Integrand size = 21, antiderivative size = 123 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {1}{16 a^2 d (1-\cos (c+d x))}-\frac {1}{12 a^2 d (1+\cos (c+d x))^3}+\frac {1}{2 a^2 d (1+\cos (c+d x))^2}-\frac {23}{16 a^2 d (1+\cos (c+d x))}-\frac {3 \log (1-\cos (c+d x))}{16 a^2 d}-\frac {13 \log (1+\cos (c+d x))}{16 a^2 d} \]
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Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3964, 90} \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {1}{16 a^2 d (1-\cos (c+d x))}-\frac {23}{16 a^2 d (\cos (c+d x)+1)}+\frac {1}{2 a^2 d (\cos (c+d x)+1)^2}-\frac {1}{12 a^2 d (\cos (c+d x)+1)^3}-\frac {3 \log (1-\cos (c+d x))}{16 a^2 d}-\frac {13 \log (\cos (c+d x)+1)}{16 a^2 d} \]
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Rule 90
Rule 3964
Rubi steps \begin{align*} \text {integral}& = -\frac {a^4 \text {Subst}\left (\int \frac {x^5}{(a-a x)^2 (a+a x)^4} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^4 \text {Subst}\left (\int \left (\frac {1}{16 a^6 (-1+x)^2}+\frac {3}{16 a^6 (-1+x)}-\frac {1}{4 a^6 (1+x)^4}+\frac {1}{a^6 (1+x)^3}-\frac {23}{16 a^6 (1+x)^2}+\frac {13}{16 a^6 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {1}{16 a^2 d (1-\cos (c+d x))}-\frac {1}{12 a^2 d (1+\cos (c+d x))^3}+\frac {1}{2 a^2 d (1+\cos (c+d x))^2}-\frac {23}{16 a^2 d (1+\cos (c+d x))}-\frac {3 \log (1-\cos (c+d x))}{16 a^2 d}-\frac {13 \log (1+\cos (c+d x))}{16 a^2 d} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+156 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+36 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+69 \sec ^2\left (\frac {1}{2} (c+d x)\right )-12 \sec ^4\left (\frac {1}{2} (c+d x)\right )+\sec ^6\left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x)}{24 a^2 d (1+\sec (c+d x))^2} \]
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Time = 0.70 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(\frac {\frac {1}{16 \cos \left (d x +c \right )-16}-\frac {3 \ln \left (\cos \left (d x +c \right )-1\right )}{16}-\frac {1}{12 \left (\cos \left (d x +c \right )+1\right )^{3}}+\frac {1}{2 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {23}{16 \left (\cos \left (d x +c \right )+1\right )}-\frac {13 \ln \left (\cos \left (d x +c \right )+1\right )}{16}}{d \,a^{2}}\) | \(79\) |
default | \(\frac {\frac {1}{16 \cos \left (d x +c \right )-16}-\frac {3 \ln \left (\cos \left (d x +c \right )-1\right )}{16}-\frac {1}{12 \left (\cos \left (d x +c \right )+1\right )^{3}}+\frac {1}{2 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {23}{16 \left (\cos \left (d x +c \right )+1\right )}-\frac {13 \ln \left (\cos \left (d x +c \right )+1\right )}{16}}{d \,a^{2}}\) | \(79\) |
risch | \(\frac {i x}{a^{2}}+\frac {2 i c}{a^{2} d}-\frac {33 \,{\mathrm e}^{7 i \left (d x +c \right )}+36 \,{\mathrm e}^{6 i \left (d x +c \right )}-49 \,{\mathrm e}^{5 i \left (d x +c \right )}-136 \,{\mathrm e}^{4 i \left (d x +c \right )}-49 \,{\mathrm e}^{3 i \left (d x +c \right )}+36 \,{\mathrm e}^{2 i \left (d x +c \right )}+33 \,{\mathrm e}^{i \left (d x +c \right )}}{12 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a^{2} d}-\frac {13 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a^{2} d}\) | \(171\) |
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Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.32 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {66 \, \cos \left (d x + c\right )^{3} + 36 \, \cos \left (d x + c\right )^{2} + 39 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 9 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 74 \, \cos \left (d x + c\right ) - 52}{48 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )}} \]
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\[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{3} + 18 \, \cos \left (d x + c\right )^{2} - 37 \, \cos \left (d x + c\right ) - 26\right )}}{a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right ) - a^{2}} + \frac {39 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {9 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{48 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.51 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (\frac {6 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {18 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac {96 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} + \frac {\frac {48 \, a^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{6}}}{96 \, d} \]
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Time = 14.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8}-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{96}}{a^2\,d} \]
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