Integrand size = 25, antiderivative size = 131 \[ \int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx=\frac {b^2 d^2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{2 f \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}-\frac {b d^2 (b \tan (e+f x))^{3/2}}{2 f \sqrt {d \sec (e+f x)}}+\frac {b (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}{3 f} \]
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Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2691, 2693, 2696, 2721, 2719} \[ \int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx=\frac {b^2 d^2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{2 f \sqrt {\sin (e+f x)} \sqrt {d \sec (e+f x)}}-\frac {b d^2 (b \tan (e+f x))^{3/2}}{2 f \sqrt {d \sec (e+f x)}}+\frac {b (b \tan (e+f x))^{3/2} (d \sec (e+f x))^{3/2}}{3 f} \]
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Rule 2691
Rule 2693
Rule 2696
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {b (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}{3 f}-\frac {1}{2} b^2 \int (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx \\ & = -\frac {b d^2 (b \tan (e+f x))^{3/2}}{2 f \sqrt {d \sec (e+f x)}}+\frac {b (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}{3 f}+\frac {1}{4} \left (b^2 d^2\right ) \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {d \sec (e+f x)}} \, dx \\ & = -\frac {b d^2 (b \tan (e+f x))^{3/2}}{2 f \sqrt {d \sec (e+f x)}}+\frac {b (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}{3 f}+\frac {\left (b^2 d^2 \sqrt {b \tan (e+f x)}\right ) \int \sqrt {b \sin (e+f x)} \, dx}{4 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}} \\ & = -\frac {b d^2 (b \tan (e+f x))^{3/2}}{2 f \sqrt {d \sec (e+f x)}}+\frac {b (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}{3 f}+\frac {\left (b^2 d^2 \sqrt {b \tan (e+f x)}\right ) \int \sqrt {\sin (e+f x)} \, dx}{4 \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}} \\ & = \frac {b^2 d^2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{2 f \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}-\frac {b d^2 (b \tan (e+f x))^{3/2}}{2 f \sqrt {d \sec (e+f x)}}+\frac {b (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{3/2}}{3 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.74 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61 \[ \int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx=\frac {b d^2 \left (-3+2 \sec ^2(e+f x)+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {7}{4},-\tan ^2(e+f x)\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) (b \tan (e+f x))^{3/2}}{6 f \sqrt {d \sec (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 1.89 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.72
method | result | size |
default | \(\frac {\left (\sec ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right ) \left (-6 \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 (\csc \left (f x +e \right )-\cot \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 ) \left (\cos ^{4}\left (f x +e \right )\right )+3 \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 (\csc \left (f x +e \right )-\cot \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 ) \left (\cos ^{4}\left (f x +e \right )\right )-6 \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 (\csc \left (f x +e \right )-\cot \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 ) \left (\cos ^{3}\left (f x +e \right )\right )+3 \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 (\csc \left (f x +e \right )-\cot \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 ) \left (\cos ^{3}\left (f x +e \right )\right )+3 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right )-5 \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right )+2 \sqrt {2}\right ) d \sqrt {d \sec \left (f x +e \right )}\, b^{2} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}}{12 f}\) | \(487\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.18 \[ \int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx=\frac {3 i \, \sqrt {-2 i \, b d} b^{2} d \cos \left (f x + e\right )^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 3 i \, \sqrt {2 i \, b d} b^{2} d \cos \left (f x + e\right )^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (3 \, b^{2} d \cos \left (f x + e\right )^{2} - 2 \, b^{2} d\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{12 \, f \cos \left (f x + e\right )^{2}} \]
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Timed out. \[ \int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} \left (b \tan \left (f x + e\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int (d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2} \, dx=\int {\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]
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