Integrand size = 25, antiderivative size = 167 \[ \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {d \sec (e+f x)}} \, dx=-\frac {2 d \csc (e+f x) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2}}+\frac {b^{3/2} d \arctan \left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}}+\frac {b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2696, 2644, 327, 335, 218, 212, 209} \[ \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {d \sec (e+f x)}} \, dx=\frac {b^{3/2} d (b \tan (e+f x))^{3/2} \arctan \left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right )}{f (b \sin (e+f x))^{3/2} (d \sec (e+f x))^{3/2}}+\frac {b^{3/2} d (b \tan (e+f x))^{3/2} \text {arctanh}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right )}{f (b \sin (e+f x))^{3/2} (d \sec (e+f x))^{3/2}}-\frac {2 d \csc (e+f x) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2}} \]
[In]
[Out]
Rule 209
Rule 212
Rule 218
Rule 327
Rule 335
Rule 2644
Rule 2696
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d (b \tan (e+f x))^{3/2}\right ) \int \sec (e+f x) (b \sin (e+f x))^{3/2} \, dx}{(d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}} \\ & = \frac {\left (d (b \tan (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {x^{3/2}}{1-\frac {x^2}{b^2}} \, dx,x,b \sin (e+f x)\right )}{b f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}} \\ & = -\frac {2 d \csc (e+f x) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2}}+\frac {\left (b d (b \tan (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{b^2}\right )} \, dx,x,b \sin (e+f x)\right )}{f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}} \\ & = -\frac {2 d \csc (e+f x) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2}}+\frac {\left (2 b d (b \tan (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}} \\ & = -\frac {2 d \csc (e+f x) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2}}+\frac {\left (b^2 d (b \tan (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}}+\frac {\left (b^2 d (b \tan (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}} \\ & = -\frac {2 d \csc (e+f x) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2}}+\frac {b^{3/2} d \arctan \left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}}+\frac {b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) (b \tan (e+f x))^{3/2}}{f (d \sec (e+f x))^{3/2} (b \sin (e+f x))^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.70 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.40 \[ \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {d \sec (e+f x)}} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {5}{4},\frac {9}{4},-\tan ^2(e+f x)\right ) \sqrt [4]{\sec ^2(e+f x)} (b \tan (e+f x))^{5/2}}{5 b f \sqrt {d \sec (e+f x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(291\) vs. \(2(139)=278\).
Time = 19.50 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.75
method | result | size |
default | \(-\frac {\sin \left (f x +e \right ) \left (\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )\right ) \sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \arctan \left (\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )\right ) \cos \left (f x +e \right )+\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )\right )-\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \arctan \left (\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )\right )-2 \sin \left (f x +e \right )\right ) \sqrt {b \tan \left (f x +e \right )}\, b}{f \left (\cos \left (f x +e \right )-1\right ) \sqrt {d \sec \left (f x +e \right )}\, \left (\cos \left (f x +e \right )+1\right )}\) | \(292\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (139) = 278\).
Time = 0.55 (sec) , antiderivative size = 741, normalized size of antiderivative = 4.44 \[ \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {d \sec (e+f x)}} \, dx=\left [-\frac {2 \, b d \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (\cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) + 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) + 4\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {-\frac {b}{d}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{4 \, {\left (b \cos \left (f x + e\right )^{2} - {\left (b \cos \left (f x + e\right ) + b\right )} \sin \left (f x + e\right ) - b\right )}}\right ) - b d \sqrt {-\frac {b}{d}} \log \left (\frac {b \cos \left (f x + e\right )^{4} - 72 \, b \cos \left (f x + e\right )^{2} + 8 \, {\left (7 \, \cos \left (f x + e\right )^{3} - {\left (\cos \left (f x + e\right )^{3} - 8 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) - 8 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {-\frac {b}{d}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} + 28 \, {\left (b \cos \left (f x + e\right )^{2} - 2 \, b\right )} \sin \left (f x + e\right ) + 72 \, b}{\cos \left (f x + e\right )^{4} - 8 \, \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} - 2\right )} \sin \left (f x + e\right ) + 8}\right ) + 16 \, b \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{8 \, d f}, \frac {2 \, b d \sqrt {\frac {b}{d}} \arctan \left (\frac {{\left (\cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) + 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) + 4\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {b}{d}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{4 \, {\left (b \cos \left (f x + e\right )^{2} + {\left (b \cos \left (f x + e\right ) + b\right )} \sin \left (f x + e\right ) - b\right )}}\right ) + b d \sqrt {\frac {b}{d}} \log \left (\frac {b \cos \left (f x + e\right )^{4} - 72 \, b \cos \left (f x + e\right )^{2} - 8 \, {\left (7 \, \cos \left (f x + e\right )^{3} + {\left (\cos \left (f x + e\right )^{3} - 8 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) - 8 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {b}{d}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} - 28 \, {\left (b \cos \left (f x + e\right )^{2} - 2 \, b\right )} \sin \left (f x + e\right ) + 72 \, b}{\cos \left (f x + e\right )^{4} - 8 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} - 2\right )} \sin \left (f x + e\right ) + 8}\right ) - 16 \, b \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{8 \, d f}\right ] \]
[In]
[Out]
\[ \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {d \sec (e+f x)}} \, dx=\int \frac {\left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {d \sec {\left (e + f x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {d \sec (e+f x)}} \, dx=\int { \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {d \sec \left (f x + e\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {d \sec (e+f x)}} \, dx=\int { \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {d \sec \left (f x + e\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {d \sec (e+f x)}} \, dx=\int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
[In]
[Out]