Integrand size = 26, antiderivative size = 97 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx=a^3 c^2 x+\frac {3 a^3 c^2 \text {arctanh}(\sin (e+f x))}{8 f}-\frac {c^2 \left (8 a^3+3 a^3 \sec (e+f x)\right ) \tan (e+f x)}{8 f}+\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f} \]
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Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3989, 3966, 3855} \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx=\frac {3 a^3 c^2 \text {arctanh}(\sin (e+f x))}{8 f}+\frac {c^2 \tan ^3(e+f x) \left (3 a^3 \sec (e+f x)+4 a^3\right )}{12 f}-\frac {c^2 \tan (e+f x) \left (3 a^3 \sec (e+f x)+8 a^3\right )}{8 f}+a^3 c^2 x \]
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Rule 3855
Rule 3966
Rule 3989
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int (a+a \sec (e+f x)) \tan ^4(e+f x) \, dx \\ & = \frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f}-\frac {1}{4} \left (a^2 c^2\right ) \int (4 a+3 a \sec (e+f x)) \tan ^2(e+f x) \, dx \\ & = -\frac {c^2 \left (8 a^3+3 a^3 \sec (e+f x)\right ) \tan (e+f x)}{8 f}+\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f}+\frac {1}{8} \left (a^2 c^2\right ) \int (8 a+3 a \sec (e+f x)) \, dx \\ & = a^3 c^2 x-\frac {c^2 \left (8 a^3+3 a^3 \sec (e+f x)\right ) \tan (e+f x)}{8 f}+\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f}+\frac {1}{8} \left (3 a^3 c^2\right ) \int \sec (e+f x) \, dx \\ & = a^3 c^2 x+\frac {3 a^3 c^2 \text {arctanh}(\sin (e+f x))}{8 f}-\frac {c^2 \left (8 a^3+3 a^3 \sec (e+f x)\right ) \tan (e+f x)}{8 f}+\frac {c^2 \left (4 a^3+3 a^3 \sec (e+f x)\right ) \tan ^3(e+f x)}{12 f} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx=\frac {a^3 c^2 \sec ^4(e+f x) \left (72 e+72 f x+72 \text {arctanh}(\sin (e+f x)) \cos ^4(e+f x)+96 (e+f x) \cos (2 (e+f x))+24 e \cos (4 (e+f x))+24 f x \cos (4 (e+f x))+18 \sin (e+f x)-32 \sin (2 (e+f x))-30 \sin (3 (e+f x))-32 \sin (4 (e+f x))\right )}{192 f} \]
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Time = 2.74 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.37
method | result | size |
parts | \(a^{3} c^{2} x -\frac {c^{2} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {c^{2} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}-\frac {2 c^{2} a^{3} \tan \left (f x +e \right )}{f}-\frac {c^{2} a^{3} \tan \left (f x +e \right ) \sec \left (f x +e \right )}{f}\) | \(133\) |
risch | \(a^{3} c^{2} x +\frac {i c^{2} a^{3} \left (15 \,{\mathrm e}^{7 i \left (f x +e \right )}-48 \,{\mathrm e}^{6 i \left (f x +e \right )}-9 \,{\mathrm e}^{5 i \left (f x +e \right )}-96 \,{\mathrm e}^{4 i \left (f x +e \right )}+9 \,{\mathrm e}^{3 i \left (f x +e \right )}-80 \,{\mathrm e}^{2 i \left (f x +e \right )}-15 \,{\mathrm e}^{i \left (f x +e \right )}-32\right )}{12 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{4}}+\frac {3 c^{2} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{8 f}-\frac {3 c^{2} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{8 f}\) | \(162\) |
derivativedivides | \(\frac {c^{2} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-c^{2} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-2 c^{2} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{2} a^{3} \tan \left (f x +e \right )+c^{2} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{2} a^{3} \left (f x +e \right )}{f}\) | \(169\) |
default | \(\frac {c^{2} a^{3} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-c^{2} a^{3} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-2 c^{2} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{2} a^{3} \tan \left (f x +e \right )+c^{2} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{2} a^{3} \left (f x +e \right )}{f}\) | \(169\) |
parallelrisch | \(\frac {a^{3} c^{2} \left (-9 \left (3+\cos \left (4 f x +4 e \right )+4 \cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )-9 \left (-\cos \left (4 f x +4 e \right )-4 \cos \left (2 f x +2 e \right )-3\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+96 f x \cos \left (2 f x +2 e \right )+24 f x \cos \left (4 f x +4 e \right )+72 f x -30 \sin \left (3 f x +3 e \right )-32 \sin \left (4 f x +4 e \right )+18 \sin \left (f x +e \right )-32 \sin \left (2 f x +2 e \right )\right )}{24 f \left (3+\cos \left (4 f x +4 e \right )+4 \cos \left (2 f x +2 e \right )\right )}\) | \(182\) |
norman | \(\frac {a^{3} c^{2} x +a^{3} c^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-4 a^{3} c^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+6 a^{3} c^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-4 a^{3} c^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-\frac {11 c^{2} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {137 c^{2} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{12 f}-\frac {71 c^{2} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{12 f}+\frac {5 c^{2} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{4}}-\frac {3 c^{2} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{8 f}+\frac {3 c^{2} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{8 f}\) | \(238\) |
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Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.52 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx=\frac {48 \, a^{3} c^{2} f x \cos \left (f x + e\right )^{4} + 9 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 9 \, a^{3} c^{2} \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (32 \, a^{3} c^{2} \cos \left (f x + e\right )^{3} + 15 \, a^{3} c^{2} \cos \left (f x + e\right )^{2} - 8 \, a^{3} c^{2} \cos \left (f x + e\right ) - 6 \, a^{3} c^{2}\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )^{4}} \]
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\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx=a^{3} c^{2} \left (\int 1\, dx + \int \sec {\left (e + f x \right )}\, dx + \int \left (- 2 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx + \int \sec ^{5}{\left (e + f x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (91) = 182\).
Time = 0.20 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.09 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx=\frac {16 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{2} + 48 \, {\left (f x + e\right )} a^{3} c^{2} - 3 \, a^{3} c^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 24 \, a^{3} c^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 48 \, a^{3} c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 96 \, a^{3} c^{2} \tan \left (f x + e\right )}{48 \, f} \]
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Time = 0.35 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.58 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx=\frac {24 \, {\left (f x + e\right )} a^{3} c^{2} + 9 \, a^{3} c^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 9 \, a^{3} c^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 71 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 137 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 33 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, f} \]
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Time = 15.66 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.68 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^2 \, dx=\frac {\frac {5\,a^3\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{4}-\frac {71\,a^3\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{12}+\frac {137\,a^3\,c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{12}-\frac {11\,a^3\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+a^3\,c^2\,x+\frac {3\,a^3\,c^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{4\,f} \]
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