Integrand size = 15, antiderivative size = 40 \[ \int \sin ^4(a+b x) \tan (a+b x) \, dx=\frac {\cos ^2(a+b x)}{b}-\frac {\cos ^4(a+b x)}{4 b}-\frac {\log (\cos (a+b x))}{b} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2670, 272, 45} \[ \int \sin ^4(a+b x) \tan (a+b x) \, dx=-\frac {\cos ^4(a+b x)}{4 b}+\frac {\cos ^2(a+b x)}{b}-\frac {\log (\cos (a+b x))}{b} \]
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Rule 45
Rule 272
Rule 2670
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x} \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {(1-x)^2}{x} \, dx,x,\cos ^2(a+b x)\right )}{2 b} \\ & = -\frac {\text {Subst}\left (\int \left (-2+\frac {1}{x}+x\right ) \, dx,x,\cos ^2(a+b x)\right )}{2 b} \\ & = \frac {\cos ^2(a+b x)}{b}-\frac {\cos ^4(a+b x)}{4 b}-\frac {\log (\cos (a+b x))}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \sin ^4(a+b x) \tan (a+b x) \, dx=-\frac {-\cos ^2(a+b x)+\frac {1}{4} \cos ^4(a+b x)+\log (\cos (a+b x))}{b} \]
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Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\sin ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{2}-\ln \left (\cos \left (b x +a \right )\right )}{b}\) | \(35\) |
default | \(\frac {-\frac {\left (\sin ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{2}-\ln \left (\cos \left (b x +a \right )\right )}{b}\) | \(35\) |
risch | \(i x +\frac {3 \,{\mathrm e}^{2 i \left (b x +a \right )}}{16 b}+\frac {3 \,{\mathrm e}^{-2 i \left (b x +a \right )}}{16 b}+\frac {2 i a}{b}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}-\frac {\cos \left (4 b x +4 a \right )}{32 b}\) | \(72\) |
parallelrisch | \(\frac {-32 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )-32 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+32 \ln \left (\sec ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-11-\cos \left (4 b x +4 a \right )+12 \cos \left (2 b x +2 a \right )}{32 b}\) | \(72\) |
norman | \(\frac {-\frac {2 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {2 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {8 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{4}}+\frac {\ln \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}\) | \(119\) |
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Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \sin ^4(a+b x) \tan (a+b x) \, dx=-\frac {\cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} + 4 \, \log \left (-\cos \left (b x + a\right )\right )}{4 \, b} \]
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Timed out. \[ \int \sin ^4(a+b x) \tan (a+b x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \sin ^4(a+b x) \tan (a+b x) \, dx=-\frac {\sin \left (b x + a\right )^{4} + 2 \, \sin \left (b x + a\right )^{2} + 2 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right )}{4 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (38) = 76\).
Time = 0.34 (sec) , antiderivative size = 226, normalized size of antiderivative = 5.65 \[ \int \sin ^4(a+b x) \tan (a+b x) \, dx=-\frac {\frac {3 \, {\left (\frac {\cos \left (b x + a\right ) + 1}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}^{2} - \frac {20 \, {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} - \frac {20 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 44}{{\left (\frac {\cos \left (b x + a\right ) + 1}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2\right )}^{2}} - 2 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) + 1}{\cos \left (b x + a\right ) - 1} - \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 2 \right |}\right ) + 2 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) + 1}{\cos \left (b x + a\right ) - 1} - \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \right |}\right )}{4 \, b} \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int \sin ^4(a+b x) \tan (a+b x) \, dx=\frac {\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )}{2\,b}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^2+\frac {3}{4}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^4+2\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]
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