Integrand size = 24, antiderivative size = 77 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=a^3 c x+\frac {a^3 c \text {arctanh}(\sin (e+f x))}{f}-\frac {a^3 c \tan (e+f x)}{f}-\frac {a^3 c \sec (e+f x) \tan (e+f x)}{f}-\frac {a^3 c \tan ^3(e+f x)}{3 f} \]
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Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3989, 3971, 3554, 8, 2691, 3855, 2687, 30} \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {a^3 c \text {arctanh}(\sin (e+f x))}{f}-\frac {a^3 c \tan ^3(e+f x)}{3 f}-\frac {a^3 c \tan (e+f x)}{f}-\frac {a^3 c \tan (e+f x) \sec (e+f x)}{f}+a^3 c x \]
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Rule 8
Rule 30
Rule 2687
Rule 2691
Rule 3554
Rule 3855
Rule 3971
Rule 3989
Rubi steps \begin{align*} \text {integral}& = -\left ((a c) \int (a+a \sec (e+f x))^2 \tan ^2(e+f x) \, dx\right ) \\ & = -\left ((a c) \int \left (a^2 \tan ^2(e+f x)+2 a^2 \sec (e+f x) \tan ^2(e+f x)+a^2 \sec ^2(e+f x) \tan ^2(e+f x)\right ) \, dx\right ) \\ & = -\left (\left (a^3 c\right ) \int \tan ^2(e+f x) \, dx\right )-\left (a^3 c\right ) \int \sec ^2(e+f x) \tan ^2(e+f x) \, dx-\left (2 a^3 c\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx \\ & = -\frac {a^3 c \tan (e+f x)}{f}-\frac {a^3 c \sec (e+f x) \tan (e+f x)}{f}+\left (a^3 c\right ) \int 1 \, dx+\left (a^3 c\right ) \int \sec (e+f x) \, dx-\frac {\left (a^3 c\right ) \text {Subst}\left (\int x^2 \, dx,x,\tan (e+f x)\right )}{f} \\ & = a^3 c x+\frac {a^3 c \text {arctanh}(\sin (e+f x))}{f}-\frac {a^3 c \tan (e+f x)}{f}-\frac {a^3 c \sec (e+f x) \tan (e+f x)}{f}-\frac {a^3 c \tan ^3(e+f x)}{3 f} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.31 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {a^3 c \sec ^3(e+f x) \left (9 (e+f x) \cos (e+f x)+12 \text {arctanh}(\sin (e+f x)) \cos ^3(e+f x)+3 e \cos (3 (e+f x))+3 f x \cos (3 (e+f x))-6 \sin (e+f x)-6 \sin (2 (e+f x))-2 \sin (3 (e+f x))\right )}{12 f} \]
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Time = 1.80 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00
method | result | size |
parts | \(a^{3} c x +\frac {a^{3} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}-\frac {a^{3} c \sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}+\frac {a^{3} c \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(77\) |
derivativedivides | \(\frac {a^{3} c \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-2 a^{3} c \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+2 a^{3} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+a^{3} c \left (f x +e \right )}{f}\) | \(96\) |
default | \(\frac {a^{3} c \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-2 a^{3} c \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+2 a^{3} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+a^{3} c \left (f x +e \right )}{f}\) | \(96\) |
risch | \(a^{3} c x +\frac {2 i a^{3} c \left (3 \,{\mathrm e}^{5 i \left (f x +e \right )}-6 \,{\mathrm e}^{2 i \left (f x +e \right )}-3 \,{\mathrm e}^{i \left (f x +e \right )}-2\right )}{3 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{3}}+\frac {a^{3} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{f}-\frac {a^{3} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{f}\) | \(109\) |
parallelrisch | \(-\frac {2 \left (\frac {3 \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2}+\frac {3 \left (-\cos \left (3 f x +3 e \right )-3 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2}-\frac {9 f x \cos \left (f x +e \right )}{2}-\frac {3 f x \cos \left (3 f x +3 e \right )}{2}+\sin \left (3 f x +3 e \right )+3 \sin \left (f x +e \right )+3 \sin \left (2 f x +2 e \right )\right ) a^{3} c}{3 f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\) | \(148\) |
norman | \(\frac {a^{3} c x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-a^{3} c x +\frac {4 a^{3} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {4 a^{3} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f}+3 a^{3} c x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-3 a^{3} c x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}+\frac {a^{3} c \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{f}-\frac {a^{3} c \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{f}\) | \(158\) |
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Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.53 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {6 \, a^{3} c f x \cos \left (f x + e\right )^{3} + 3 \, a^{3} c \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, a^{3} c \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, a^{3} c \cos \left (f x + e\right )^{2} + 3 \, a^{3} c \cos \left (f x + e\right ) + a^{3} c\right )} \sin \left (f x + e\right )}{6 \, f \cos \left (f x + e\right )^{3}} \]
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\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=- a^{3} c \left (\int \left (-1\right )\, dx + \int \left (- 2 \sec {\left (e + f x \right )}\right )\, dx + \int 2 \sec ^{3}{\left (e + f x \right )}\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
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Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=-\frac {2 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c - 6 \, {\left (f x + e\right )} a^{3} c - 3 \, a^{3} c {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{3} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right )}{6 \, f} \]
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Time = 0.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {3 \, {\left (f x + e\right )} a^{3} c + 3 \, a^{3} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, a^{3} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {4 \, {\left (a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}}{3 \, f} \]
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Time = 15.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {4\,a^3\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-\frac {4\,a^3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}+a^3\,c\,x+\frac {2\,a^3\,c\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{f} \]
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