Integrand size = 19, antiderivative size = 67 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {4 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
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Time = 0.16 (sec) , antiderivative size = 67, 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, 2915, 12, 90} \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {4 a^3 \log (\cos (c+d x))}{d} \]
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \csc (c+d x) \sec ^3(c+d x) \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {a^3 (-a+x)^2}{(-a-x) x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^4 \text {Subst}\left (\int \frac {(-a+x)^2}{(-a-x) x^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^4 \text {Subst}\left (\int \left (-\frac {a}{x^3}+\frac {3}{x^2}-\frac {4}{a x}+\frac {4}{a (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {4 a^3 \log (1-\cos (c+d x))}{d}-\frac {4 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 = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3 \left (1+6 \cos (c+d x)-4 \log (\cos (c+d x))-4 \cos (2 (c+d x)) \left (\log (\cos (c+d x))-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sec ^2(c+d x)}{2 d} \]
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Time = 0.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.34
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+3 a^{3} \ln \left (\tan \left (d x +c \right )\right )+a^{3} \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d}\) | \(90\) |
default | \(\frac {a^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+3 a^{3} \ln \left (\tan \left (d x +c \right )\right )+a^{3} \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d}\) | \(90\) |
risch | \(\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {8 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(95\) |
norman | \(\frac {\frac {6 a^{3}}{d}-\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {4 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(104\) |
parallelrisch | \(\frac {-8 a^{3} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-a^{3} \left (-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (2 d x +2 c \right )+8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (2 d x +2 c \right )-12 \cos \left (d x +c \right )-5 \cos \left (2 d x +2 c \right )-16 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-7\right )}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(144\) |
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.13 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {8 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 8 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 6 \, a^{3} \cos \left (d x + c\right ) - a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=a^{3} \left (\int 3 \csc {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \csc {\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {8 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 8 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {6 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (65) = 130\).
Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.12 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {2 \, {\left (2 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {6 \, a^{3} + \frac {8 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}\right )}}{d} \]
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Time = 13.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{d\,{\cos \left (c+d\,x\right )}^2}-\frac {8\,a^3\,\mathrm {atanh}\left (2\,\cos \left (c+d\,x\right )-1\right )}{d} \]
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