Integrand size = 23, antiderivative size = 129 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\left (8 a^2+12 a b+5 b^2\right ) \text {arctanh}(\sin (e+f x))}{16 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \sec (e+f x) \tan (e+f x)}{16 f}+\frac {b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f} \]
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Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4232, 424, 393, 205, 212} \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\left (8 a^2+12 a b+5 b^2\right ) \text {arctanh}(\sin (e+f x))}{16 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \tan (e+f x) \sec (e+f x)}{16 f}+\frac {b (8 a+5 b) \tan (e+f x) \sec ^3(e+f x)}{24 f}+\frac {b \tan (e+f x) \sec ^5(e+f x) \left (-a \sin ^2(e+f x)+a+b\right )}{6 f} \]
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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 )^4} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}-\frac {\text {Subst}\left (\int \frac {-((a+b) (6 a+5 b))+3 a (2 a+b) x^2}{\left (1-x^2\right )^3} \, dx,x,\sin (e+f x)\right )}{6 f} \\ & = \frac {b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{8 f} \\ & = \frac {\left (8 a^2+12 a b+5 b^2\right ) \sec (e+f x) \tan (e+f x)}{16 f}+\frac {b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (e+f x)\right )}{16 f} \\ & = \frac {\left (8 a^2+12 a b+5 b^2\right ) \text {arctanh}(\sin (e+f x))}{16 f}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \sec (e+f x) \tan (e+f x)}{16 f}+\frac {b (8 a+5 b) \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b \sec ^5(e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{6 f} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.45 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {a^2 \text {arctanh}(\sin (e+f x))}{2 f}+\frac {3 a b \text {arctanh}(\sin (e+f x))}{4 f}+\frac {5 b^2 \text {arctanh}(\sin (e+f x))}{16 f}+\frac {a^2 \sec (e+f x) \tan (e+f x)}{2 f}+\frac {3 a b \sec (e+f x) \tan (e+f x)}{4 f}+\frac {5 b^2 \sec (e+f x) \tan (e+f x)}{16 f}+\frac {a b \sec ^3(e+f x) \tan (e+f x)}{2 f}+\frac {5 b^2 \sec ^3(e+f x) \tan (e+f x)}{24 f}+\frac {b^2 \sec ^5(e+f x) \tan (e+f x)}{6 f} \]
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Time = 1.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\tan \left (f x +e \right ) \sec \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}\) | \(147\) |
default | \(\frac {a^{2} \left (\frac {\tan \left (f x +e \right ) \sec \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )+b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}\) | \(147\) |
parts | \(\frac {a^{2} \left (\frac {\tan \left (f x +e \right ) \sec \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}+\frac {b^{2} \left (-\left (-\frac {\sec \left (f x +e \right )^{5}}{6}-\frac {5 \sec \left (f x +e \right )^{3}}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )}{f}+\frac {2 a b \left (-\left (-\frac {\sec \left (f x +e \right )^{3}}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )}{f}\) | \(152\) |
parallelrisch | \(\frac {-360 \left (a^{2}+\frac {3}{2} a b +\frac {5}{8} b^{2}\right ) \left (\frac {2}{3}+\frac {\cos \left (6 f x +6 e \right )}{15}+\frac {2 \cos \left (4 f x +4 e \right )}{5}+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+360 \left (a^{2}+\frac {3}{2} a b +\frac {5}{8} b^{2}\right ) \left (\frac {2}{3}+\frac {\cos \left (6 f x +6 e \right )}{15}+\frac {2 \cos \left (4 f x +4 e \right )}{5}+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\left (144 a^{2}+408 a b +170 b^{2}\right ) \sin \left (3 f x +3 e \right )+\left (48 a^{2}+72 a b +30 b^{2}\right ) \sin \left (5 f x +5 e \right )+96 \sin \left (f x +e \right ) \left (a^{2}+\frac {7}{2} a b +\frac {33}{8} b^{2}\right )}{48 f \left (10+\cos \left (6 f x +6 e \right )+6 \cos \left (4 f x +4 e \right )+15 \cos \left (2 f x +2 e \right )\right )}\) | \(233\) |
norman | \(\frac {\frac {\left (8 a^{2}+4 a b +15 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{4 f}+\frac {\left (8 a^{2}+4 a b +15 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 f}+\frac {\left (8 a^{2}+20 a b +11 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {\left (8 a^{2}+20 a b +11 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 f}-\frac {\left (72 a^{2}+84 a b -5 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{24 f}-\frac {\left (72 a^{2}+84 a b -5 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{24 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{6}}-\frac {\left (8 a^{2}+12 a b +5 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}+\frac {\left (8 a^{2}+12 a b +5 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\) | \(267\) |
risch | \(-\frac {i {\mathrm e}^{i \left (f x +e \right )} \left (24 a^{2} {\mathrm e}^{10 i \left (f x +e \right )}+36 a b \,{\mathrm e}^{10 i \left (f x +e \right )}+15 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}+72 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}+204 a b \,{\mathrm e}^{8 i \left (f x +e \right )}+85 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}+48 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+168 a b \,{\mathrm e}^{6 i \left (f x +e \right )}+198 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-48 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-168 a b \,{\mathrm e}^{4 i \left (f x +e \right )}-198 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-72 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-204 a b \,{\mathrm e}^{2 i \left (f x +e \right )}-85 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-24 a^{2}-36 a b -15 b^{2}\right )}{24 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a^{2}}{2 f}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a b}{4 f}+\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b^{2}}{16 f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a^{2}}{2 f}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a b}{4 f}-\frac {5 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b^{2}}{16 f}\) | \(406\) |
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Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (12 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, b^{2}\right )} \sin \left (f x + e\right )}{96 \, f \cos \left (f x + e\right )^{6}} \]
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\[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \sec ^{3}{\left (e + f x \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.29 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \sin \left (f x + e\right )^{5} - 8 \, {\left (6 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \sin \left (f x + e\right )^{3} + 3 \, {\left (8 \, a^{2} + 20 \, a b + 11 \, b^{2}\right )} \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1}}{96 \, f} \]
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Time = 0.31 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.42 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, a^{2} \sin \left (f x + e\right )^{5} + 36 \, a b \sin \left (f x + e\right )^{5} + 15 \, b^{2} \sin \left (f x + e\right )^{5} - 48 \, a^{2} \sin \left (f x + e\right )^{3} - 96 \, a b \sin \left (f x + e\right )^{3} - 40 \, b^{2} \sin \left (f x + e\right )^{3} + 24 \, a^{2} \sin \left (f x + e\right ) + 60 \, a b \sin \left (f x + e\right ) + 33 \, b^{2} \sin \left (f x + e\right )\right )}}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{3}}}{96 \, f} \]
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Time = 18.61 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.04 \[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\mathrm {atanh}\left (\sin \left (e+f\,x\right )\right )\,\left (\frac {a^2}{2}+\frac {3\,a\,b}{4}+\frac {5\,b^2}{16}\right )}{f}-\frac {\left (\frac {a^2}{2}+\frac {3\,a\,b}{4}+\frac {5\,b^2}{16}\right )\,{\sin \left (e+f\,x\right )}^5+\left (-a^2-2\,a\,b-\frac {5\,b^2}{6}\right )\,{\sin \left (e+f\,x\right )}^3+\left (\frac {a^2}{2}+\frac {5\,a\,b}{4}+\frac {11\,b^2}{16}\right )\,\sin \left (e+f\,x\right )}{f\,\left ({\sin \left (e+f\,x\right )}^6-3\,{\sin \left (e+f\,x\right )}^4+3\,{\sin \left (e+f\,x\right )}^2-1\right )} \]
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