Integrand size = 17, antiderivative size = 74 \[ \int \frac {\tan ^3(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {b}}+\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}} \]
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Time = 0.15 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 1266, 858, 223, 212, 739} \[ \int \frac {\tan ^3(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {b}}+\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}} \]
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Rule 212
Rule 223
Rule 739
Rule 858
Rule 1266
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^3}{\left (1+x^2\right ) \sqrt {a+b x^4}} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {b}}+\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^3(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {b}}+\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}} \]
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Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {\ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )}{2 \sqrt {b}}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \sqrt {a +b}}\) | \(91\) |
default | \(\frac {\ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )}{2 \sqrt {b}}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \sqrt {a +b}}\) | \(91\) |
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Time = 0.40 (sec) , antiderivative size = 483, normalized size of antiderivative = 6.53 \[ \int \frac {\tan ^3(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\left [\frac {{\left (a + b\right )} \sqrt {b} \log \left (-2 \, b \tan \left (x\right )^{4} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {b} \tan \left (x\right )^{2} - a\right ) + \sqrt {a + b} b \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right )}{4 \, {\left (a b + b^{2}\right )}}, -\frac {2 \, {\left (a + b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) - \sqrt {a + b} b \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right )}{4 \, {\left (a b + b^{2}\right )}}, \frac {2 \, \sqrt {-a - b} b \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) + {\left (a + b\right )} \sqrt {b} \log \left (-2 \, b \tan \left (x\right )^{4} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {b} \tan \left (x\right )^{2} - a\right )}{4 \, {\left (a b + b^{2}\right )}}, \frac {\sqrt {-a - b} b \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) - {\left (a + b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right )}{2 \, {\left (a b + b^{2}\right )}}\right ] \]
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\[ \int \frac {\tan ^3(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\int \frac {\tan ^{3}{\left (x \right )}}{\sqrt {a + b \tan ^{4}{\left (x \right )}}}\, dx \]
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\[ \int \frac {\tan ^3(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\int { \frac {\tan \left (x\right )^{3}}{\sqrt {b \tan \left (x\right )^{4} + a}} \,d x } \]
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Exception generated. \[ \int \frac {\tan ^3(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\tan ^3(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\int \frac {{\mathrm {tan}\left (x\right )}^3}{\sqrt {b\,{\mathrm {tan}\left (x\right )}^4+a}} \,d x \]
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