Integrand size = 21, antiderivative size = 88 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {2 a^4}{d (a-a \cos (c+d x))}+\frac {5 a^3 \log (1-\cos (c+d x))}{d}-\frac {5 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
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Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 78} \[ \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {2 a^4}{d (a-a \cos (c+d x))}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {5 a^3 \log (1-\cos (c+d x))}{d}-\frac {5 a^3 \log (\cos (c+d x))}{d} \]
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Rule 12
Rule 78
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \csc ^3(c+d x) \sec ^3(c+d x) \, dx \\ & = \frac {a^3 \text {Subst}\left (\int \frac {a^3 (-a+x)}{(-a-x)^2 x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^6 \text {Subst}\left (\int \frac {-a+x}{(-a-x)^2 x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^6 \text {Subst}\left (\int \left (-\frac {1}{a x^3}+\frac {3}{a^2 x^2}-\frac {5}{a^3 x}+\frac {2}{a^2 (a+x)^2}+\frac {5}{a^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {2 a^4}{d (a-a \cos (c+d x))}+\frac {5 a^3 \log (1-\cos (c+d x))}{d}-\frac {5 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+10 \left (\log (\cos (c+d x))-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-6 \sec (c+d x)-\sec ^2(c+d x)\right )}{16 d} \]
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Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.50
method | result | size |
risch | \(\frac {2 a^{3} \left (5 \,{\mathrm e}^{5 i \left (d x +c \right )}-5 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{3 i \left (d x +c \right )}-5 \,{\mathrm e}^{2 i \left (d x +c \right )}+5 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}+\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {5 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(132\) |
norman | \(\frac {\frac {8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {a^{3}}{d}-\frac {5 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {10 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {5 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {5 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(134\) |
derivativedivides | \(\frac {a^{3} \left (\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {1}{\sin \left (d x +c \right )^{2}}+2 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cot \left (d x +c \right ) \csc \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(160\) |
default | \(\frac {a^{3} \left (\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {1}{\sin \left (d x +c \right )^{2}}+2 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\cot \left (d x +c \right ) \csc \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(160\) |
parallelrisch | \(\frac {a^{3} \left (9 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (2 d x +2 c \right )-56 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (d x +c \right )+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (2 d x +2 c \right )-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (2 d x +2 c \right )-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (2 d x +2 c \right )+75 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-32 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right )}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(197\) |
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Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.50 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {10 \, a^{3} \cos \left (d x + c\right )^{2} - 5 \, a^{3} \cos \left (d x + c\right ) - a^{3} - 10 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 10 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2}\right )}} \]
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\[ \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx=a^{3} \left (\int 3 \csc ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \csc ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \csc ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {10 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 10 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {10 \, a^{3} \cos \left (d x + c\right )^{2} - 5 \, a^{3} \cos \left (d x + c\right ) - a^{3}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2}}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (87) = 174\).
Time = 0.38 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.15 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {10 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 10 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {2 \, {\left (a^{3} - \frac {5 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1} + \frac {27 \, a^{3} + \frac {38 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {15 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \]
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Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85 \[ \int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {-5\,a^3\,{\cos \left (c+d\,x\right )}^2+\frac {5\,a^3\,\cos \left (c+d\,x\right )}{2}+\frac {a^3}{2}}{d\,\left ({\cos \left (c+d\,x\right )}^2-{\cos \left (c+d\,x\right )}^3\right )}-\frac {10\,a^3\,\mathrm {atanh}\left (2\,\cos \left (c+d\,x\right )-1\right )}{d} \]
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