Integrand size = 21, antiderivative size = 30 \[ \int \frac {\tan (c+d x)}{\sqrt {a \sin ^2(c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a \sin ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3284, 65, 212} \[ \int \frac {\tan (c+d x)}{\sqrt {a \sin ^2(c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a \sin ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rule 65
Rule 212
Rule 3284
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \sin ^2(c+d x)}\right )}{a d} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a \sin ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\tan (c+d x)}{\sqrt {a \sin ^2(c+d x)}} \, dx=\frac {\text {arctanh}(\sin (c+d x)) \sin (c+d x)}{d \sqrt {a \sin ^2(c+d x)}} \]
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Time = 0.66 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\sin \left (d x +c \right ) \operatorname {arctanh}\left (\sin \left (d x +c \right )\right )}{\sqrt {a \sin \left (d x +c \right )^{2}}\, d}\) | \(30\) |
risch | \(-\frac {2 \ln \left ({\mathrm e}^{i d x}-i {\mathrm e}^{-i c}\right ) \sin \left (d x +c \right )}{d \sqrt {-\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} a \,{\mathrm e}^{-2 i \left (d x +c \right )}}}+\frac {2 \ln \left ({\mathrm e}^{i d x}+i {\mathrm e}^{-i c}\right ) \sin \left (d x +c \right )}{d \sqrt {-\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} a \,{\mathrm e}^{-2 i \left (d x +c \right )}}}\) | \(110\) |
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none
Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.03 \[ \int \frac {\tan (c+d x)}{\sqrt {a \sin ^2(c+d x)}} \, dx=\left [\frac {\sqrt {-a \cos \left (d x + c\right )^{2} + a} \log \left (-\frac {\sin \left (d x + c\right ) + 1}{\sin \left (d x + c\right ) - 1}\right )}{2 \, a d \sin \left (d x + c\right )}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a \cos \left (d x + c\right )^{2} + a} \sqrt {-a}}{a}\right )}{a d}\right ] \]
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\[ \int \frac {\tan (c+d x)}{\sqrt {a \sin ^2(c+d x)}} \, dx=\int \frac {\tan {\left (c + d x \right )}}{\sqrt {a \sin ^{2}{\left (c + d x \right )}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).
Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.53 \[ \int \frac {\tan (c+d x)}{\sqrt {a \sin ^2(c+d x)}} \, dx=\frac {\frac {\left (-1\right )^{2 \, a \sin \left (d x + c\right )} \log \left (-\frac {a \sin \left (d x + c\right )}{\sin \left (d x + c\right ) + 1}\right )}{\sqrt {a}} + \frac {\left (-1\right )^{2 \, a \sin \left (d x + c\right )} \log \left (-\frac {a \sin \left (d x + c\right )}{\sin \left (d x + c\right ) - 1}\right )}{\sqrt {a}}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.41 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {\tan (c+d x)}{\sqrt {a \sin ^2(c+d x)}} \, dx=\frac {\log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) + 2 \right |}\right ) - \log \left ({\left | \frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right ) - 2 \right |}\right )}{4 \, \sqrt {a} d \mathrm {sgn}\left (\sin \left (d x + c\right )\right )} \]
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Timed out. \[ \int \frac {\tan (c+d x)}{\sqrt {a \sin ^2(c+d x)}} \, dx=\int \frac {\mathrm {tan}\left (c+d\,x\right )}{\sqrt {a\,{\sin \left (c+d\,x\right )}^2}} \,d x \]
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