Integrand size = 10, antiderivative size = 3 \[ \int \sqrt {1+\tan ^2(x)} \, dx=\text {arcsinh}(\tan (x)) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3738, 4207, 221} \[ \int \sqrt {1+\tan ^2(x)} \, dx=\text {arcsinh}(\tan (x)) \]
[In]
[Out]
Rule 221
Rule 3738
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\sec ^2(x)} \, dx \\ & = \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tan (x)\right ) \\ & = \text {arcsinh}(\tan (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(14\) vs. \(2(3)=6\).
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 4.67 \[ \int \sqrt {1+\tan ^2(x)} \, dx=\text {arctanh}(\sin (x)) \cos (x) \sqrt {\sec ^2(x)} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\operatorname {arcsinh}\left (\tan \left (x \right )\right )\) | \(4\) |
default | \(\operatorname {arcsinh}\left (\tan \left (x \right )\right )\) | \(4\) |
risch | \(2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )-2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )\) | \(62\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (3) = 6\).
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 20.00 \[ \int \sqrt {1+\tan ^2(x)} \, dx=\frac {1}{2} \, \log \left (\frac {\tan \left (x\right )^{2} + \sqrt {\tan \left (x\right )^{2} + 1} \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {\tan \left (x\right )^{2} - \sqrt {\tan \left (x\right )^{2} + 1} \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) \]
[In]
[Out]
\[ \int \sqrt {1+\tan ^2(x)} \, dx=\int \sqrt {\tan ^{2}{\left (x \right )} + 1}\, dx \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \sqrt {1+\tan ^2(x)} \, dx=\operatorname {arsinh}\left (\tan \left (x\right )\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (3) = 6\).
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 5.33 \[ \int \sqrt {1+\tan ^2(x)} \, dx=-\log \left (\sqrt {\tan \left (x\right )^{2} + 1} - \tan \left (x\right )\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \sqrt {1+\tan ^2(x)} \, dx=\mathrm {asinh}\left (\mathrm {tan}\left (x\right )\right ) \]
[In]
[Out]