Integrand size = 30, antiderivative size = 17 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx=a x-\frac {b \text {arctanh}(\sin (c+d x))}{d} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4127, 3855} \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx=a x-\frac {b \text {arctanh}(\sin (c+d x))}{d} \]
[In]
[Out]
Rule 3855
Rule 4127
Rubi steps \begin{align*} \text {integral}& = -\int (-a+b \sec (c+d x)) \, dx \\ & = a x-b \int \sec (c+d x) \, dx \\ & = a x-\frac {b \text {arctanh}(\sin (c+d x))}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx=a x-\frac {b \text {arctanh}(\sin (c+d x))}{d} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76
method | result | size |
derivativedivides | \(\frac {a \left (d x +c \right )-b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(30\) |
default | \(\frac {a \left (d x +c \right )-b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(30\) |
parallelrisch | \(\frac {a x d +b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(39\) |
risch | \(a x +\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(42\) |
norman | \(\frac {a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-a x}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(72\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.12 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {2 \, a d x - b \log \left (\sin \left (d x + c\right ) + 1\right ) + b \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
[In]
[Out]
Time = 1.40 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx=a x - b \left (\begin {cases} \frac {x \left (\tan {\left (c \right )} \sec {\left (c \right )} + \sec ^{2}{\left (c \right )}\right )}{\tan {\left (c \right )} + \sec {\left (c \right )}} & \text {for}\: d = 0 \\\frac {\log {\left (\tan {\left (c + d x \right )} + \sec {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
Exception generated. \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (17) = 34\).
Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.53 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {{\left (d x + c\right )} a - b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{d} \]
[In]
[Out]
Time = 16.98 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.35 \[ \int \frac {a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {2\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
[In]
[Out]