Integrand size = 14, antiderivative size = 255 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)} \]
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Time = 0.13 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3739, 3554, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\sqrt {b \tan ^3(e+f x)} \log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\sqrt {b \tan ^3(e+f x)} \log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b \tan ^3(e+f x)} \int \tan ^{\frac {3}{2}}(e+f x) \, dx}{\tan ^{\frac {3}{2}}(e+f x)} \\ & = \frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}-\frac {\sqrt {b \tan ^3(e+f x)} \int \frac {1}{\sqrt {\tan (e+f x)}} \, dx}{\tan ^{\frac {3}{2}}(e+f x)} \\ & = \frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}-\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f \tan ^{\frac {3}{2}}(e+f x)} \\ & = \frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}-\frac {\left (2 \sqrt {b \tan ^3(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)} \\ & = \frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}-\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (e+f x)}\right )}{f \tan ^{\frac {3}{2}}(e+f x)} \\ & = \frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}-\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (e+f x)}\right )}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)} \\ & = \frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\sqrt {b \tan ^3(e+f x)} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)} \\ & = \frac {2 \cot (e+f x) \sqrt {b \tan ^3(e+f x)}}{f}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{\sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \sqrt {b \tan ^3(e+f x)}}{2 \sqrt {2} f \tan ^{\frac {3}{2}}(e+f x)} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.64 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\frac {\left (\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )}{2 \sqrt {2}}+2 \sqrt {\tan (e+f x)}\right ) \sqrt {b \tan ^3(e+f x)}}{f \tan ^{\frac {3}{2}}(e+f x)} \]
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Time = 0.04 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \tan \left (f x +e \right )^{3}}\, \left (\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {b \tan \left (f x +e \right )}\right )}{4 f \tan \left (f x +e \right ) \sqrt {b \tan \left (f x +e \right )}}\) | \(208\) |
default | \(-\frac {\sqrt {b \tan \left (f x +e \right )^{3}}\, \left (\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {b \tan \left (f x +e \right )}\right )}{4 f \tan \left (f x +e \right ) \sqrt {b \tan \left (f x +e \right )}}\) | \(208\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.07 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=-\frac {f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) - f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) - \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) + i \, f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) - i \, f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, f \left (-\frac {b^{2}}{f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) - 4 \, \sqrt {b \tan \left (f x + e\right )^{3}}}{2 \, f \tan \left (f x + e\right )} \]
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\[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\int \sqrt {b \tan ^{3}{\left (e + f x \right )}}\, dx \]
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Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.52 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=-\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + \sqrt {2} \sqrt {b} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) - \sqrt {2} \sqrt {b} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) - 8 \, \sqrt {b} \sqrt {\tan \left (f x + e\right )}}{4 \, f} \]
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Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.76 \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{f} + \frac {2 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{f} + \frac {\sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (f x + e\right ) + \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{f} - \frac {\sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (f x + e\right ) - \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{f} - \frac {8 \, \sqrt {b \tan \left (f x + e\right )}}{f}\right )} \mathrm {sgn}\left (\tan \left (f x + e\right )\right ) \]
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Timed out. \[ \int \sqrt {b \tan ^3(e+f x)} \, dx=\int \sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^3} \,d x \]
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