Integrand size = 23, antiderivative size = 165 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\left (48 a^2+80 a b+35 b^2\right ) \text {arctanh}(\sin (e+f x))}{128 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \sec (e+f x) \tan (e+f x)}{128 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \sec ^3(e+f x) \tan (e+f x)}{192 f}+\frac {b (10 a+7 b) \sec ^5(e+f x) \tan (e+f x)}{48 f}+\frac {b \sec ^7(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{8 f} \]
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Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4232, 424, 393, 205, 212} \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\left (48 a^2+80 a b+35 b^2\right ) \text {arctanh}(\sin (e+f x))}{128 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \tan (e+f x) \sec ^3(e+f x)}{192 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \tan (e+f x) \sec (e+f x)}{128 f}+\frac {b (10 a+7 b) \tan (e+f x) \sec ^5(e+f x)}{48 f}+\frac {b \tan (e+f x) \sec ^7(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{8 f} \]
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Rule 205
Rule 212
Rule 393
Rule 424
Rule 4232
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b-a x^2\right )^2}{\left (1-x^2\right )^5} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {b \sec ^7(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{8 f}-\frac {\text {Subst}\left (\int \frac {-((a+b) (8 a+7 b))+a (8 a+5 b) x^2}{\left (1-x^2\right )^4} \, dx,x,\sin (e+f x)\right )}{8 f} \\ & = \frac {b (10 a+7 b) \sec ^5(e+f x) \tan (e+f x)}{48 f}+\frac {b \sec ^7(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{8 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{48 f} \\ & = \frac {\left (48 a^2+80 a b+35 b^2\right ) \sec ^3(e+f x) \tan (e+f x)}{192 f}+\frac {b (10 a+7 b) \sec ^5(e+f x) \tan (e+f x)}{48 f}+\frac {b \sec ^7(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{8 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{64 f} \\ & = \frac {\left (48 a^2+80 a b+35 b^2\right ) \sec (e+f x) \tan (e+f x)}{128 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \sec ^3(e+f x) \tan (e+f x)}{192 f}+\frac {b (10 a+7 b) \sec ^5(e+f x) \tan (e+f x)}{48 f}+\frac {b \sec ^7(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{8 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (e+f x)\right )}{128 f} \\ & = \frac {\left (48 a^2+80 a b+35 b^2\right ) \text {arctanh}(\sin (e+f x))}{128 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \sec (e+f x) \tan (e+f x)}{128 f}+\frac {\left (48 a^2+80 a b+35 b^2\right ) \sec ^3(e+f x) \tan (e+f x)}{192 f}+\frac {b (10 a+7 b) \sec ^5(e+f x) \tan (e+f x)}{48 f}+\frac {b \sec ^7(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{8 f} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.56 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 a^2 \text {arctanh}(\sin (e+f x))}{8 f}+\frac {5 a b \text {arctanh}(\sin (e+f x))}{8 f}+\frac {35 b^2 \text {arctanh}(\sin (e+f x))}{128 f}+\frac {3 a^2 \sec (e+f x) \tan (e+f x)}{8 f}+\frac {5 a b \sec (e+f x) \tan (e+f x)}{8 f}+\frac {35 b^2 \sec (e+f x) \tan (e+f x)}{128 f}+\frac {a^2 \sec ^3(e+f x) \tan (e+f x)}{4 f}+\frac {5 a b \sec ^3(e+f x) \tan (e+f x)}{12 f}+\frac {35 b^2 \sec ^3(e+f x) \tan (e+f x)}{192 f}+\frac {a b \sec ^5(e+f x) \tan (e+f x)}{3 f}+\frac {7 b^2 \sec ^5(e+f x) \tan (e+f x)}{48 f}+\frac {b^2 \sec ^7(e+f x) \tan (e+f x)}{8 f} \]
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Time = 1.38 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\) | \(180\) |
default | \(\frac {a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}\) | \(180\) |
parts | \(\frac {a^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}+\frac {b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{7}}{8}-\frac {7 \sec \left (f x +e \right )^{5}}{48}-\frac {35 \sec \left (f x +e \right )^{3}}{192}-\frac {35 \sec \left (f x +e \right )}{128}\right ) \tan \left (f x +e \right )+\frac {35 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{128}\right )}{f}+\frac {2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}\) | \(185\) |
parallelrisch | \(\frac {-8064 \left (\frac {\cos \left (8 f x +8 e \right )}{56}+\frac {\cos \left (6 f x +6 e \right )}{7}+\frac {\cos \left (4 f x +4 e \right )}{2}+\cos \left (2 f x +2 e \right )+\frac {5}{8}\right ) \left (a^{2}+\frac {5}{3} a b +\frac {35}{48} b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+8064 \left (\frac {\cos \left (8 f x +8 e \right )}{56}+\frac {\cos \left (6 f x +6 e \right )}{7}+\frac {\cos \left (4 f x +4 e \right )}{2}+\cos \left (2 f x +2 e \right )+\frac {5}{8}\right ) \left (a^{2}+\frac {5}{3} a b +\frac {35}{48} b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\left (4896 a^{2}+12256 a b +5362 b^{2}\right ) \sin \left (3 f x +3 e \right )+\left (2208 a^{2}+3680 a b +1610 b^{2}\right ) \sin \left (5 f x +5 e \right )+\left (288 a^{2}+480 a b +210 b^{2}\right ) \sin \left (7 f x +7 e \right )+2976 \sin \left (f x +e \right ) \left (a^{2}+\frac {283}{93} a b +\frac {163}{48} b^{2}\right )}{384 f \left (\cos \left (8 f x +8 e \right )+8 \cos \left (6 f x +6 e \right )+28 \cos \left (4 f x +4 e \right )+56 \cos \left (2 f x +2 e \right )+35\right )}\) | \(291\) |
