Integrand size = 29, antiderivative size = 115 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx=\frac {\left (2 b c d+a \left (2 c^2+d^2\right )\right ) \text {arctanh}(\sin (e+f x))}{2 f}+\frac {2 \left (3 a c d+b \left (c^2+d^2\right )\right ) \tan (e+f x)}{3 f}+\frac {d (2 b c+3 a d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac {b (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f} \]
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Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx=\frac {\left (a \left (2 c^2+d^2\right )+2 b c d\right ) \text {arctanh}(\sin (e+f x))}{2 f}+\frac {2 \left (3 a c d+b \left (c^2+d^2\right )\right ) \tan (e+f x)}{3 f}+\frac {d (3 a d+2 b c) \tan (e+f x) \sec (e+f x)}{6 f}+\frac {b \tan (e+f x) (c+d \sec (e+f x))^2}{3 f} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rubi steps \begin{align*} \text {integral}& = \frac {b (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac {1}{3} \int \sec (e+f x) (c+d \sec (e+f x)) (3 a c+2 b d+(2 b c+3 a d) \sec (e+f x)) \, dx \\ & = \frac {d (2 b c+3 a d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac {b (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac {1}{6} \int \sec (e+f x) \left (3 \left (2 b c d+a \left (2 c^2+d^2\right )\right )+4 \left (3 a c d+b \left (c^2+d^2\right )\right ) \sec (e+f x)\right ) \, dx \\ & = \frac {d (2 b c+3 a d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac {b (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac {1}{3} \left (2 \left (3 a c d+b \left (c^2+d^2\right )\right )\right ) \int \sec ^2(e+f x) \, dx+\frac {1}{2} \left (2 b c d+a \left (2 c^2+d^2\right )\right ) \int \sec (e+f x) \, dx \\ & = \frac {\left (2 b c d+a \left (2 c^2+d^2\right )\right ) \text {arctanh}(\sin (e+f x))}{2 f}+\frac {d (2 b c+3 a d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac {b (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f}-\frac {\left (2 \left (3 a c d+b \left (c^2+d^2\right )\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 f} \\ & = \frac {\left (2 b c d+a \left (2 c^2+d^2\right )\right ) \text {arctanh}(\sin (e+f x))}{2 f}+\frac {2 \left (3 a c d+b \left (c^2+d^2\right )\right ) \tan (e+f x)}{3 f}+\frac {d (2 b c+3 a d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac {b (c+d \sec (e+f x))^2 \tan (e+f x)}{3 f} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx=\frac {3 \left (2 b c d+a \left (2 c^2+d^2\right )\right ) \text {arctanh}(\sin (e+f x))+\tan (e+f x) \left (12 a c d+6 b \left (c^2+d^2\right )+3 d (2 b c+a d) \sec (e+f x)+2 b d^2 \tan ^2(e+f x)\right )}{6 f} \]
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Time = 3.40 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03
method | result | size |
parts | \(\frac {\left (a \,d^{2}+2 b c d \right ) \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {\left (2 a c d +b \,c^{2}\right ) \tan \left (f x +e \right )}{f}+\frac {a \,c^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}-\frac {b \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(118\) |
derivativedivides | \(\frac {a \,c^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+2 a c d \tan \left (f x +e \right )+a \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+b \,c^{2} \tan \left (f x +e \right )+2 b c d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-b \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(143\) |
default | \(\frac {a \,c^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+2 a c d \tan \left (f x +e \right )+a \,d^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+b \,c^{2} \tan \left (f x +e \right )+2 b c d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-b \,d^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(143\) |
parallelrisch | \(\frac {-9 \left (\cos \left (f x +e \right )+\frac {\cos \left (3 f x +3 e \right )}{3}\right ) \left (a \,c^{2}+\frac {1}{2} a \,d^{2}+b c d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+9 \left (\cos \left (f x +e \right )+\frac {\cos \left (3 f x +3 e \right )}{3}\right ) \left (a \,c^{2}+\frac {1}{2} a \,d^{2}+b c d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\left (6 a c d +3 b \,c^{2}+2 b \,d^{2}\right ) \sin \left (3 f x +3 e \right )+3 \left (a \,d^{2}+2 b c d \right ) \sin \left (2 f x +2 e \right )+6 \sin \left (f x +e \right ) \left (a c d +\frac {1}{2} b \,c^{2}+b \,d^{2}\right )}{3 f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\) | \(197\) |
