Integrand size = 19, antiderivative size = 58 \[ \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx=-\frac {a \cos (c+d x)}{d}+\frac {a \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {a \log (\cos (c+d x))}{d} \]
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Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3957, 2915, 12, 76} \[ \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx=\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^2(c+d x)}{2 d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rule 12
Rule 76
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x)) \sin ^2(c+d x) \tan (c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a (-a-x) (-a+x)^2}{x} \, dx,x,-a \cos (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x) (-a+x)^2}{x} \, dx,x,-a \cos (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (a^2-\frac {a^3}{x}+a x-x^2\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d} \\ & = -\frac {a \cos (c+d x)}{d}+\frac {a \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {a \log (\cos (c+d x))}{d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx=-\frac {3 a \cos (c+d x)}{4 d}+\frac {a \cos (3 (c+d x))}{12 d}-\frac {a \left (-\frac {1}{2} \cos ^2(c+d x)+\log (\cos (c+d x))\right )}{d} \]
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Time = 1.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(47\) |
default | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(47\) |
parts | \(-\frac {a \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3 d}+\frac {a \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(49\) |
parallelrisch | \(\frac {a \left (12 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-11+\cos \left (3 d x +3 c \right )+3 \cos \left (2 d x +2 c \right )-9 \cos \left (d x +c \right )\right )}{12 d}\) | \(79\) |
risch | \(i a x +\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i a c}{d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {3 a \cos \left (d x +c \right )}{4 d}+\frac {a \cos \left (3 d x +3 c \right )}{12 d}\) | \(90\) |
norman | \(\frac {-\frac {4 a}{3 d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {6 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {a \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(114\) |
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Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx=\frac {2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )^{2} - 6 \, a \cos \left (d x + c\right ) - 6 \, a \log \left (-\cos \left (d x + c\right )\right )}{6 \, d} \]
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\[ \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx=a \left (\int \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx=\frac {2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )^{2} - 6 \, a \cos \left (d x + c\right ) - 6 \, a \log \left (\cos \left (d x + c\right )\right )}{6 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.14 \[ \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx=-\frac {a \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {2 \, a d^{2} \cos \left (d x + c\right )^{3} + 3 \, a d^{2} \cos \left (d x + c\right )^{2} - 6 \, a d^{2} \cos \left (d x + c\right )}{6 \, d^{3}} \]
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Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.78 \[ \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx=-\frac {a\,\cos \left (c+d\,x\right )-\frac {a\,{\cos \left (c+d\,x\right )}^2}{2}-\frac {a\,{\cos \left (c+d\,x\right )}^3}{3}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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