Integrand size = 21, antiderivative size = 115 \[ \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx=-\frac {9 a^2 x}{8}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \tan (c+d x)}{d} \]
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Time = 0.35 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2951, 2717, 2715, 8, 2713, 3855, 3852} \[ \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \tan (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {9 a^2 x}{8} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2717
Rule 2951
Rule 3852
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int (-a-a \cos (c+d x))^2 \sin ^2(c+d x) \tan ^2(c+d x) \, dx \\ & = \frac {\int \left (-a^6-4 a^6 \cos (c+d x)-a^6 \cos ^2(c+d x)+2 a^6 \cos ^3(c+d x)+a^6 \cos ^4(c+d x)+2 a^6 \sec (c+d x)+a^6 \sec ^2(c+d x)\right ) \, dx}{a^4} \\ & = -a^2 x-a^2 \int \cos ^2(c+d x) \, dx+a^2 \int \cos ^4(c+d x) \, dx+a^2 \int \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^3(c+d x) \, dx+\left (2 a^2\right ) \int \sec (c+d x) \, dx-\left (4 a^2\right ) \int \cos (c+d x) \, dx \\ & = -a^2 x+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {4 a^2 \sin (c+d x)}{d}-\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{2} a^2 \int 1 \, dx+\frac {1}{4} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac {a^2 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = -\frac {3 a^2 x}{2}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \tan (c+d x)}{d}+\frac {1}{8} \left (3 a^2\right ) \int 1 \, dx \\ & = -\frac {9 a^2 x}{8}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \tan (c+d x)}{d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (48 c+48 d x+60 \arctan (\tan (c+d x))-192 \text {arctanh}(\sin (c+d x))+192 \sin (c+d x)+64 \sin ^3(c+d x)-3 \sin (4 (c+d x))-96 \tan (c+d x)\right )}{384 d} \]
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Time = 2.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01
method | result | size |
parallelrisch | \(\frac {a^{2} \left (-72 d x \cos \left (d x +c \right )-128 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+128 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+\sin \left (5 d x +5 c \right )+64 \sin \left (d x +c \right )-\frac {224 \sin \left (2 d x +2 c \right )}{3}+\sin \left (3 d x +3 c \right )+\frac {16 \sin \left (4 d x +4 c \right )}{3}\right )}{64 d \cos \left (d x +c \right )}\) | \(116\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(134\) |
default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(134\) |
parts | \(\frac {a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}+\frac {2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(139\) |
risch | \(-\frac {9 a^{2} x}{8}+\frac {5 i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{4 d}-\frac {5 i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{4 d}+\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{2} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{2} \sin \left (3 d x +3 c \right )}{6 d}\) | \(142\) |
norman | \(\frac {\frac {9 a^{2} x}{8}+\frac {7 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {22 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {31 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 d}-\frac {58 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {25 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}+\frac {27 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {9 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}-\frac {9 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}-\frac {27 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}-\frac {9 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(258\) |
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Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.16 \[ \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx=-\frac {27 \, a^{2} d x \cos \left (d x + c\right ) - 24 \, a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + 24 \, a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (6 \, a^{2} \cos \left (d x + c\right )^{4} + 16 \, a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )^{2} - 64 \, a^{2} \cos \left (d x + c\right ) + 24 \, a^{2}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]
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\[ \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx=a^{2} \left (\int 2 \sin ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.10 \[ \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx=-\frac {32 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{2} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 48 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{2}}{96 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.40 \[ \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx=-\frac {27 \, {\left (d x + c\right )} a^{2} - 48 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 48 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {48 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (51 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 187 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 229 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 14.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.54 \[ \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx=\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {9\,a^2\,x}{8}+\frac {\frac {25\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+\frac {58\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {31\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}-\frac {22\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {7\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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