Integrand size = 17, antiderivative size = 66 \[ \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {5 \text {arctanh}(\sin (a+b x))}{2 b}+\frac {5 \sin (a+b x)}{2 b}+\frac {5 \sin ^3(a+b x)}{6 b}+\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2672, 294, 308, 212} \[ \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {5 \text {arctanh}(\sin (a+b x))}{2 b}+\frac {5 \sin ^3(a+b x)}{6 b}+\frac {5 \sin (a+b x)}{2 b}+\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b} \]
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Rule 212
Rule 294
Rule 308
Rule 2672
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b} \\ & = \frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\sin (a+b x)\right )}{2 b} \\ & = \frac {5 \sin (a+b x)}{2 b}+\frac {5 \sin ^3(a+b x)}{6 b}+\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac {5 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (a+b x)\right )}{2 b} \\ & = -\frac {5 \text {arctanh}(\sin (a+b x))}{2 b}+\frac {5 \sin (a+b x)}{2 b}+\frac {5 \sin ^3(a+b x)}{6 b}+\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{2 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18 \[ \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {5 \text {arctanh}(\sin (a+b x))}{2 b}+\frac {5 \sec (a+b x) \tan (a+b x)}{2 b}-\frac {5 \sin (a+b x) \tan ^2(a+b x)}{3 b}-\frac {\sin ^3(a+b x) \tan ^2(a+b x)}{3 b} \]
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Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {\frac {\sin ^{7}\left (b x +a \right )}{2 \cos \left (b x +a \right )^{2}}+\frac {\left (\sin ^{5}\left (b x +a \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (b x +a \right )\right )}{6}+\frac {5 \sin \left (b x +a \right )}{2}-\frac {5 \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2}}{b}\) | \(68\) |
default | \(\frac {\frac {\sin ^{7}\left (b x +a \right )}{2 \cos \left (b x +a \right )^{2}}+\frac {\left (\sin ^{5}\left (b x +a \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (b x +a \right )\right )}{6}+\frac {5 \sin \left (b x +a \right )}{2}-\frac {5 \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2}}{b}\) | \(68\) |
parallelrisch | \(\frac {\left (60 \cos \left (2 b x +2 a \right )+60\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-60 \cos \left (2 b x +2 a \right )-60\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+50 \sin \left (b x +a \right )+25 \sin \left (3 b x +3 a \right )-\sin \left (5 b x +5 a \right )}{24 b \left (1+\cos \left (2 b x +2 a \right )\right )}\) | \(102\) |
risch | \(\frac {i {\mathrm e}^{3 i \left (b x +a \right )}}{24 b}-\frac {9 i {\mathrm e}^{i \left (b x +a \right )}}{8 b}+\frac {9 i {\mathrm e}^{-i \left (b x +a \right )}}{8 b}-\frac {i {\mathrm e}^{-3 i \left (b x +a \right )}}{24 b}-\frac {i \left ({\mathrm e}^{3 i \left (b x +a \right )}-{\mathrm e}^{i \left (b x +a \right )}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2}}-\frac {5 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{2 b}+\frac {5 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{2 b}\) | \(138\) |
norman | \(\frac {\frac {5 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {20 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}-\frac {22 \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {20 \left (\tan ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {5 \left (\tan ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{2}}+\frac {5 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{2 b}-\frac {5 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{2 b}\) | \(146\) |
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Time = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.27 \[ \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx=-\frac {15 \, \cos \left (b x + a\right )^{2} \log \left (\sin \left (b x + a\right ) + 1\right ) - 15 \, \cos \left (b x + a\right )^{2} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, {\left (2 \, \cos \left (b x + a\right )^{4} - 14 \, \cos \left (b x + a\right )^{2} - 3\right )} \sin \left (b x + a\right )}{12 \, b \cos \left (b x + a\right )^{2}} \]
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Timed out. \[ \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx=\frac {4 \, \sin \left (b x + a\right )^{3} - \frac {6 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} - 15 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\sin \left (b x + a\right ) - 1\right ) + 24 \, \sin \left (b x + a\right )}{12 \, b} \]
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Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx=\frac {4 \, \sin \left (b x + a\right )^{3} - \frac {6 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} - 15 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 15 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right ) + 24 \, \sin \left (b x + a\right )}{12 \, b} \]
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Time = 7.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.23 \[ \int \sin ^3(a+b x) \tan ^3(a+b x) \, dx=\frac {5\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^9+\frac {20\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^7}{3}-\frac {22\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^5}{3}+\frac {20\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^3}{3}+5\,\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^{10}+{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^8-2\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6-2\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2+1\right )}-\frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{b} \]
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