Optimal. Leaf size=55 \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]
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Rubi [A] time = 0.0332414, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031, 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*}
Mathematica [C] time = 21.9791, size = 11586, normalized size = 210.65 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 1.319, size = 113, normalized size = 2.1 \begin{align*} 2\,{\frac{ \left ( 1+\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) \sqrt{{b}^{2}+{c}^{2}} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -1 \right ) }{\cos \left ( ex+d-\arctan \left ( -b,c \right ) \right ) e}{\frac{1}{\sqrt{{\frac{{b}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +{c}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +{b}^{2}+{c}^{2}}{\sqrt{{b}^{2}+{c}^{2}}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64086, size = 201, normalized size = 3.65 \begin{align*} \frac{2 \, \sqrt{b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt{b^{2} + c^{2}}}{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) - \sqrt{b^{2} + c^{2}}\right )}}{c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cos{\left (d + e x \right )} + c \sin{\left (d + e x \right )} + \sqrt{b^{2} + c^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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