Integrand size = 17, antiderivative size = 55 \[ \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx=-\frac {\text {arctanh}(\sin (a+b x))}{8 b}-\frac {\sec (a+b x) \tan (a+b x)}{8 b}+\frac {\sec ^3(a+b x) \tan (a+b x)}{4 b} \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2691, 3853, 3855} \[ \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx=-\frac {\text {arctanh}(\sin (a+b x))}{8 b}+\frac {\tan (a+b x) \sec ^3(a+b x)}{4 b}-\frac {\tan (a+b x) \sec (a+b x)}{8 b} \]
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Rule 2691
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(a+b x) \tan (a+b x)}{4 b}-\frac {1}{4} \int \sec ^3(a+b x) \, dx \\ & = -\frac {\sec (a+b x) \tan (a+b x)}{8 b}+\frac {\sec ^3(a+b x) \tan (a+b x)}{4 b}-\frac {1}{8} \int \sec (a+b x) \, dx \\ & = -\frac {\text {arctanh}(\sin (a+b x))}{8 b}-\frac {\sec (a+b x) \tan (a+b x)}{8 b}+\frac {\sec ^3(a+b x) \tan (a+b x)}{4 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx=-\frac {\text {arctanh}(\sin (a+b x))}{8 b}-\frac {\sec (a+b x) \tan (a+b x)}{8 b}+\frac {\sec ^3(a+b x) \tan (a+b x)}{4 b} \]
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Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{4 \cos \left (b x +a \right )^{4}}+\frac {\sin ^{3}\left (b x +a \right )}{8 \cos \left (b x +a \right )^{2}}+\frac {\sin \left (b x +a \right )}{8}-\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{8}}{b}\) | \(66\) |
default | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{4 \cos \left (b x +a \right )^{4}}+\frac {\sin ^{3}\left (b x +a \right )}{8 \cos \left (b x +a \right )^{2}}+\frac {\sin \left (b x +a \right )}{8}-\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{8}}{b}\) | \(66\) |
risch | \(\frac {i \left ({\mathrm e}^{7 i \left (b x +a \right )}-7 \,{\mathrm e}^{5 i \left (b x +a \right )}+7 \,{\mathrm e}^{3 i \left (b x +a \right )}-{\mathrm e}^{i \left (b x +a \right )}\right )}{4 b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{8 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{8 b}\) | \(100\) |
norman | \(\frac {\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{4 b}+\frac {7 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {7 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )}{4 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{4}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{8 b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{8 b}\) | \(115\) |
parallelrisch | \(\frac {\left (\cos \left (4 b x +4 a \right )+4 \cos \left (2 b x +2 a \right )+3\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-\cos \left (4 b x +4 a \right )-4 \cos \left (2 b x +2 a \right )-3\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+14 \sin \left (b x +a \right )-2 \sin \left (3 b x +3 a \right )}{8 b \left (\cos \left (4 b x +4 a \right )+4 \cos \left (2 b x +2 a \right )+3\right )}\) | \(122\) |
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Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.29 \[ \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx=-\frac {\cos \left (b x + a\right )^{4} \log \left (\sin \left (b x + a\right ) + 1\right ) - \cos \left (b x + a\right )^{4} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, {\left (\cos \left (b x + a\right )^{2} - 2\right )} \sin \left (b x + a\right )}{16 \, b \cos \left (b x + a\right )^{4}} \]
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\[ \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx=\int \sin ^{2}{\left (a + b x \right )} \sec ^{5}{\left (a + b x \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx=\frac {\frac {2 \, {\left (\sin \left (b x + a\right )^{3} + \sin \left (b x + a\right )\right )}}{\sin \left (b x + a\right )^{4} - 2 \, \sin \left (b x + a\right )^{2} + 1} - \log \left (\sin \left (b x + a\right ) + 1\right ) + \log \left (\sin \left (b x + a\right ) - 1\right )}{16 \, b} \]
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Time = 0.35 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.49 \[ \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx=\frac {\frac {4 \, {\left (\frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right )\right )}}{{\left (\frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right )\right )}^{2} - 4} - \log \left ({\left | \frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) + 2 \right |}\right ) + \log \left ({\left | \frac {1}{\sin \left (b x + a\right )} + \sin \left (b x + a\right ) - 2 \right |}\right )}{32 \, b} \]
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Time = 5.68 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.27 \[ \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^7}{4}+\frac {7\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^5}{4}+\frac {7\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^3}{4}+\frac {\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}{4}}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2+1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{4\,b} \]
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