Optimal. Leaf size=79 \[ \frac{\sqrt{a+(b \sin (e+f x)+c \cos (e+f x))^2} E\left (e+f x+\tan ^{-1}(b,c)|-\frac{b^2+c^2}{a}\right )}{f \sqrt{\frac{(b \sin (e+f x)+c \cos (e+f x))^2}{a}+1}} \]
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Rubi [F] time = 0.698582, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+\frac{(c+b x)^2}{1+x^2}}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{i \sqrt{a+\frac{(c+b x)^2}{1+x^2}}}{2 (i-x)}+\frac{i \sqrt{a+\frac{(c+b x)^2}{1+x^2}}}{2 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{\sqrt{a+\frac{(c+b x)^2}{1+x^2}}}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac{i \operatorname{Subst}\left (\int \frac{\sqrt{a+\frac{(c+b x)^2}{1+x^2}}}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f}\\ \end{align*}
Mathematica [B] time = 1.63885, size = 325, normalized size = 4.11 \[ -\frac{\left (\left (b^2-c^2\right ) \sin (2 (e+f x))+2 b c \cos (2 (e+f x))\right ) \sqrt{2 a+\left (c^2-b^2\right ) \cos (2 (e+f x))+b^2+2 b c \sin (2 (e+f x))+c^2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (b^2-c^2\right ) \cos (2 (e+f x))-2 b c \sin (2 (e+f x))+\sqrt{\left (b^2+c^2\right )^2}}{\sqrt{\left (b^2+c^2\right )^2}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{\left (b^2+c^2\right )^2}}{b^2+c^2+2 a+\sqrt{\left (b^2+c^2\right )^2}}\right )}{\sqrt{2} f \sqrt{\left (b^2+c^2\right )^2} \sqrt{\frac{\left (\left (b^2-c^2\right ) \sin (2 (e+f x))+2 b c \cos (2 (e+f x))\right )^2}{\left (b^2+c^2\right )^2}} \sqrt{\frac{2 a+\left (c^2-b^2\right ) \cos (2 (e+f x))+b^2+2 b c \sin (2 (e+f x))+c^2}{2 a+\sqrt{\left (b^2+c^2\right )^2}+b^2+c^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.51, size = 4061599, normalized size = 51412.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\left (c \cos \left (f x + e\right ) + b \sin \left (f x + e\right )\right )}^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{2 \, b c \cos \left (f x + e\right ) \sin \left (f x + e\right ) -{\left (b^{2} - c^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2} + a}, x\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{a + \left (b \sin{\left (e + f x \right )} + c \cos{\left (e + f x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\left (c \cos \left (f x + e\right ) + b \sin \left (f x + e\right )\right )}^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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