Optimal. Leaf size=91 \[ -\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )-\sqrt{b^2+c^2}}}\right )}{e \sqrt [4]{b^2+c^2}} \]
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Rubi [A] time = 0.0978455, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {3115, 2649, 204} \[ -\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )-\sqrt{b^2+c^2}}}\right )}{e \sqrt [4]{b^2+c^2}} \]
Antiderivative was successfully verified.
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Rule 3115
Rule 2649
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx &=\int \frac{1}{\sqrt{-\sqrt{b^2+c^2}+\sqrt{b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}} \, dx\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-2 \sqrt{b^2+c^2}-x^2} \, dx,x,-\frac{\sqrt{b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt{-\sqrt{b^2+c^2}+\sqrt{b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{e}\\ &=-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt [4]{b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt{2} \sqrt{-\sqrt{b^2+c^2}+\sqrt{b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{\sqrt [4]{b^2+c^2} e}\\ \end{align*}
Mathematica [C] time = 33.8815, size = 61904, normalized size = 680.26 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 1.314, size = 175, normalized size = 1.9 \begin{align*}{\frac{ \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -1 \right ) \sqrt{2}}{\cos \left ( ex+d-\arctan \left ( -b,c \right ) \right ) e}\sqrt{-\sqrt{{b}^{2}+{c}^{2}} \left ( 1+\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) }\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-\sqrt{{b}^{2}+{c}^{2}} \left ( 1+\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) }{\frac{1}{\sqrt [4]{{b}^{2}+{c}^{2}}}}} \right ){\frac{1}{\sqrt [4]{{b}^{2}+{c}^{2}}}}{\frac{1}{\sqrt{{({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}){\frac{1}{\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 [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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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