Integrand size = 21, antiderivative size = 75 \[ \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right ) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{d \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3157, 2719} \[ \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {2 \sqrt {a \cos (c+d x)+b \sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{d \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}} \]
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Rule 2719
Rule 3157
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a \cos (c+d x)+b \sin (c+d x)} \int \sqrt {\cos \left (c+d x-\tan ^{-1}(a,b)\right )} \, dx}{\sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}} \\ & = \frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right ) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{d \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.88 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.57 \[ \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\cos \left (c+d x-\arctan \left (\frac {b}{a}\right )\right ) \left (-b \left (a^2+b^2\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2\left (c+d x-\arctan \left (\frac {b}{a}\right )\right )\right ) \sin \left (c+d x-\arctan \left (\frac {b}{a}\right )\right )+\sqrt {\sin ^2\left (c+d x-\arctan \left (\frac {b}{a}\right )\right )} \left (-2 a \left (a^2+b^2\right ) \cos \left (c+d x-\arctan \left (\frac {b}{a}\right )\right )+2 a^2 \sqrt {1+\frac {b^2}{a^2}} \sqrt {a \sqrt {1+\frac {b^2}{a^2}} \cos \left (c+d x-\arctan \left (\frac {b}{a}\right )\right )} \sqrt {a \cos (c+d x)+b \sin (c+d x)}+b \left (a^2+b^2\right ) \sin \left (c+d x-\arctan \left (\frac {b}{a}\right )\right )\right )\right )}{b d \left (a \sqrt {1+\frac {b^2}{a^2}} \cos \left (c+d x-\arctan \left (\frac {b}{a}\right )\right )\right )^{3/2} \sqrt {\sin ^2\left (c+d x-\arctan \left (\frac {b}{a}\right )\right )}} \]
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Time = 3.51 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.17
method | result | size |
default | \(-\frac {\sqrt {a^{2}+b^{2}}\, \sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )+1}\, \sqrt {2 \sin \left (d x +c -\arctan \left (-a , b\right )\right )+2}\, \sqrt {\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {\sin \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {a^{2}+b^{2}}}\, d}\) | \(163\) |
risch | \(\text {Expression too large to display}\) | \(1175\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.17 \[ \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {i \, \sqrt {2} \sqrt {a - i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - i \, \sqrt {2} \sqrt {a + i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{d} \]
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\[ \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx=\int \sqrt {a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx=\int { \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx=\int { \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx=\int \sqrt {a\,\cos \left (c+d\,x\right )+b\,\sin \left (c+d\,x\right )} \,d x \]
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