Integrand size = 19, antiderivative size = 46 \[ \int \frac {\sec (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\text {arctanh}\left (\frac {\cos (c+d x) (b-a \tan (c+d x))}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d} \]
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Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3590, 212} \[ \int \frac {\sec (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\text {arctanh}\left (\frac {\cos (c+d x) (b-a \tan (c+d x))}{\sqrt {a^2+b^2}}\right )}{d \sqrt {a^2+b^2}} \]
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Rule 212
Rule 3590
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,\cos (c+d x) (b-a \tan (c+d x))\right )}{d} \\ & = -\frac {\text {arctanh}\left (\frac {\cos (c+d x) (b-a \tan (c+d x))}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {\sec (c+d x)}{a+b \tan (c+d x)} \, dx=\frac {2 \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d} \]
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Time = 1.34 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {2 \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \sqrt {a^{2}+b^{2}}}\) | \(43\) |
default | \(\frac {2 \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \sqrt {a^{2}+b^{2}}}\) | \(43\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}\, d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}\, d}\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (44) = 88\).
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.85 \[ \int \frac {\sec (c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right )}{2 \, \sqrt {a^{2} + b^{2}} d} \]
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\[ \int \frac {\sec (c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\sec {\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
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none
Time = 0.51 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74 \[ \int \frac {\sec (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} d} \]
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Time = 0.38 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.61 \[ \int \frac {\sec (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} d} \]
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Time = 4.57 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \frac {\sec (c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {b-a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{d\,\sqrt {a^2+b^2}} \]
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