Integrand size = 19, antiderivative size = 64 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^2 b \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
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Time = 0.11 (sec) , antiderivative size = 64, 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} \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^2 b \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
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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))^3 \sec ^2(c+d x) \tan (c+d x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {a^3 (-b+x)^3}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a d} \\ & = \frac {a^2 \text {Subst}\left (\int \frac {(-b+x)^3}{x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^2 \text {Subst}\left (\int \left (1-\frac {b^3}{x^3}+\frac {3 b^2}{x^2}-\frac {3 b}{x}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^2 b \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=\frac {-2 a^3 \cos (c+d x)+b \left (-6 a^2 \log (\cos (c+d x))+6 a b \sec (c+d x)+b^2 \sec ^2(c+d x)\right )}{2 d} \]
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Time = 0.82 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} \sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right ) a \,b^{2}+3 a^{2} b \ln \left (\sec \left (d x +c \right )\right )-\frac {a^{3}}{\sec \left (d x +c \right )}}{d}\) | \(57\) |
default | \(\frac {\frac {b^{3} \sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right ) a \,b^{2}+3 a^{2} b \ln \left (\sec \left (d x +c \right )\right )-\frac {a^{3}}{\sec \left (d x +c \right )}}{d}\) | \(57\) |
parts | \(-\frac {a^{3} \cos \left (d x +c \right )}{d}+\frac {b^{3} \sec \left (d x +c \right )^{2}}{2 d}+\frac {3 a^{2} b \ln \left (\sec \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \sec \left (d x +c \right )}{d}\) | \(63\) |
risch | \(3 i a^{2} b x -\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {6 i b \,a^{2} c}{d}+\frac {2 b^{2} \left (3 a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )} a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}\) | \(133\) |
norman | \(\frac {\frac {\left (4 a^{3}+2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {2 a^{3}-6 a \,b^{2}}{d}-\frac {\left (2 a^{3}+6 a \,b^{2}-2 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {3 a^{2} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {3 a^{2} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {3 a^{2} b \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(175\) |
parallelrisch | \(\frac {6 a^{2} b \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-6 a^{2} b \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-6 a^{2} b \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-2 a^{3}+6 a \,b^{2}-b^{3}\right ) \cos \left (2 d x +2 c \right )-a^{3} \cos \left (3 d x +3 c \right )+\left (-3 a^{3}+12 a \,b^{2}\right ) \cos \left (d x +c \right )-2 a^{3}+6 a \,b^{2}+b^{3}}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(181\) |
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {2 \, a^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, a b^{2} \cos \left (d x + c\right ) - b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sin {\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {2 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{2} b \log \left (\cos \left (d x + c\right )\right ) - \frac {6 \, a b^{2}}{\cos \left (d x + c\right )} - \frac {b^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.03 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^{3} \cos \left (d x + c\right )}{d} - \frac {3 \, a^{2} b \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
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Time = 13.58 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^3\,\cos \left (c+d\,x\right )-\frac {\frac {b^3}{2}+3\,a\,\cos \left (c+d\,x\right )\,b^2}{{\cos \left (c+d\,x\right )}^2}+3\,a^2\,b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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