Integrand size = 23, antiderivative size = 42 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{\sqrt {a+b} f} \]
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Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3269, 385, 212} \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f \sqrt {a+b}} \]
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Rule 212
Rule 385
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{f} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{\sqrt {a+b} f} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{\sqrt {a+b} f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(36)=72\).
Time = 1.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.50
method | result | size |
default | \(\frac {\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right )-\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right )}{2 \sqrt {a +b}\, f}\) | \(105\) |
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (36) = 72\).
Time = 0.36 (sec) , antiderivative size = 240, normalized size of antiderivative = 5.71 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\left [\frac {\log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 8 \, {\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - 2 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a + b} \sin \left (f x + e\right ) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right )}{4 \, \sqrt {a + b} f}, -\frac {\sqrt {-a - b} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - 2 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a - b}}{2 \, {\left ({\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sin \left (f x + e\right )}\right )}{2 \, {\left (a + b\right )} f}\right ] \]
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\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (36) = 72\).
Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.50 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {\frac {\operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b} {\left (\sin \left (f x + e\right ) + 1\right )}} - \frac {a}{\sqrt {a b} {\left (\sin \left (f x + e\right ) + 1\right )}}\right )}{\sqrt {a + b}} + \frac {\operatorname {arsinh}\left (-\frac {b \sin \left (f x + e\right )}{\sqrt {a b} {\left (\sin \left (f x + e\right ) - 1\right )}} - \frac {a}{\sqrt {a b} {\left (\sin \left (f x + e\right ) - 1\right )}}\right )}{\sqrt {a + b}}}{2 \, f} \]
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\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]
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