norman | \(\frac {\frac {4 \left (6 a c d +3 b \,c^{2}+b \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f}-\frac {\left (4 a c d -a \,d^{2}+2 b \,c^{2}-2 b c d +2 b \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{f}-\frac {\left (4 a c d +a \,d^{2}+2 b \,c^{2}+2 b c d +2 b \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}-\frac {\left (2 a \,c^{2}+a \,d^{2}+2 b c d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 f}+\frac {\left (2 a \,c^{2}+a \,d^{2}+2 b c d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 f}\) | \(207\) |
risch | \(-\frac {i \left (3 a \,d^{2} {\mathrm e}^{5 i \left (f x +e \right )}+6 b c d \,{\mathrm e}^{5 i \left (f x +e \right )}-12 a c d \,{\mathrm e}^{4 i \left (f x +e \right )}-6 b \,c^{2} {\mathrm e}^{4 i \left (f x +e \right )}-24 a c d \,{\mathrm e}^{2 i \left (f x +e \right )}-12 b \,c^{2} {\mathrm e}^{2 i \left (f x +e \right )}-12 b \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-3 a \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-6 b c d \,{\mathrm e}^{i \left (f x +e \right )}-12 a c d -6 b \,c^{2}-4 b \,d^{2}\right )}{3 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{3}}-\frac {a \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{f}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) d^{2}}{2 f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b c d}{f}+\frac {a \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{f}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) d^{2}}{2 f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b c d}{f}\) | \(298\) |
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Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.30 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx=\frac {3 \, {\left (2 \, a c^{2} + 2 \, b c d + a d^{2}\right )} \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (2 \, a c^{2} + 2 \, b c d + a d^{2}\right )} \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (2 \, b d^{2} + 2 \, {\left (3 \, b c^{2} + 6 \, a c d + 2 \, b d^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (2 \, b c d + a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, f \cos \left (f x + e\right )^{3}} \]
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\[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx=\int \left (a + b \sec {\left (e + f x \right )}\right ) \left (c + d \sec {\left (e + f x \right )}\right )^{2} \sec {\left (e + f x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.43 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx=\frac {4 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} b d^{2} - 6 \, b c d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 3 \, a d^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 12 \, a c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 12 \, b c^{2} \tan \left (f x + e\right ) + 24 \, a c d \tan \left (f x + e\right )}{12 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (107) = 214\).
Time = 0.34 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.56 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx=\frac {3 \, {\left (2 \, a c^{2} + 2 \, b c d + a d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, a c^{2} + 2 \, b c d + a d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, b c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, a c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, b c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, a d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, b d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, b c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, a c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, b c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, a c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}}{6 \, f} \]
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Time = 17.02 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.97 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,c^2+b\,c\,d+\frac {a\,d^2}{2}\right )}{4\,a\,c^2+4\,b\,c\,d+2\,a\,d^2}\right )\,\left (2\,a\,c^2+2\,b\,c\,d+a\,d^2\right )}{f}-\frac {\left (2\,b\,c^2-a\,d^2+2\,b\,d^2+4\,a\,c\,d-2\,b\,c\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-4\,b\,c^2-8\,a\,c\,d-\frac {4\,b\,d^2}{3}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (a\,d^2+2\,b\,c^2+2\,b\,d^2+4\,a\,c\,d+2\,b\,c\,d\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
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