Integrand size = 17, antiderivative size = 46 \[ \int \sec ^6(a+b x) \tan ^2(a+b x) \, dx=\frac {\tan ^3(a+b x)}{3 b}+\frac {2 \tan ^5(a+b x)}{5 b}+\frac {\tan ^7(a+b x)}{7 b} \]
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Time = 0.02 (sec) , antiderivative size = 46, 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} \[ \int \sec ^6(a+b x) \tan ^2(a+b x) \, dx=\frac {\tan ^7(a+b x)}{7 b}+\frac {2 \tan ^5(a+b x)}{5 b}+\frac {\tan ^3(a+b x)}{3 b} \]
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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 )^2 \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\tan ^3(a+b x)}{3 b}+\frac {2 \tan ^5(a+b x)}{5 b}+\frac {\tan ^7(a+b x)}{7 b} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.67 \[ \int \sec ^6(a+b x) \tan ^2(a+b x) \, dx=-\frac {8 \tan (a+b x)}{105 b}-\frac {4 \sec ^2(a+b x) \tan (a+b x)}{105 b}-\frac {\sec ^4(a+b x) \tan (a+b x)}{35 b}+\frac {\sec ^6(a+b x) \tan (a+b x)}{7 b} \]
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Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30
method | result | size |
derivativedivides | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{7 \cos \left (b x +a \right )^{7}}+\frac {4 \left (\sin ^{3}\left (b x +a \right )\right )}{35 \cos \left (b x +a \right )^{5}}+\frac {8 \left (\sin ^{3}\left (b x +a \right )\right )}{105 \cos \left (b x +a \right )^{3}}}{b}\) | \(60\) |
default | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{7 \cos \left (b x +a \right )^{7}}+\frac {4 \left (\sin ^{3}\left (b x +a \right )\right )}{35 \cos \left (b x +a \right )^{5}}+\frac {8 \left (\sin ^{3}\left (b x +a \right )\right )}{105 \cos \left (b x +a \right )^{3}}}{b}\) | \(60\) |
risch | \(-\frac {16 i \left (70 \,{\mathrm e}^{8 i \left (b x +a \right )}-35 \,{\mathrm e}^{6 i \left (b x +a \right )}+21 \,{\mathrm e}^{4 i \left (b x +a \right )}+7 \,{\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{105 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{7}}\) | \(66\) |
parallelrisch | \(-\frac {8 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (35 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+28 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+114 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+28 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+35\right )}{105 b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{7}}\) | \(86\) |
norman | \(\frac {-\frac {8 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {32 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{15 b}-\frac {304 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{35 b}-\frac {32 \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{15 b}-\frac {8 \left (\tan ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{7}}\) | \(98\) |
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.11 \[ \int \sec ^6(a+b x) \tan ^2(a+b x) \, dx=-\frac {{\left (8 \, \cos \left (b x + a\right )^{6} + 4 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 15\right )} \sin \left (b x + a\right )}{105 \, b \cos \left (b x + a\right )^{7}} \]
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Timed out. \[ \int \sec ^6(a+b x) \tan ^2(a+b x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sec ^6(a+b x) \tan ^2(a+b x) \, dx=\frac {15 \, \tan \left (b x + a\right )^{7} + 42 \, \tan \left (b x + a\right )^{5} + 35 \, \tan \left (b x + a\right )^{3}}{105 \, b} \]
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Time = 0.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sec ^6(a+b x) \tan ^2(a+b x) \, dx=\frac {15 \, \tan \left (b x + a\right )^{7} + 42 \, \tan \left (b x + a\right )^{5} + 35 \, \tan \left (b x + a\right )^{3}}{105 \, b} \]
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Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \sec ^6(a+b x) \tan ^2(a+b x) \, dx=\frac {{\mathrm {tan}\left (a+b\,x\right )}^3\,\left (15\,{\mathrm {tan}\left (a+b\,x\right )}^4+42\,{\mathrm {tan}\left (a+b\,x\right )}^2+35\right )}{105\,b} \]
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