Integrand size = 19, antiderivative size = 43 \[ \int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \]
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Time = 0.10 (sec) , antiderivative size = 43, 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, 2912, 12, 45} \[ \int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \sec (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \]
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Rule 12
Rule 45
Rule 2912
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int (-a-a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a^2 (-a+x)^2}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a d} \\ & = \frac {a \text {Subst}\left (\int \frac {(-a+x)^2}{x^2} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \left (1+\frac {a^2}{x^2}-\frac {2 a}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72 \[ \int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx=\frac {a^2 (1-2 \log (\cos (c+d x))+\sin (c+d x) \tan (c+d x))}{d} \]
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Time = 0.83 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\sec \left (d x +c \right )-\frac {1}{\sec \left (d x +c \right )}+2 \ln \left (\sec \left (d x +c \right )\right )\right )}{d}\) | \(34\) |
default | \(\frac {a^{2} \left (\sec \left (d x +c \right )-\frac {1}{\sec \left (d x +c \right )}+2 \ln \left (\sec \left (d x +c \right )\right )\right )}{d}\) | \(34\) |
parts | \(-\frac {a^{2} \cos \left (d x +c \right )}{d}+\frac {a^{2} \sec \left (d x +c \right )}{d}+\frac {2 a^{2} \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(44\) |
parallelrisch | \(-\frac {a^{2} \left (4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )-4 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \cos \left (d x +c \right )+\cos \left (2 d x +2 c \right )-1\right )}{2 d \cos \left (d x +c \right )}\) | \(88\) |
risch | \(2 i a^{2} x -\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {4 i a^{2} c}{d}+\frac {2 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(103\) |
norman | \(-\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a^{2} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(113\) |
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - a^{2}}{d \cos \left (d x + c\right )} \]
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\[ \int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx=a^{2} \left (\int 2 \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^{2} \cos \left (d x + c\right ) + 2 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {a^{2}}{\cos \left (d x + c\right )}}{d} \]
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Time = 0.32 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^{2} \cos \left (d x + c\right )}{d} - \frac {2 \, a^{2} \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {a^{2}}{d \cos \left (d x + c\right )} \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^2\,\left (2\,\cos \left (c+d\,x\right )\,\ln \left (\cos \left (c+d\,x\right )\right )+{\cos \left (c+d\,x\right )}^2-1\right )}{d\,\cos \left (c+d\,x\right )} \]
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