Integrand size = 23, antiderivative size = 36 \[ \int \frac {\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b} f} \]
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Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4231, 211} \[ \int \frac {\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {b} f \sqrt {a+b}} \]
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Rule 211
Rule 4231
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b} f} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{\sqrt {b} \sqrt {a+b} f} \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{f \sqrt {\left (a +b \right ) b}}\) | \(28\) |
default | \(\frac {\arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{f \sqrt {\left (a +b \right ) b}}\) | \(28\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i b a +2 i b^{2}+a \sqrt {-a b -b^{2}}+2 b \sqrt {-a b -b^{2}}}{a \sqrt {-a b -b^{2}}}\right )}{2 \sqrt {-a b -b^{2}}\, f}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {-2 i b a -2 i b^{2}+a \sqrt {-a b -b^{2}}+2 b \sqrt {-a b -b^{2}}}{a \sqrt {-a b -b^{2}}}\right )}{2 \sqrt {-a b -b^{2}}\, f}\) | \(172\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 5.81 \[ \int \frac {\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\left [-\frac {\sqrt {-a b - b^{2}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {-a b - b^{2}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right )}{4 \, {\left (a b + b^{2}\right )} f}, -\frac {\arctan \left (\frac {{\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b}{2 \, \sqrt {a b + b^{2}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right )}{2 \, \sqrt {a b + b^{2}} f}\right ] \]
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\[ \int \frac {\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\int \frac {\sec ^{2}{\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} f} \]
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none
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )}{\sqrt {a b + b^{2}} f} \]
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Time = 18.86 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {\sec ^2(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {b^2+a\,b}}\right )}{f\,\sqrt {b^2+a\,b}} \]
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