Integrand size = 13, antiderivative size = 5 \[ \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx=-\arctan (\cos (x)) \]
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Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4424, 209} \[ \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx=-\arctan (\cos (x)) \]
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Rule 209
Rule 4424
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cos (x)\right ) \\ & = -\arctan (\cos (x)) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx=-\arctan (\cos (x)) \]
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Time = 2.26 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\arctan \left (\sec \left (x \right )\right )\) | \(4\) |
default | \(\arctan \left (\sec \left (x \right )\right )\) | \(4\) |
risch | \(-\frac {i \ln \left (2 i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}+1\right )}{2}+\frac {i \ln \left (-2 i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}+1\right )}{2}\) | \(40\) |
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none
Time = 0.24 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx=-\arctan \left (\cos \left (x\right )\right ) \]
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Time = 0.09 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.60 \[ \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx=\operatorname {atan}{\left (\sec {\left (x \right )} \right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx=-\arctan \left (\cos \left (x\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx=-\arctan \left (\cos \left (x\right )\right ) \]
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Time = 26.47 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.40 \[ \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx=\mathrm {atan}\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2\right ) \]
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