Integrand size = 23, antiderivative size = 121 \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\frac {\sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f} \]
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Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3960, 3917} \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\frac {\sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}-\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f} \]
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Rule 3917
Rule 3960
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f}+\frac {1}{2} b \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx \\ & = \frac {\sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) \sqrt {a+b \sec (e+f x)}}{f} \\ \end{align*}
Time = 1.74 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\frac {-\left ((b+a \cos (e+f x)) \csc (e+f x) \sqrt {\frac {1}{1+\sec (e+f x)}}\right )+b \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}}{f \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}} \]
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Time = 6.64 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.78
method | result | size |
default | \(-\frac {\sqrt {a +b \sec \left (f x +e \right )}\, \left (\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, b \cos \left (f x +e \right )+\operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, b +a \cos \left (f x +e \right ) \cot \left (f x +e \right )+b \cot \left (f x +e \right )\right )}{f \left (b +a \cos \left (f x +e \right )\right )}\) | \(215\) |
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\[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2} \,d x } \]
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\[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int \sqrt {a + b \sec {\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}\, dx \]
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\[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2} \,d x } \]
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\[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \csc ^2(e+f x) \sqrt {a+b \sec (e+f x)} \, dx=\int \frac {\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^2} \,d x \]
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