Integrand size = 21, antiderivative size = 98 \[ \int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx=-\frac {5 a^3 x}{2}+\frac {5 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^3 \sin (c+d x)}{d}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2951, 2717, 2715, 8, 3855, 3852, 3853} \[ \int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx=\frac {5 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \tan (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {5 a^3 x}{2} \]
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Rule 8
Rule 2715
Rule 2717
Rule 2951
Rule 3852
Rule 3853
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \sec (c+d x) \tan ^2(c+d x) \, dx \\ & = -\frac {\int \left (2 a^5+3 a^5 \cos (c+d x)+a^5 \cos ^2(c+d x)-2 a^5 \sec (c+d x)-3 a^5 \sec ^2(c+d x)-a^5 \sec ^3(c+d x)\right ) \, dx}{a^2} \\ & = -2 a^3 x-a^3 \int \cos ^2(c+d x) \, dx+a^3 \int \sec ^3(c+d x) \, dx+\left (2 a^3\right ) \int \sec (c+d x) \, dx-\left (3 a^3\right ) \int \cos (c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx \\ & = -2 a^3 x+\frac {2 a^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {3 a^3 \sin (c+d x)}{d}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}-\frac {1}{2} a^3 \int 1 \, dx+\frac {1}{2} a^3 \int \sec (c+d x) \, dx-\frac {\left (3 a^3\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = -\frac {5 a^3 x}{2}+\frac {5 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^3 \sin (c+d x)}{d}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(300\) vs. \(2(98)=196\).
Time = 2.20 (sec) , antiderivative size = 300, normalized size of antiderivative = 3.06 \[ \int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx=\frac {1}{32} a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (-10 x-\frac {10 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {10 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {12 \cos (d x) \sin (c)}{d}-\frac {\cos (2 d x) \sin (2 c)}{d}-\frac {12 \cos (c) \sin (d x)}{d}-\frac {\cos (2 c) \sin (2 d x)}{d}+\frac {1}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {1}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \]
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Time = 1.55 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(126\) |
default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+3 a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(126\) |
parts | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(134\) |
parallelrisch | \(\frac {a^{3} \left (-20 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-20 d x \cos \left (2 d x +2 c \right )+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (2 d x +2 c \right )-20 d x +20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-4 \sin \left (d x +c \right )+22 \sin \left (2 d x +2 c \right )-12 \sin \left (3 d x +3 c \right )-\sin \left (4 d x +4 c \right )\right )}{8 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(143\) |
norman | \(\frac {-\frac {5 a^{3} x}{2}+5 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {5 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {18 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-\frac {10 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {5 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {5 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(152\) |
risch | \(-\frac {5 a^{3} x}{2}+\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {3 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {3 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {i a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}-6\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\) | \(177\) |
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Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.28 \[ \int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx=-\frac {10 \, a^{3} d x \cos \left (d x + c\right )^{2} - 5 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 5 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{3} \cos \left (d x + c\right )^{2} - 6 \, a^{3} \cos \left (d x + c\right ) - a^{3}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx=a^{3} \left (\int 3 \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.30 \[ \int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx=\frac {{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 12 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{3} - a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )}}{4 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04 \[ \int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx=-\frac {5 \, {\left (d x + c\right )} a^{3} - 5 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 5 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {4 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1\right )}^{2}}}{2 \, d} \]
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Time = 13.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx=\frac {5\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {5\,a^3\,x}{2}+\frac {18\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+1\right )} \]
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