Integrand size = 23, antiderivative size = 106 \[ \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx=\frac {a^2 \tan (e+f x)}{f}+\frac {2 a (2 a+b) \tan ^3(e+f x)}{3 f}+\frac {\left (6 a^2+6 a b+b^2\right ) \tan ^5(e+f x)}{5 f}+\frac {2 (a+b) (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {(a+b)^2 \tan ^9(e+f x)}{9 f} \]
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Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3270, 380} \[ \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx=\frac {\left (6 a^2+6 a b+b^2\right ) \tan ^5(e+f x)}{5 f}+\frac {a^2 \tan (e+f x)}{f}+\frac {(a+b)^2 \tan ^9(e+f x)}{9 f}+\frac {2 (a+b) (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {2 a (2 a+b) \tan ^3(e+f x)}{3 f} \]
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Rule 380
Rule 3270
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (1+x^2\right )^2 \left (a+(a+b) x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (a^2+2 a (2 a+b) x^2+\left (6 a^2+6 a b+b^2\right ) x^4+2 (a+b) (2 a+b) x^6+(a+b)^2 x^8\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a^2 \tan (e+f x)}{f}+\frac {2 a (2 a+b) \tan ^3(e+f x)}{3 f}+\frac {\left (6 a^2+6 a b+b^2\right ) \tan ^5(e+f x)}{5 f}+\frac {2 (a+b) (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {(a+b)^2 \tan ^9(e+f x)}{9 f} \\ \end{align*}
Time = 6.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.01 \[ \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx=\frac {\sec ^9(e+f x) \left (252 \left (8 a^2+8 a b+3 b^2\right ) \sin (e+f x)+336 \left (4 a^2-a b-b^2\right ) \sin (3 (e+f x))+\left (16 a^2-4 a b+b^2\right ) (36 \sin (5 (e+f x))+9 \sin (7 (e+f x))+\sin (9 (e+f x)))\right )}{10080 f} \]
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Time = 1.69 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.84
method | result | size |
derivativedivides | \(\frac {-a^{2} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (f x +e \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (f x +e \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (f x +e \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (f x +e \right )\right )}{315}\right ) \tan \left (f x +e \right )+2 a b \left (\frac {\sin ^{3}\left (f x +e \right )}{9 \cos \left (f x +e \right )^{9}}+\frac {2 \left (\sin ^{3}\left (f x +e \right )\right )}{21 \cos \left (f x +e \right )^{7}}+\frac {8 \left (\sin ^{3}\left (f x +e \right )\right )}{105 \cos \left (f x +e \right )^{5}}+\frac {16 \left (\sin ^{3}\left (f x +e \right )\right )}{315 \cos \left (f x +e \right )^{3}}\right )+b^{2} \left (\frac {\sin ^{5}\left (f x +e \right )}{9 \cos \left (f x +e \right )^{9}}+\frac {4 \left (\sin ^{5}\left (f x +e \right )\right )}{63 \cos \left (f x +e \right )^{7}}+\frac {8 \left (\sin ^{5}\left (f x +e \right )\right )}{315 \cos \left (f x +e \right )^{5}}\right )}{f}\) | \(195\) |
default | \(\frac {-a^{2} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (f x +e \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (f x +e \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (f x +e \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (f x +e \right )\right )}{315}\right ) \tan \left (f x +e \right )+2 a b \left (\frac {\sin ^{3}\left (f x +e \right )}{9 \cos \left (f x +e \right )^{9}}+\frac {2 \left (\sin ^{3}\left (f x +e \right )\right )}{21 \cos \left (f x +e \right )^{7}}+\frac {8 \left (\sin ^{3}\left (f x +e \right )\right )}{105 \cos \left (f x +e \right )^{5}}+\frac {16 \left (\sin ^{3}\left (f x +e \right )\right )}{315 \cos \left (f x +e \right )^{3}}\right )+b^{2} \left (\frac {\sin ^{5}\left (f x +e \right )}{9 \cos \left (f x +e \right )^{9}}+\frac {4 \left (\sin ^{5}\left (f x +e \right )\right )}{63 \cos \left (f x +e \right )^{7}}+\frac {8 \left (\sin ^{5}\left (f x +e \right )\right )}{315 \cos \left (f x +e \right )^{5}}\right )}{f}\) | \(195\) |
