Integrand size = 21, antiderivative size = 36 \[ \int \frac {\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b} f} \]
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Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4232, 214} \[ \int \frac {\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {a} f \sqrt {a+b}} \]
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Rule 214
Rule 4232
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+b-a x^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b} f} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b} f} \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{f \sqrt {a \left (a +b \right )}}\) | \(28\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {a \sin \left (f x +e \right )}{\sqrt {a \left (a +b \right )}}\right )}{f \sqrt {a \left (a +b \right )}}\) | \(28\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, f}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \left (a +b \right ) {\mathrm e}^{i \left (f x +e \right )}}{\sqrt {a^{2}+a b}}-1\right )}{2 \sqrt {a^{2}+a b}\, f}\) | \(102\) |
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Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.25 \[ \int \frac {\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx=\left [\frac {\log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} + a b} \sin \left (f x + e\right ) - 2 \, a - b}{a \cos \left (f x + e\right )^{2} + b}\right )}{2 \, \sqrt {a^{2} + a b} f}, -\frac {\sqrt {-a^{2} - a b} \arctan \left (\frac {\sqrt {-a^{2} - a b} \sin \left (f x + e\right )}{a + b}\right )}{{\left (a^{2} + a b\right )} f}\right ] \]
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\[ \int \frac {\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx=\int \frac {\sec {\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx=-\frac {\log \left (\frac {a \sin \left (f x + e\right ) - \sqrt {{\left (a + b\right )} a}}{a \sin \left (f x + e\right ) + \sqrt {{\left (a + b\right )} a}}\right )}{2 \, \sqrt {{\left (a + b\right )} a} f} \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx=-\frac {\arctan \left (\frac {a \sin \left (f x + e\right )}{\sqrt {-a^{2} - a b}}\right )}{\sqrt {-a^{2} - a b} f} \]
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Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {\sec (e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {a}\,\sin \left (e+f\,x\right )}{\sqrt {a+b}}\right )}{\sqrt {a}\,f\,\sqrt {a+b}} \]
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