Integrand size = 12, antiderivative size = 192 \[ \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}-\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d} \]
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Time = 0.14 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}+1\right )}{\sqrt {2} \sqrt {b} d}-\frac {\log \left (\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {\log \left (\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt {b} d} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {(2 b) \text {Subst}\left (\int \frac {1}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {b-x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d}+\frac {\text {Subst}\left (\int \frac {b+x^2}{b^2+x^4} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{b-\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {1}{b+\sqrt {2} \sqrt {b} x+x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 d}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b}+2 x}{-b-\sqrt {2} \sqrt {b} x-x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b}-2 x}{-b+\sqrt {2} \sqrt {b} x-x^2} \, dx,x,\sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d} \\ & = -\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {b \tan (c+d x)}}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {b} d}-\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)-\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d}+\frac {\log \left (\sqrt {b}+\sqrt {b} \tan (c+d x)+\sqrt {2} \sqrt {b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {b} d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx=\frac {\left (-2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )+2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )-\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} d \sqrt {b \tan (c+d x)}} \]
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Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d b}\) | \(138\) |
default | \(\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\left (b^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d b}\) | \(138\) |
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx=\frac {1}{2} \, \left (-\frac {1}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (b d \left (-\frac {1}{b^{2} d^{4}}\right )^{\frac {1}{4}} + \sqrt {b \tan \left (d x + c\right )}\right ) + \frac {1}{2} i \, \left (-\frac {1}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (i \, b d \left (-\frac {1}{b^{2} d^{4}}\right )^{\frac {1}{4}} + \sqrt {b \tan \left (d x + c\right )}\right ) - \frac {1}{2} i \, \left (-\frac {1}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (-i \, b d \left (-\frac {1}{b^{2} d^{4}}\right )^{\frac {1}{4}} + \sqrt {b \tan \left (d x + c\right )}\right ) - \frac {1}{2} \, \left (-\frac {1}{b^{2} d^{4}}\right )^{\frac {1}{4}} \log \left (-b d \left (-\frac {1}{b^{2} d^{4}}\right )^{\frac {1}{4}} + \sqrt {b \tan \left (d x + c\right )}\right ) \]
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\[ \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {b \tan {\left (c + d x \right )}}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx=\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right ) + 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {b} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {b}}\right ) + \sqrt {2} \sqrt {b} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right ) - \sqrt {2} \sqrt {b} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {b} + b\right )}{4 \, b d} \]
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Time = 0.39 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx=\frac {\sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{2 \, b d} + \frac {\sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{2 \, b d} + \frac {\sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{4 \, b d} - \frac {\sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{4 \, b d} \]
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Time = 3.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt {b \tan (c+d x)}} \, dx=-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}}{\sqrt {b}}\right )\,1{}\mathrm {i}}{\sqrt {b}\,d}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {b\,\mathrm {tan}\left (c+d\,x\right )}}{\sqrt {b}}\right )\,1{}\mathrm {i}}{\sqrt {b}\,d} \]
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