Integrand size = 23, antiderivative size = 228 \[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=-\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}+\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{f}-\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f} \]
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Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3960, 3914, 3917, 4089} \[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}-\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{f}-\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f} \]
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Rule 3914
Rule 3917
Rule 3960
Rule 4089
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f}+\frac {1}{2} (3 b) \int \sec (e+f x) \sqrt {a+b \sec (e+f x)} \, dx \\ & = -\frac {\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f}+\frac {1}{2} (3 (a-b) b) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx+\frac {1}{2} \left (3 b^2\right ) \int \frac {\sec (e+f x) (1+\sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx \\ & = -\frac {3 (a-b) \sqrt {a+b} \cot (e+f x) E\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 {3 (a-b) \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) (a+b \sec (e+f x))^{3/2}}{f} \\ \end{align*}
Time = 11.45 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.21 \[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\frac {\cos (e+f x) (a+b \sec (e+f x))^{3/2} ((-b-a \cos (e+f x)) \csc (e+f x)+3 b \sin (e+f x))}{f (b+a \cos (e+f x))}+\frac {3 b (a+b \sec (e+f x))^{3/2} \left (-\frac {(a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left (E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )-\operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}-(b+a \cos (e+f x)) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b+a \cos (e+f x))^2 \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sec ^{\frac {3}{2}}(e+f x) \sqrt {\cos ^2\left (\frac {1}{2} (e+f x)\right ) \sec (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(762\) vs. \(2(208)=416\).
Time = 9.18 (sec) , antiderivative size = 763, normalized size of antiderivative = 3.35
method | result | size |
default | \(-\frac {\sqrt {a +b \sec \left (f x +e \right )}\, \left (3 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a b \cos \left (f x +e \right )+3 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, b^{2} \cos \left (f x +e \right )-3 \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a b \cos \left (f x +e \right )-3 \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) b^{2} \cos \left (f x +e \right )+3 \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\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}}\, a b +3 \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\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}}\, b^{2}-3 \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticE}\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}}\, a b -3 \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticE}\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}}\, b^{2}+a^{2} \cos \left (f x +e \right ) \cot \left (f x +e \right )+3 a b \cos \left (f x +e \right ) \cot \left (f x +e \right )-a b \cot \left (f x +e \right )+3 b^{2} \cot \left (f x +e \right )-2 b^{2} \csc \left (f x +e \right )\right )}{f \left (b +a \cos \left (f x +e \right )\right )}\) | \(763\) |
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\[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{2} \,d x } \]
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\[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^2} \,d x \]
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