Integrand size = 25, antiderivative size = 132 \[ \int \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \, dx=-\frac {\sqrt {b} d \arctan \left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}}+\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}} \]
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Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2696, 2644, 335, 304, 209, 212} \[ \int \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \, dx=\frac {\sqrt {b} d \sqrt {b \tan (e+f x)} \text {arctanh}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right )}{f \sqrt {b \sin (e+f x)} \sqrt {d \sec (e+f x)}}-\frac {\sqrt {b} d \sqrt {b \tan (e+f x)} \arctan \left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right )}{f \sqrt {b \sin (e+f x)} \sqrt {d \sec (e+f x)}} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 2644
Rule 2696
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d \sqrt {b \tan (e+f x)}\right ) \int \sec (e+f x) \sqrt {b \sin (e+f x)} \, dx}{\sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}} \\ & = \frac {\left (d \sqrt {b \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{b^2}} \, dx,x,b \sin (e+f x)\right )}{b f \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}} \\ & = \frac {\left (2 d \sqrt {b \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{b f \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}} \\ & = \frac {\left (b d \sqrt {b \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{f \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}}-\frac {\left (b d \sqrt {b \tan (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{f \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}} \\ & = -\frac {\sqrt {b} d \arctan \left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}}+\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.77 \[ \int \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \, dx=\frac {\left (-\arctan \left (\frac {\sqrt {\tan (e+f x)}}{\sqrt [4]{\sec ^2(e+f x)}}\right )+\text {arctanh}\left (\frac {\sqrt {\tan (e+f x)}}{\sqrt [4]{\sec ^2(e+f x)}}\right )\right ) \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)}}{f \sqrt [4]{\sec ^2(e+f x)} \sqrt {\tan (e+f x)}} \]
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Time = 17.51 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\sqrt {d \sec \left (f x +e \right )}\, \sqrt {b \tan \left (f x +e \right )}\, \left (\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )\right )+\arctan \left (\sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )\right )\right ) \cos \left (f x +e \right )}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(129\) |
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Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (108) = 216\).
Time = 0.38 (sec) , antiderivative size = 654, normalized size of antiderivative = 4.95 \[ \int \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \, dx=\left [-\frac {2 \, \sqrt {-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 {-b d} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{4 \, {\left (b d \cos \left (f x + e\right )^{2} - b d - {\left (b d \cos \left (f x + e\right ) + b d\right )} \sin \left (f x + e\right )\right )}}\right ) - \sqrt {-b d} \log \left (\frac {b d \cos \left (f x + e\right )^{4} - 72 \, b d \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 {-b d} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} + 72 \, b d + 28 \, {\left (b d \cos \left (f x + e\right )^{2} - 2 \, b d\right )} \sin \left (f x + e\right )}{\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 )}{8 \, f}, -\frac {2 \, \sqrt {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 {b d} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{4 \, {\left (b d \cos \left (f x + e\right )^{2} - b d + {\left (b d \cos \left (f x + e\right ) + b d\right )} \sin \left (f x + e\right )\right )}}\right ) - \sqrt {b d} \log \left (\frac {b d \cos \left (f x + e\right )^{4} - 72 \, b d \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 {b d} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} + 72 \, b d - 28 \, {\left (b d \cos \left (f x + e\right )^{2} - 2 \, b d\right )} \sin \left (f x + e\right )}{\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 )}{8 \, f}\right ] \]
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\[ \int \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \, dx=\int \sqrt {b \tan {\left (e + f x \right )}} \sqrt {d \sec {\left (e + f x \right )}}\, dx \]
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\[ \int \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \, dx=\int { \sqrt {d \sec \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )} \,d x } \]
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\[ \int \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \, dx=\int { \sqrt {d \sec \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )} \,d x } \]
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Timed out. \[ \int \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \, dx=\int \sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}} \,d x \]
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