Integrand size = 21, antiderivative size = 62 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \]
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Time = 0.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 76} \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {a^2 \cos ^3(c+d x)}{3 d}+\frac {a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \sec (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \]
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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))^2 \sin (c+d x) \tan ^2(c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a^2 (-a-x) (-a+x)^3}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x) (-a+x)^3}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^4}{x^2}-\frac {2 a^3}{x}+2 a x-x^2\right ) \, dx,x,-a \cos (c+d x)\right )}{a d} \\ & = \frac {a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {a^2 (27+4 \cos (2 (c+d x))+6 \cos (3 (c+d x))+\cos (4 (c+d x))-6 \cos (c+d x) (1+8 \log (\cos (c+d x)))) \sec (c+d x)}{24 d} \]
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Time = 2.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.47
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(91\) |
default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(91\) |
parts | \(-\frac {a^{2} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3 d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )}{d}-\frac {a^{2} \sin \left (d x +c \right )^{2}}{d}-\frac {2 a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}\) | \(99\) |
parallelrisch | \(\frac {a^{2} \left (-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+48 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \cos \left (d x +c \right )+26 \cos \left (d x +c \right )+6 \cos \left (3 d x +3 c \right )+\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+27\right )}{24 d \cos \left (d x +c \right )}\) | \(118\) |
risch | \(2 i a^{2} x +\frac {a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}+\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}+\frac {4 i a^{2} c}{d}+\frac {2 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}+\frac {a^{2} \cos \left (3 d x +3 c \right )}{12 d}\) | \(154\) |
norman | \(\frac {-\frac {8 a^{2}}{3 d}-\frac {8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a^{2} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(161\) |
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {2 \, a^{2} \cos \left (d x + c\right )^{4} + 6 \, a^{2} \cos \left (d x + c\right )^{3} - 12 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 3 \, a^{2} \cos \left (d x + c\right ) + 6 \, a^{2}}{6 \, d \cos \left (d x + c\right )} \]
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\[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=a^{2} \left (\int 2 \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) + \frac {3 \, a^{2}}{\cos \left (d x + c\right )}}{3 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.19 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=-\frac {2 \, a^{2} \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {a^{2}}{d \cos \left (d x + c\right )} + \frac {a^{2} d^{5} \cos \left (d x + c\right )^{3} + 3 \, a^{2} d^{5} \cos \left (d x + c\right )^{2}}{3 \, d^{6}} \]
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Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {\frac {a^2}{\cos \left (c+d\,x\right )}+a^2\,{\cos \left (c+d\,x\right )}^2+\frac {a^2\,{\cos \left (c+d\,x\right )}^3}{3}-2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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