Integrand size = 27, antiderivative size = 74 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {a \text {arctanh}(\sin (c+d x))}{8 d}-\frac {a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b \tan ^4(c+d x)}{4 d} \]
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Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2913, 2691, 3853, 3855, 2687, 30} \[ \int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac {a \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b \tan ^4(c+d x)}{4 d} \]
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Rule 30
Rule 2687
Rule 2691
Rule 2913
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = a \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx+b \int \sec ^2(c+d x) \tan ^3(c+d x) \, dx \\ & = \frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{4} a \int \sec ^3(c+d x) \, dx+\frac {b \text {Subst}\left (\int x^3 \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b \tan ^4(c+d x)}{4 d}-\frac {1}{8} a \int \sec (c+d x) \, dx \\ & = -\frac {a \text {arctanh}(\sin (c+d x))}{8 d}-\frac {a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b \tan ^4(c+d x)}{4 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {a \text {arctanh}(\sin (c+d x))}{8 d}-\frac {a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b \tan ^4(c+d x)}{4 d} \]
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Time = 0.52 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {b \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(88\) |
default | \(\frac {a \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {b \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(88\) |
risch | \(\frac {i \left (a \,{\mathrm e}^{7 i \left (d x +c \right )}-7 a \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i b \,{\mathrm e}^{6 i \left (d x +c \right )}+7 a \,{\mathrm e}^{3 i \left (d x +c \right )}-a \,{\mathrm e}^{i \left (d x +c \right )}+8 i b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(133\) |
parallelrisch | \(\frac {2 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+7 a \sin \left (d x +c \right )-a \sin \left (3 d x +3 c \right )-4 b \cos \left (2 d x +2 c \right )+\cos \left (4 d x +4 c \right ) b +3 b}{4 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(152\) |
norman | \(\frac {\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {7 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {2 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {4 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(187\) |
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Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.23 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {a \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \, b \cos \left (d x + c\right )^{2} + 2 \, {\left (a \cos \left (d x + c\right )^{2} - 2 \, a\right )} \sin \left (d x + c\right ) - 4 \, b}{16 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (a \sin \left (d x + c\right )^{3} + 4 \, b \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) - 2 \, b\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.05 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=-\frac {a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (a \sin \left (d x + c\right )^{3} + 4 \, b \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) - 2 \, b\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
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Time = 17.84 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.95 \[ \int \sec ^3(c+d x) (a+b \sin (c+d x)) \tan ^2(c+d x) \, dx=\frac {\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d} \]
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