Integrand size = 25, antiderivative size = 135 \[ \int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx=\frac {15 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \cos ^2(c+d x)}{2 d}-\frac {3 b \log (\cos (c+d x))}{d}-\frac {3 b \sec ^2(c+d x)}{2 d}+\frac {b \sec ^4(c+d x)}{4 d}-\frac {15 a \sin (c+d x)}{8 d}-\frac {5 a \sin (c+d x) \tan ^2(c+d x)}{8 d}+\frac {a \sin (c+d x) \tan ^4(c+d x)}{4 d} \]
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Time = 0.09 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2913, 2672, 294, 327, 212, 2670, 272, 45} \[ \int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx=\frac {15 a \text {arctanh}(\sin (c+d x))}{8 d}-\frac {15 a \sin (c+d x)}{8 d}+\frac {a \sin (c+d x) \tan ^4(c+d x)}{4 d}-\frac {5 a \sin (c+d x) \tan ^2(c+d x)}{8 d}+\frac {b \cos ^2(c+d x)}{2 d}+\frac {b \sec ^4(c+d x)}{4 d}-\frac {3 b \sec ^2(c+d x)}{2 d}-\frac {3 b \log (\cos (c+d x))}{d} \]
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Rule 45
Rule 212
Rule 272
Rule 294
Rule 327
Rule 2670
Rule 2672
Rule 2913
Rubi steps \begin{align*} \text {integral}& = a \int \sin (c+d x) \tan ^5(c+d x) \, dx+b \int \sin ^2(c+d x) \tan ^5(c+d x) \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3} \, dx,x,\sin (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^5} \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a \sin (c+d x) \tan ^4(c+d x)}{4 d}-\frac {(5 a) \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{4 d}-\frac {b \text {Subst}\left (\int \frac {(1-x)^3}{x^3} \, dx,x,\cos ^2(c+d x)\right )}{2 d} \\ & = -\frac {5 a \sin (c+d x) \tan ^2(c+d x)}{8 d}+\frac {a \sin (c+d x) \tan ^4(c+d x)}{4 d}+\frac {(15 a) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\sin (c+d x)\right )}{8 d}-\frac {b \text {Subst}\left (\int \left (-1+\frac {1}{x^3}-\frac {3}{x^2}+\frac {3}{x}\right ) \, dx,x,\cos ^2(c+d x)\right )}{2 d} \\ & = \frac {b \cos ^2(c+d x)}{2 d}-\frac {3 b \log (\cos (c+d x))}{d}-\frac {3 b \sec ^2(c+d x)}{2 d}+\frac {b \sec ^4(c+d x)}{4 d}-\frac {15 a \sin (c+d x)}{8 d}-\frac {5 a \sin (c+d x) \tan ^2(c+d x)}{8 d}+\frac {a \sin (c+d x) \tan ^4(c+d x)}{4 d}+\frac {(15 a) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{8 d} \\ & = \frac {15 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \cos ^2(c+d x)}{2 d}-\frac {3 b \log (\cos (c+d x))}{d}-\frac {3 b \sec ^2(c+d x)}{2 d}+\frac {b \sec ^4(c+d x)}{4 d}-\frac {15 a \sin (c+d x)}{8 d}-\frac {5 a \sin (c+d x) \tan ^2(c+d x)}{8 d}+\frac {a \sin (c+d x) \tan ^4(c+d x)}{4 d} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.08 \[ \int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx=\frac {15 a \text {arctanh}(\sin (c+d x))}{8 d}-\frac {b \left (12 \log (\cos (c+d x))+6 \sec ^2(c+d x)-\sec ^4(c+d x)+2 \sin ^2(c+d x)\right )}{4 d}+\frac {15 a \sec (c+d x) \tan (c+d x)}{8 d}-\frac {15 a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {5 a \sec (c+d x) \tan ^3(c+d x)}{d}-\frac {a \sin (c+d x) \tan ^4(c+d x)}{d} \]
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Time = 0.83 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(167\) |
default | \(\frac {a \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(167\) |
parallelrisch | \(\frac {96 b \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 \left (a +\frac {8 b}{5}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+60 \left (a -\frac {8 b}{5}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-30 a \sin \left (3 d x +3 c \right )-4 a \sin \left (5 d x +5 c \right )-10 a \sin \left (d x +c \right )-9 b \cos \left (2 d x +2 c \right )+12 \cos \left (4 d x +4 c \right ) b +b \cos \left (6 d x +6 c \right )-4 b}{8 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(221\) |
risch | \(3 i x b +\frac {b \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {6 i b c}{d}+\frac {i \left (9 a \,{\mathrm e}^{7 i \left (d x +c \right )}+a \,{\mathrm e}^{5 i \left (d x +c \right )}+24 i b \,{\mathrm e}^{6 i \left (d x +c \right )}-a \,{\mathrm e}^{3 i \left (d x +c \right )}+32 i b \,{\mathrm e}^{4 i \left (d x +c \right )}-9 a \,{\mathrm e}^{i \left (d x +c \right )}+24 i b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b}{d}-\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b}{d}\) | \(259\) |
norman | \(\frac {\frac {12 b}{d}+\frac {12 b \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {15 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {25 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {11 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {11 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {25 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {15 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {52 b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {30 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {30 b \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {3 b \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 \left (5 a -8 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}-\frac {3 \left (8 b +5 a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}\) | \(276\) |
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Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.02 \[ \int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx=\frac {8 \, b \cos \left (d x + c\right )^{6} + 3 \, {\left (5 \, a - 8 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, a + 8 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 4 \, b \cos \left (d x + c\right )^{4} - 24 \, b \cos \left (d x + c\right )^{2} - 2 \, {\left (8 \, a \cos \left (d x + c\right )^{4} + 9 \, a \cos \left (d x + c\right )^{2} - 2 \, a\right )} \sin \left (d x + c\right ) + 4 \, b}{16 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90 \[ \int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx=-\frac {8 \, b \sin \left (d x + c\right )^{2} - 3 \, {\left (5 \, a - 8 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (5 \, a + 8 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, a \sin \left (d x + c\right ) - \frac {2 \, {\left (9 \, a \sin \left (d x + c\right )^{3} + 12 \, b \sin \left (d x + c\right )^{2} - 7 \, a \sin \left (d x + c\right ) - 10 \, b\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.92 \[ \int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx=-\frac {8 \, b \sin \left (d x + c\right )^{2} - 3 \, {\left (5 \, a - 8 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \, {\left (5 \, a + 8 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 16 \, a \sin \left (d x + c\right ) - \frac {2 \, {\left (18 \, b \sin \left (d x + c\right )^{4} + 9 \, a \sin \left (d x + c\right )^{3} - 24 \, b \sin \left (d x + c\right )^{2} - 7 \, a \sin \left (d x + c\right ) + 8 \, b\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
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Time = 12.35 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.25 \[ \int \sin (c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx=\frac {3\,b\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (\frac {15\,a}{8}+3\,b\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (\frac {15\,a}{8}-3\,b\right )}{d}-\frac {-\frac {15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{4}-6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {25\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+12\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}+12\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {25\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {15\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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