norman | \(\frac {\frac {\left (80 a^{2}+176 a b +93 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{64 f}+\frac {\left (80 a^{2}+176 a b +93 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{64 f}-\frac {\left (432 a^{2}+1360 a b -1085 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{192 f}-\frac {\left (432 a^{2}+1360 a b -1085 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{192 f}-\frac {\left (816 a^{2}+976 a b +91 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{192 f}-\frac {\left (816 a^{2}+976 a b +91 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{192 f}+\frac {\left (1008 a^{2}+1808 a b +1799 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{192 f}+\frac {\left (1008 a^{2}+1808 a b +1799 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{192 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{8}}-\frac {\left (48 a^{2}+80 a b +35 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{128 f}+\frac {\left (48 a^{2}+80 a b +35 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{128 f}\) | \(329\) |
risch | \(-\frac {i {\mathrm e}^{i \left (f x +e \right )} \left (144 a^{2} {\mathrm e}^{14 i \left (f x +e \right )}+240 a b \,{\mathrm e}^{14 i \left (f x +e \right )}+105 b^{2} {\mathrm e}^{14 i \left (f x +e \right )}+1104 a^{2} {\mathrm e}^{12 i \left (f x +e \right )}+1840 a b \,{\mathrm e}^{12 i \left (f x +e \right )}+805 b^{2} {\mathrm e}^{12 i \left (f x +e \right )}+2448 a^{2} {\mathrm e}^{10 i \left (f x +e \right )}+6128 a b \,{\mathrm e}^{10 i \left (f x +e \right )}+2681 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}+1488 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}+4528 a b \,{\mathrm e}^{8 i \left (f x +e \right )}+5053 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}-1488 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}-4528 a b \,{\mathrm e}^{6 i \left (f x +e \right )}-5053 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-2448 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-6128 a b \,{\mathrm e}^{4 i \left (f x +e \right )}-2681 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-1104 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-1840 a b \,{\mathrm e}^{2 i \left (f x +e \right )}-805 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-144 a^{2}-240 a b -105 b^{2}\right )}{192 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{8}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a^{2}}{8 f}-\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a b}{8 f}-\frac {35 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b^{2}}{128 f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a^{2}}{8 f}+\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a b}{8 f}+\frac {35 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b^{2}}{128 f}\) | \(500\) |
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Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.02 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{8} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{8} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{6} + 2 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (16 \, a b + 7 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 48 \, b^{2}\right )} \sin \left (f x + e\right )}{768 \, f \cos \left (f x + e\right )^{8}} \]
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\[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \sec ^{5}{\left (e + f x \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.21 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \sin \left (f x + e\right )^{7} - 11 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \sin \left (f x + e\right )^{5} + {\left (624 \, a^{2} + 1168 \, a b + 511 \, b^{2}\right )} \sin \left (f x + e\right )^{3} - 3 \, {\left (80 \, a^{2} + 176 \, a b + 93 \, b^{2}\right )} \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{8} - 4 \, \sin \left (f x + e\right )^{6} + 6 \, \sin \left (f x + e\right )^{4} - 4 \, \sin \left (f x + e\right )^{2} + 1}}{768 \, f} \]
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Time = 0.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.34 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right ) - 3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (144 \, a^{2} \sin \left (f x + e\right )^{7} + 240 \, a b \sin \left (f x + e\right )^{7} + 105 \, b^{2} \sin \left (f x + e\right )^{7} - 528 \, a^{2} \sin \left (f x + e\right )^{5} - 880 \, a b \sin \left (f x + e\right )^{5} - 385 \, b^{2} \sin \left (f x + e\right )^{5} + 624 \, a^{2} \sin \left (f x + e\right )^{3} + 1168 \, a b \sin \left (f x + e\right )^{3} + 511 \, b^{2} \sin \left (f x + e\right )^{3} - 240 \, a^{2} \sin \left (f x + e\right ) - 528 \, a b \sin \left (f x + e\right ) - 279 \, b^{2} \sin \left (f x + e\right )\right )}}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{4}}}{768 \, f} \]
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Time = 18.83 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.03 \[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\left (-\frac {3\,a^2}{8}-\frac {5\,a\,b}{8}-\frac {35\,b^2}{128}\right )\,{\sin \left (e+f\,x\right )}^7+\left (\frac {11\,a^2}{8}+\frac {55\,a\,b}{24}+\frac {385\,b^2}{384}\right )\,{\sin \left (e+f\,x\right )}^5+\left (-\frac {13\,a^2}{8}-\frac {73\,a\,b}{24}-\frac {511\,b^2}{384}\right )\,{\sin \left (e+f\,x\right )}^3+\left (\frac {5\,a^2}{8}+\frac {11\,a\,b}{8}+\frac {93\,b^2}{128}\right )\,\sin \left (e+f\,x\right )}{f\,\left ({\sin \left (e+f\,x\right )}^8-4\,{\sin \left (e+f\,x\right )}^6+6\,{\sin \left (e+f\,x\right )}^4-4\,{\sin \left (e+f\,x\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\sin \left (e+f\,x\right )\right )\,\left (\frac {3\,a^2}{8}+\frac {5\,a\,b}{8}+\frac {35\,b^2}{128}\right )}{f} \]
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