Integrand size = 19, antiderivative size = 42 \[ \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^2 \cos (c+d x)}{d}-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \sec (c+d x)}{d} \]
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Time = 0.10 (sec) , antiderivative size = 42, 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} \[ \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^2 \cos (c+d x)}{d}-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \sec (c+d x)}{d} \]
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Rule 12
Rule 45
Rule 2912
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int (-b-a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a^2 (-b+x)^2}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a d} \\ & = \frac {a \text {Subst}\left (\int \frac {(-b+x)^2}{x^2} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \left (1+\frac {b^2}{x^2}-\frac {2 b}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^2 \cos (c+d x)}{d}-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {b^2 \sec (c+d x)}{d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx=\frac {-a^2 \cos (c+d x)+b (-2 a \log (\cos (c+d x))+b \sec (c+d x))}{d} \]
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Time = 0.54 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\sec \left (d x +c \right ) b^{2}-\frac {a^{2}}{\sec \left (d x +c \right )}+2 a b \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(40\) |
default | \(\frac {\sec \left (d x +c \right ) b^{2}-\frac {a^{2}}{\sec \left (d x +c \right )}+2 a b \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(40\) |
parts | \(-\frac {a^{2} \cos \left (d x +c \right )}{d}+\frac {b^{2} \sec \left (d x +c \right )}{d}+\frac {2 a b \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(43\) |
risch | \(2 i a b x -\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 d}+\frac {4 i a b c}{d}+\frac {2 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(100\) |
parallelrisch | \(\frac {4 a b \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \cos \left (d x +c \right )-4 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-4 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )-\cos \left (2 d x +2 c \right ) a^{2}+\left (-2 a^{2}+2 b^{2}\right ) \cos \left (d x +c \right )-a^{2}+2 b^{2}}{2 d \cos \left (d x +c \right )}\) | \(123\) |
norman | \(\frac {\frac {2 a^{2}-2 b^{2}}{d}-\frac {2 \left (a^{2}+b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a b \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(131\) |
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - b^{2}}{d \cos \left (d x + c\right )} \]
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\[ \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \sin {\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^{2} \cos \left (d x + c\right ) + 2 \, a b \log \left (\cos \left (d x + c\right )\right ) - \frac {b^{2}}{\cos \left (d x + c\right )}}{d} \]
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Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^{2} \cos \left (d x + c\right )}{d} - \frac {2 \, a b \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {b^{2}}{d \cos \left (d x + c\right )} \]
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Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx=-\frac {a^2\,\cos \left (c+d\,x\right )-\frac {b^2}{\cos \left (c+d\,x\right )}+2\,a\,b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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