Integrand size = 25, antiderivative size = 88 \[ \int \frac {(b \tan (e+f x))^{5/2}}{\sqrt {d \sec (e+f x)}} \, dx=-\frac {3 b^2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}+\frac {b (b \tan (e+f x))^{3/2}}{f \sqrt {d \sec (e+f x)}} \]
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Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2691, 2696, 2721, 2719} \[ \int \frac {(b \tan (e+f x))^{5/2}}{\sqrt {d \sec (e+f x)}} \, dx=\frac {b (b \tan (e+f x))^{3/2}}{f \sqrt {d \sec (e+f x)}}-\frac {3 b^2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {\sin (e+f x)} \sqrt {d \sec (e+f x)}} \]
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Rule 2691
Rule 2696
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {b (b \tan (e+f x))^{3/2}}{f \sqrt {d \sec (e+f x)}}-\frac {1}{2} \left (3 b^2\right ) \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {d \sec (e+f x)}} \, dx \\ & = \frac {b (b \tan (e+f x))^{3/2}}{f \sqrt {d \sec (e+f x)}}-\frac {\left (3 b^2 \sqrt {b \tan (e+f x)}\right ) \int \sqrt {b \sin (e+f x)} \, dx}{2 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}} \\ & = \frac {b (b \tan (e+f x))^{3/2}}{f \sqrt {d \sec (e+f x)}}-\frac {\left (3 b^2 \sqrt {b \tan (e+f x)}\right ) \int \sqrt {\sin (e+f x)} \, dx}{2 \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}} \\ & = -\frac {3 b^2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}+\frac {b (b \tan (e+f x))^{3/2}}{f \sqrt {d \sec (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.63 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int \frac {(b \tan (e+f x))^{5/2}}{\sqrt {d \sec (e+f x)}} \, dx=-\frac {b \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {7}{4},-\tan ^2(e+f x)\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) (b \tan (e+f x))^{3/2}}{f \sqrt {d \sec (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 2.21 (sec) , antiderivative size = 471, normalized size of antiderivative = 5.35
method | result | size |
default | \(\frac {b^{2} \sqrt {b \tan \left (f x +e \right )}\, \left (6 \cot \left (f x +e \right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )+i\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-3 \cot \left (f x +e \right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )+i\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+6 \csc \left (f x +e \right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )+i\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )-3 \csc \left (f x +e \right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )+i\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {2}\, \cot \left (f x +e \right )-3 \csc \left (f x +e \right ) \sqrt {2}+\sec \left (f x +e \right ) \csc \left (f x +e \right ) \sqrt {2}\right ) \sqrt {2}}{2 f \sqrt {d \sec \left (f x +e \right )}}\) | \(471\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.30 \[ \int \frac {(b \tan (e+f x))^{5/2}}{\sqrt {d \sec (e+f x)}} \, dx=\frac {2 \, b^{2} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 3 i \, \sqrt {-2 i \, b d} b^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 i \, \sqrt {2 i \, b d} b^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{2 \, d f} \]
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Timed out. \[ \int \frac {(b \tan (e+f x))^{5/2}}{\sqrt {d \sec (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(b \tan (e+f x))^{5/2}}{\sqrt {d \sec (e+f x)}} \, dx=\int { \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {d \sec \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {(b \tan (e+f x))^{5/2}}{\sqrt {d \sec (e+f x)}} \, dx=\int { \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {5}{2}}}{\sqrt {d \sec \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {(b \tan (e+f x))^{5/2}}{\sqrt {d \sec (e+f x)}} \, dx=\int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
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