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 = 0.10, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 204
Rule 2649
Rule 3115
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*}
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Mathematica [C] time = 34.47, size = 61904, normalized size = 680.26 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.15, size = 107, normalized size = 1.18 \[ \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) - \sqrt {b^{2} + c^{2}}}}{2 \, {\left (b^{2} + c^{2}\right )}^{\frac {1}{4}} {\left (c \cos \left (e x + d\right ) - b \sin \left (e x + d\right )\right )}}\right )}{{\left (b^{2} + c^{2}\right )}^{\frac {1}{4}} e} \]
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 = 0.30, size = 175, normalized size = 1.92 \[ \frac {\left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \sqrt {-\sqrt {b^{2}+c^{2}}\, \left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right )}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right )}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right )}{\left (b^{2}+c^{2}\right )^{\frac {1}{4}} \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )-b^{2}-c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e} \]
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 = -0.01 \[ \int \frac {1}{\sqrt {b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )-\sqrt {b^2+c^2}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} - \sqrt {b^{2} + c^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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