Integrand size = 29, antiderivative size = 65 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx=\frac {(c-d) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {(c+2 d) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )} \]
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Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {4085, 3879} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx=\frac {(c+2 d) \tan (e+f x)}{3 f \left (a^2 \sec (e+f x)+a^2\right )}+\frac {(c-d) \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2} \]
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Rule 3879
Rule 4085
Rubi steps \begin{align*} \text {integral}& = \frac {(c-d) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {(c+2 d) \int \frac {\sec (e+f x)}{a+a \sec (e+f x)} \, dx}{3 a} \\ & = \frac {(c-d) \tan (e+f x)}{3 f (a+a \sec (e+f x))^2}+\frac {(c+2 d) \tan (e+f x)}{3 f \left (a^2+a^2 \sec (e+f x)\right )} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.17 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx=\frac {\cos \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \left (3 (c+d) \sin \left (\frac {f x}{2}\right )-3 c \sin \left (e+\frac {f x}{2}\right )+(2 c+d) \sin \left (e+\frac {3 f x}{2}\right )\right )}{3 a^2 f (1+\cos (e+f x))^2} \]
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Time = 0.69 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(-\frac {\left (-3 c -3 d +\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{6 a^{2} f}\) | \(42\) |
derivativedivides | \(\frac {-\frac {c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f \,a^{2}}\) | \(60\) |
default | \(\frac {-\frac {c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}+c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f \,a^{2}}\) | \(60\) |
risch | \(\frac {2 i \left (3 \,{\mathrm e}^{2 i \left (f x +e \right )} c +3 \,{\mathrm e}^{i \left (f x +e \right )} c +3 d \,{\mathrm e}^{i \left (f x +e \right )}+2 c +d \right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}\) | \(64\) |
norman | \(\frac {-\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{6 a f}-\frac {\left (c +d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a f}+\frac {\left (2 c +d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )}\) | \(89\) |
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx=\frac {{\left ({\left (2 \, c + d\right )} \cos \left (f x + e\right ) + c + 2 \, d\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )}} \]
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\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx=\frac {\int \frac {c \sec {\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.43 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx=\frac {\frac {d {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} + \frac {c {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx=-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{6 \, a^{2} f} \]
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Time = 13.58 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))}{(a+a \sec (e+f x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (c+d\right )}{2\,a^2\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (c-d\right )}{6\,a^2\,f} \]
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