Integrand size = 32, antiderivative size = 88 \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt {2} \sqrt {\sqrt {b^2+c^2}+\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{\sqrt [4]{b^2+c^2} e} \]
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Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3194, 2728, 212} \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt {2} \sqrt {\sqrt {b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )+\sqrt {b^2+c^2}}}\right )}{e \sqrt [4]{b^2+c^2}} \]
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Rule 212
Rule 2728
Rule 3194
Rubi steps \begin{align*} \text {integral}& = \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 \text {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} \text {arctanh}\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*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 54.77 (sec) , antiderivative size = 63264, normalized size of antiderivative = 718.91 \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(171\) vs. \(2(75)=150\).
Time = 5.57 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.95
method | result | size |
default | \(-\frac {\left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )+1\right ) \sqrt {-\sqrt {b^{2}+c^{2}}\, \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right )}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right )}{\left (b^{2}+c^{2}\right )^{\frac {1}{4}} \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+b^{2}+c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e}\) | \(172\) |
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Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (81) = 162\).
Time = 0.42 (sec) , antiderivative size = 349, normalized size of antiderivative = 3.97 \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\frac {\sqrt {2} \log \left (\frac {{\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} + {\left (b^{2} c + 4 \, c^{3}\right )} \cos \left (e x + d\right ) - {\left (3 \, b^{3} + 4 \, b c^{2} + {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right ) + \frac {2 \, \sqrt {2} {\left (2 \, {\left (b^{3} + b c^{2}\right )} \cos \left (e x + d\right ) + 2 \, {\left (b^{2} c + c^{3}\right )} \sin \left (e x + d\right ) - {\left (2 \, b c \cos \left (e x + d\right ) \sin \left (e x + d\right ) + {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} + b^{2} + 2 \, c^{2}\right )} \sqrt {b^{2} + c^{2}}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}}}{{\left (b^{2} + c^{2}\right )}^{\frac {1}{4}}} - 4 \, {\left (2 \, b c \cos \left (e x + d\right )^{2} - {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) - b c\right )} \sqrt {b^{2} + c^{2}}}{3 \, b^{2} c \cos \left (e x + d\right ) - {\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} - {\left (b^{3} - {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )}\right )}{2 \, {\left (b^{2} + c^{2}\right )}^{\frac {1}{4}} e} \]
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\[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} + \sqrt {b^{2} + c^{2}}}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )+\sqrt {b^2+c^2}}} \,d x \]
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