Integrand size = 17, antiderivative size = 61 \[ \int \sec ^8(a+b x) \tan ^2(a+b x) \, dx=\frac {\tan ^3(a+b x)}{3 b}+\frac {3 \tan ^5(a+b x)}{5 b}+\frac {3 \tan ^7(a+b x)}{7 b}+\frac {\tan ^9(a+b x)}{9 b} \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2687, 276} \[ \int \sec ^8(a+b x) \tan ^2(a+b x) \, dx=\frac {\tan ^9(a+b x)}{9 b}+\frac {3 \tan ^7(a+b x)}{7 b}+\frac {3 \tan ^5(a+b x)}{5 b}+\frac {\tan ^3(a+b x)}{3 b} \]
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Rule 276
Rule 2687
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^2 \left (1+x^2\right )^3 \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (x^2+3 x^4+3 x^6+x^8\right ) \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\tan ^3(a+b x)}{3 b}+\frac {3 \tan ^5(a+b x)}{5 b}+\frac {3 \tan ^7(a+b x)}{7 b}+\frac {\tan ^9(a+b x)}{9 b} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.61 \[ \int \sec ^8(a+b x) \tan ^2(a+b x) \, dx=-\frac {16 \tan (a+b x)}{315 b}-\frac {8 \sec ^2(a+b x) \tan (a+b x)}{315 b}-\frac {2 \sec ^4(a+b x) \tan (a+b x)}{105 b}-\frac {\sec ^6(a+b x) \tan (a+b x)}{63 b}+\frac {\sec ^8(a+b x) \tan (a+b x)}{9 b} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.26
method | result | size |
risch | \(-\frac {32 i \left (315 \,{\mathrm e}^{10 i \left (b x +a \right )}-189 \,{\mathrm e}^{8 i \left (b x +a \right )}+84 \,{\mathrm e}^{6 i \left (b x +a \right )}+36 \,{\mathrm e}^{4 i \left (b x +a \right )}+9 \,{\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{315 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{9}}\) | \(77\) |
derivativedivides | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{9 \cos \left (b x +a \right )^{9}}+\frac {2 \left (\sin ^{3}\left (b x +a \right )\right )}{21 \cos \left (b x +a \right )^{7}}+\frac {8 \left (\sin ^{3}\left (b x +a \right )\right )}{105 \cos \left (b x +a \right )^{5}}+\frac {16 \left (\sin ^{3}\left (b x +a \right )\right )}{315 \cos \left (b x +a \right )^{3}}}{b}\) | \(78\) |
default | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{9 \cos \left (b x +a \right )^{9}}+\frac {2 \left (\sin ^{3}\left (b x +a \right )\right )}{21 \cos \left (b x +a \right )^{7}}+\frac {8 \left (\sin ^{3}\left (b x +a \right )\right )}{105 \cos \left (b x +a \right )^{5}}+\frac {16 \left (\sin ^{3}\left (b x +a \right )\right )}{315 \cos \left (b x +a \right )^{3}}}{b}\) | \(78\) |
parallelrisch | \(-\frac {8 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (105 \left (\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+126 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+711 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+356 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+711 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+126 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+105\right )}{315 b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{9}}\) | \(112\) |
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Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \sec ^8(a+b x) \tan ^2(a+b x) \, dx=-\frac {{\left (16 \, \cos \left (b x + a\right )^{8} + 8 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 5 \, \cos \left (b x + a\right )^{2} - 35\right )} \sin \left (b x + a\right )}{315 \, b \cos \left (b x + a\right )^{9}} \]
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Timed out. \[ \int \sec ^8(a+b x) \tan ^2(a+b x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \sec ^8(a+b x) \tan ^2(a+b x) \, dx=\frac {35 \, \tan \left (b x + a\right )^{9} + 135 \, \tan \left (b x + a\right )^{7} + 189 \, \tan \left (b x + a\right )^{5} + 105 \, \tan \left (b x + a\right )^{3}}{315 \, b} \]
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Time = 0.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \sec ^8(a+b x) \tan ^2(a+b x) \, dx=\frac {35 \, \tan \left (b x + a\right )^{9} + 135 \, \tan \left (b x + a\right )^{7} + 189 \, \tan \left (b x + a\right )^{5} + 105 \, \tan \left (b x + a\right )^{3}}{315 \, b} \]
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Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \sec ^8(a+b x) \tan ^2(a+b x) \, dx=\frac {\frac {{\mathrm {tan}\left (a+b\,x\right )}^9}{9}+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^7}{7}+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^5}{5}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{3}}{b} \]
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