Integrand size = 15, antiderivative size = 90 \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=-\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b \tan ^4(x)} \]
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Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 1262, 749, 858, 223, 212, 739} \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=-\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b \tan ^4(x)} \]
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Rule 212
Rule 223
Rule 739
Rule 749
Rule 858
Rule 1262
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x \sqrt {a+b x^4}}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{1+x} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {1}{2} \sqrt {a+b \tan ^4(x)}+\frac {1}{2} \text {Subst}\left (\int \frac {a-b x}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {1}{2} \sqrt {a+b \tan ^4(x)}-\frac {1}{2} b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )+\frac {1}{2} (a+b) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {1}{2} \sqrt {a+b \tan ^4(x)}+\frac {1}{2} (-a-b) \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} b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right ) \\ & = -\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b \tan ^4(x)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {1}{2} \left (-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-\sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\sqrt {a+b \tan ^4(x)}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.54
method | result | size |
derivativedivides | \(\frac {\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tan \left (x \right )^{2}\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2}\) | \(139\) |
default | \(\frac {\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tan \left (x \right )^{2}\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2}\) | \(139\) |
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Time = 0.38 (sec) , antiderivative size = 475, normalized size of antiderivative = 5.28 \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\left [\frac {1}{4} \, \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 ) + \frac {1}{4} \, \sqrt {a + 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 ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a}, \frac {1}{2} \, \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) + \frac {1}{4} \, \sqrt {a + 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 ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a}, -\frac {1}{2} \, \sqrt {-a - 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 ) + \frac {1}{4} \, \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 ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a}, -\frac {1}{2} \, \sqrt {-a - 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 ) + \frac {1}{2} \, \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a}\right ] \]
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\[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\int \sqrt {a + b \tan ^{4}{\left (x \right )}} \tan {\left (x \right )}\, dx \]
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\[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\int { \sqrt {b \tan \left (x\right )^{4} + a} \tan \left (x\right ) \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.99 \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {{\left (a + b\right )} \arctan \left (-\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{\sqrt {-a - b}} + \frac {1}{2} \, \sqrt {b} \log \left ({\left | -\sqrt {b} \tan \left (x\right )^{2} + \sqrt {b \tan \left (x\right )^{4} + a} \right |}\right ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a} \]
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Timed out. \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\int \mathrm {tan}\left (x\right )\,\sqrt {b\,{\mathrm {tan}\left (x\right )}^4+a} \,d x \]
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