Integrand size = 16, antiderivative size = 51 \[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3257, 3256} \[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}} \]
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Rule 3256
Rule 3257
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b \sin ^2(e+f x)} \int \sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \, dx}{\sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \\ & = \frac {E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20 \[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {a \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
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Time = 1.38 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.39
method | result | size |
default | \(\frac {a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )}{\cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(71\) |
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\[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \,d x } \]
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\[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}}\, dx \]
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\[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \,d x } \]
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\[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \,d x } \]
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Timed out. \[ \int \sqrt {a+b \sin ^2(e+f x)} \, dx=\left \{\begin {array}{cl} \frac {\sqrt {a}\,\mathrm {E}\left (e+f\,x\middle |-\frac {b}{a}\right )}{f} & \text {\ if\ \ }0<a\\ \int \sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x & \text {\ if\ \ }\neg 0<a \end {array}\right . \]
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