Integrand size = 21, antiderivative size = 110 \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {11 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}-\frac {17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.27 (sec) , antiderivative size = 110, 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, 2729, 2727, 3855, 3852, 8, 3853} \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {11 a^3 \text {arctanh}(\sin (c+d x))}{2 d}+\frac {3 a^3 \tan (c+d x)}{d}-\frac {17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}-\frac {2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rule 8
Rule 2727
Rule 2729
Rule 2951
Rule 3852
Rule 3853
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \csc ^4(c+d x) \sec ^3(c+d x) \, dx \\ & = a^4 \int \left (\frac {2}{a (1-\cos (c+d x))^2}+\frac {5}{a (1-\cos (c+d x))}+\frac {5 \sec (c+d x)}{a}+\frac {3 \sec ^2(c+d x)}{a}+\frac {\sec ^3(c+d x)}{a}\right ) \, dx \\ & = a^3 \int \sec ^3(c+d x) \, dx+\left (2 a^3\right ) \int \frac {1}{(1-\cos (c+d x))^2} \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx+\left (5 a^3\right ) \int \frac {1}{1-\cos (c+d x)} \, dx+\left (5 a^3\right ) \int \sec (c+d x) \, dx \\ & = \frac {5 a^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}-\frac {5 a^3 \sin (c+d x)}{d (1-\cos (c+d x))}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} a^3 \int \sec (c+d x) \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {1}{1-\cos (c+d x)} \, dx-\frac {\left (3 a^3\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {11 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}-\frac {17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}+\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. \(290\) vs. \(2(110)=220\).
Time = 6.71 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.64 \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (-4 \cot \left (\frac {c}{2}\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )-66 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+66 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 (-19+17 \cos (c+d x)) \csc \left (\frac {c}{2}\right ) \csc ^3\left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )+\frac {3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {36 \sin \left (\frac {d x}{2}\right )}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {36 \sin \left (\frac {d x}{2}\right )}{\left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{96 d} \]
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Time = 1.49 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(-\frac {11 a^{3} \left (\left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-\cos \left (2 d x +2 c \right )-1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {17 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\cos \left (d x +c \right )-\frac {71 \cos \left (2 d x +2 c \right )}{102}+\frac {13 \cos \left (3 d x +3 c \right )}{51}-\frac {65}{102}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{11}\right )}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(125\) |
norman | \(\frac {-\frac {a^{3}}{3 d}-\frac {16 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d}+\frac {56 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d}-\frac {11 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {11 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {11 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(135\) |
risch | \(-\frac {i a^{3} \left (33 \,{\mathrm e}^{6 i \left (d x +c \right )}-99 \,{\mathrm e}^{5 i \left (d x +c \right )}+154 \,{\mathrm e}^{4 i \left (d x +c \right )}-210 \,{\mathrm e}^{3 i \left (d x +c \right )}+161 \,{\mathrm e}^{2 i \left (d x +c \right )}-123 \,{\mathrm e}^{i \left (d x +c \right )}+52\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\) | \(147\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {5}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )}+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+3 a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(185\) |
default | \(\frac {a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {5}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )}+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+3 a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(185\) |
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Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.62 \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {104 \, a^{3} \cos \left (d x + c\right )^{4} - 38 \, a^{3} \cos \left (d x + c\right )^{3} - 118 \, a^{3} \cos \left (d x + c\right )^{2} + 30 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{3} - 33 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]
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\[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx=a^{3} \left (\int 3 \csc ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \csc ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \csc ^{4}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.71 \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{3}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{3} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} a^{3}}{\tan \left (d x + c\right )^{3}}}{12 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.12 \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {33 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 33 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {2 \, {\left (18 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{6 \, d} \]
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Time = 18.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {11\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {56\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {16\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {a^3}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \]
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