parallelrisch | \(-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (a^{2} \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {8 a \left (a -b \right ) \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {4 \left (19 a^{2}+4 a b +4 b^{2}\right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {8 \left (-71 a^{2}+79 a b +24 b^{2}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}+\frac {2 \left (5329 a^{2}+1424 a b +1744 b^{2}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315}+\frac {8 \left (-71 a^{2}+79 a b +24 b^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}+\frac {4 \left (19 a^{2}+4 a b +4 b^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}-\frac {8 a \left (a -b \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+a^{2}\right )}{f \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(227\) |
risch | \(\frac {16 i \left (210 b^{2} {\mathrm e}^{12 i \left (f x +e \right )}-1260 a b \,{\mathrm e}^{10 i \left (f x +e \right )}-315 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}+2016 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}+756 a b \,{\mathrm e}^{8 i \left (f x +e \right )}+441 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}+1344 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}-336 a b \,{\mathrm e}^{6 i \left (f x +e \right )}-126 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+576 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-144 a b \,{\mathrm e}^{4 i \left (f x +e \right )}+36 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+144 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-36 a b \,{\mathrm e}^{2 i \left (f x +e \right )}+9 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+16 a^{2}-4 a b +b^{2}\right )}{315 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{9}}\) | \(238\) |
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Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.21 \[ \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx=\frac {{\left (8 \, {\left (16 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{8} + 4 \, {\left (16 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{6} + 3 \, {\left (16 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 10 \, {\left (4 \, a^{2} - a b - 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 35 \, a^{2} + 70 \, a b + 35 \, b^{2}\right )} \sin \left (f x + e\right )}{315 \, f \cos \left (f x + e\right )^{9}} \]
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Timed out. \[ \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.97 \[ \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx=\frac {35 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{9} + 90 \, {\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{7} + 63 \, {\left (6 \, a^{2} + 6 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} + 210 \, {\left (2 \, a^{2} + a b\right )} \tan \left (f x + e\right )^{3} + 315 \, a^{2} \tan \left (f x + e\right )}{315 \, f} \]
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Time = 0.37 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.47 \[ \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx=\frac {35 \, a^{2} \tan \left (f x + e\right )^{9} + 70 \, a b \tan \left (f x + e\right )^{9} + 35 \, b^{2} \tan \left (f x + e\right )^{9} + 180 \, a^{2} \tan \left (f x + e\right )^{7} + 270 \, a b \tan \left (f x + e\right )^{7} + 90 \, b^{2} \tan \left (f x + e\right )^{7} + 378 \, a^{2} \tan \left (f x + e\right )^{5} + 378 \, a b \tan \left (f x + e\right )^{5} + 63 \, b^{2} \tan \left (f x + e\right )^{5} + 420 \, a^{2} \tan \left (f x + e\right )^{3} + 210 \, a b \tan \left (f x + e\right )^{3} + 315 \, a^{2} \tan \left (f x + e\right )}{315 \, f} \]
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Time = 14.00 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89 \[ \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx=\frac {a^2\,\mathrm {tan}\left (e+f\,x\right )+\frac {{\mathrm {tan}\left (e+f\,x\right )}^9\,{\left (a+b\right )}^2}{9}+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {6\,a^2}{5}+\frac {6\,a\,b}{5}+\frac {b^2}{5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (\frac {4\,a^2}{7}+\frac {6\,a\,b}{7}+\frac {2\,b^2}{7}\right )+\frac {2\,a\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (2\,a+b\right )}{3}}{f} \]
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