Integrand size = 25, antiderivative size = 88 \[ \int \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}-\frac {\sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b f} \]
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Time = 0.14 (sec) , antiderivative size = 88, 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 = {3751, 457, 81, 52, 65, 214} \[ \int \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b f}-\frac {\sqrt {a+b \tan ^2(e+f x)}}{f} \]
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3 \sqrt {a+b x^2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {x \sqrt {a+b x}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f} \\ & = \frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b f}-\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f} \\ & = -\frac {\sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b f}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f} \\ & = -\frac {\sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b f}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^2(e+f x)}\right )}{b f} \\ & = \frac {\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}-\frac {\sqrt {a+b \tan ^2(e+f x)}}{f}+\frac {\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b f} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {3 \sqrt {a-b} b \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+\sqrt {a+b \tan ^2(e+f x)} \left (a-3 b+b \tan ^2(e+f x)\right )}{3 b f} \]
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Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.30
method | result | size |
derivativedivides | \(\frac {\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{3 b f}-\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{f}+\frac {b \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}-\frac {a \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}\) | \(114\) |
default | \(\frac {\left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{3 b f}-\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{f}+\frac {b \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}-\frac {a \arctan \left (\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{\sqrt {-a +b}}\right )}{f \sqrt {-a +b}}\) | \(114\) |
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Time = 0.32 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.89 \[ \int \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\left [\frac {3 \, \sqrt {a - b} b \log \left (-\frac {b^{2} \tan \left (f x + e\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 4 \, {\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) + 4 \, {\left (b \tan \left (f x + e\right )^{2} + a - 3 \, b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{12 \, b f}, -\frac {3 \, \sqrt {-a + b} b \arctan \left (\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) - 2 \, {\left (b \tan \left (f x + e\right )^{2} + a - 3 \, b\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, b f}\right ] \]
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\[ \int \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int \sqrt {a + b \tan ^{2}{\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )}\, dx \]
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\[ \int \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int { \sqrt {b \tan \left (f x + e\right )^{2} + a} \tan \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\text {Timed out} \]
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Time = 13.57 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86 \[ \int \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a-b}}\right )\,\sqrt {a-b}}{f}-\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{f}+\frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{3\,b\,f} \]
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