Integrand size = 21, antiderivative size = 72 \[ \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx=\frac {a \tan (e+f x)}{f}+\frac {(3 a+b) \tan ^3(e+f x)}{3 f}+\frac {(3 a+2 b) \tan ^5(e+f x)}{5 f}+\frac {(a+b) \tan ^7(e+f x)}{7 f} \]
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Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3270, 380} \[ \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx=\frac {(a+b) \tan ^7(e+f x)}{7 f}+\frac {(3 a+2 b) \tan ^5(e+f x)}{5 f}+\frac {(3 a+b) \tan ^3(e+f x)}{3 f}+\frac {a \tan (e+f x)}{f} \]
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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 ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (a+(3 a+b) x^2+(3 a+2 b) x^4+(a+b) x^6\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a \tan (e+f x)}{f}+\frac {(3 a+b) \tan ^3(e+f x)}{3 f}+\frac {(3 a+2 b) \tan ^5(e+f x)}{5 f}+\frac {(a+b) \tan ^7(e+f x)}{7 f} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.19 \[ \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx=\frac {\tan (e+f x) \left (105 a-8 b-4 b \sec ^2(e+f x)-3 b \sec ^4(e+f x)+15 b \sec ^6(e+f x)+105 a \tan ^2(e+f x)+63 a \tan ^4(e+f x)+15 a \tan ^6(e+f x)\right )}{105 f} \]
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Time = 1.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(\frac {-a \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (f x +e \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (f x +e \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (f x +e \right )\right )}{35}\right ) \tan \left (f x +e \right )+b \left (\frac {\sin ^{3}\left (f x +e \right )}{7 \cos \left (f x +e \right )^{7}}+\frac {4 \left (\sin ^{3}\left (f x +e \right )\right )}{35 \cos \left (f x +e \right )^{5}}+\frac {8 \left (\sin ^{3}\left (f x +e \right )\right )}{105 \cos \left (f x +e \right )^{3}}\right )}{f}\) | \(104\) |
default | \(\frac {-a \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (f x +e \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (f x +e \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (f x +e \right )\right )}{35}\right ) \tan \left (f x +e \right )+b \left (\frac {\sin ^{3}\left (f x +e \right )}{7 \cos \left (f x +e \right )^{7}}+\frac {4 \left (\sin ^{3}\left (f x +e \right )\right )}{35 \cos \left (f x +e \right )^{5}}+\frac {8 \left (\sin ^{3}\left (f x +e \right )\right )}{105 \cos \left (f x +e \right )^{3}}\right )}{f}\) | \(104\) |
risch | \(-\frac {16 i \left (70 b \,{\mathrm e}^{8 i \left (f x +e \right )}-210 a \,{\mathrm e}^{6 i \left (f x +e \right )}-35 b \,{\mathrm e}^{6 i \left (f x +e \right )}-126 a \,{\mathrm e}^{4 i \left (f x +e \right )}+21 \,{\mathrm e}^{4 i \left (f x +e \right )} b -42 a \,{\mathrm e}^{2 i \left (f x +e \right )}+7 \,{\mathrm e}^{2 i \left (f x +e \right )} b -6 a +b \right )}{105 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}\) | \(109\) |
parallelrisch | \(-\frac {2 \left (a \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-2 a +\frac {4 b}{3}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {43 a}{5}+\frac {16 b}{15}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {212 a}{35}+\frac {152 b}{35}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {43 a}{5}+\frac {16 b}{15}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-2 a +\frac {4 b}{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(140\) |
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Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07 \[ \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx=\frac {{\left (8 \, {\left (6 \, a - b\right )} \cos \left (f x + e\right )^{6} + 4 \, {\left (6 \, a - b\right )} \cos \left (f x + e\right )^{4} + 3 \, {\left (6 \, a - b\right )} \cos \left (f x + e\right )^{2} + 15 \, a + 15 \, b\right )} \sin \left (f x + e\right )}{105 \, f \cos \left (f x + e\right )^{7}} \]
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Timed out. \[ \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx=\frac {15 \, {\left (a + b\right )} \tan \left (f x + e\right )^{7} + 21 \, {\left (3 \, a + 2 \, b\right )} \tan \left (f x + e\right )^{5} + 35 \, {\left (3 \, a + b\right )} \tan \left (f x + e\right )^{3} + 105 \, a \tan \left (f x + e\right )}{105 \, f} \]
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Time = 0.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.12 \[ \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx=\frac {15 \, a \tan \left (f x + e\right )^{7} + 15 \, b \tan \left (f x + e\right )^{7} + 63 \, a \tan \left (f x + e\right )^{5} + 42 \, b \tan \left (f x + e\right )^{5} + 105 \, a \tan \left (f x + e\right )^{3} + 35 \, b \tan \left (f x + e\right )^{3} + 105 \, a \tan \left (f x + e\right )}{105 \, f} \]
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Time = 13.65 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \sec ^8(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx=\frac {\left (\frac {a}{7}+\frac {b}{7}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^7+\left (\frac {3\,a}{5}+\frac {2\,b}{5}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^5+\left (a+\frac {b}{3}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+a\,\mathrm {tan}\left (e+f\,x\right )}{f} \]
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