 11415, normalized size = 200.26 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 2.89, size = 80, normalized size = 1.40 \[ \frac {2 \, {\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) - \sqrt {b^{2} + c^{2}}}}{c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 117, normalized size = 2.05 \[ \frac {2 \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \sqrt {b^{2}+c^{2}}\, \left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right )}{\cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )-b^{2}-c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \sqrt {b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )-\sqrt {b^2+c^2}} \,d x \]
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 \[ \int \sqrt {b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} - \sqrt {b^{2} + c^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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