Integrand size = 31, antiderivative size = 88 \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {-c+c \sec (2 a+2 b x)}}\right )}{\sqrt {2} b \sqrt {c}}+\frac {\tan (2 a+2 b x)}{b \sqrt {-c+c \sec (2 a+2 b x)}} \]
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Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4482, 3883, 3880, 213} \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\frac {\tan (2 a+2 b x)}{b \sqrt {c \sec (2 a+2 b x)-c}}-\frac {\text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {c \sec (2 a+2 b x)-c}}\right )}{\sqrt {2} b \sqrt {c}} \]
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Rule 213
Rule 3880
Rule 3883
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^2(2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}} \, dx \\ & = \frac {\tan (2 a+2 b x)}{b \sqrt {-c+c \sec (2 a+2 b x)}}+\int \frac {\sec (2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}} \, dx \\ & = \frac {\tan (2 a+2 b x)}{b \sqrt {-c+c \sec (2 a+2 b x)}}-\frac {\text {Subst}\left (\int \frac {1}{-2 c+x^2} \, dx,x,-\frac {c \tan (2 a+2 b x)}{\sqrt {-c+c \sec (2 a+2 b x)}}\right )}{b} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {c} \tan (2 a+2 b x)}{\sqrt {2} \sqrt {-c+c \sec (2 a+2 b x)}}\right )}{\sqrt {2} b \sqrt {c}}+\frac {\tan (2 a+2 b x)}{b \sqrt {-c+c \sec (2 a+2 b x)}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.76 \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\frac {\left (2+\arctan \left (\sqrt {-1+\tan ^2(a+b x)}\right ) \sqrt {-1+\tan ^2(a+b x)}\right ) \tan (2 (a+b x))}{2 b \sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(77)=154\).
Time = 4.09 (sec) , antiderivative size = 370, normalized size of antiderivative = 4.20
method | result | size |
default | \(\frac {\sqrt {2}\, \left (\cos \left (x b +a \right )-1\right ) \left (\operatorname {arctanh}\left (\frac {2 \cos \left (x b +a \right )-1}{\left (1+\cos \left (x b +a \right )\right ) \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}}\right ) \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}\, \sin \left (x b +a \right )^{2}-\ln \left (\frac {2 \cos \left (x b +a \right ) \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}+2 \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}-4 \cos \left (x b +a \right )-2}{1+\cos \left (x b +a \right )}\right ) \sqrt {\frac {2 \cos \left (x b +a \right )^{2}-1}{\left (1+\cos \left (x b +a \right )\right )^{2}}}\, \sin \left (x b +a \right )^{2}-2 \sin \left (x b +a \right )^{2}+2 \cos \left (x b +a \right )^{2}-4 \cos \left (x b +a \right )+2\right ) \sin \left (x b +a \right ) \sqrt {4}}{2 b \sqrt {\frac {c \sin \left (x b +a \right )^{2}}{2 \cos \left (x b +a \right )^{2}-1}}\, \left (\sin \left (x b +a \right )^{4}-6 \cos \left (x b +a \right )^{2} \sin \left (x b +a \right )^{2}+\cos \left (x b +a \right )^{4}+12 \cos \left (x b +a \right ) \sin \left (x b +a \right )^{2}-4 \cos \left (x b +a \right )^{3}-6 \sin \left (x b +a \right )^{2}+6 \cos \left (x b +a \right )^{2}-4 \cos \left (x b +a \right )+1\right )}\) | \(370\) |
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Time = 0.29 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.78 \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\left [\frac {\sqrt {2} \sqrt {c} \log \left (\frac {\tan \left (b x + a\right )^{3} - \frac {2 \, \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )}}{\sqrt {c}} - 2 \, \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{3}}\right ) \tan \left (b x + a\right ) + 4 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{4 \, b c \tan \left (b x + a\right )}, -\frac {\sqrt {2} c \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} {\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt {-\frac {1}{c}}}{\tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) - 2 \, \sqrt {2} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{2 \, b c \tan \left (b x + a\right )}\right ] \]
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Timed out. \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\int { \frac {\sec \left (2 \, b x + 2 \, a\right )^{2}}{\sqrt {c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sec ^2(2 (a+b x))}{\sqrt {c \tan (a+b x) \tan (2 (a+b x))}} \, dx=\int \frac {1}{{\cos \left (2\,a+2\,b\,x\right )}^2\,\sqrt {c\,\mathrm {tan}\left (a+b\,x\right )\,\mathrm {tan}\left (2\,a+2\,b\,x\right )}} \,d x \]
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