Integrand size = 21, antiderivative size = 143 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {1}{32 a^3 d (1-\cos (c+d x))}+\frac {1}{16 a^3 d (1+\cos (c+d x))^4}-\frac {5}{12 a^3 d (1+\cos (c+d x))^3}+\frac {39}{32 a^3 d (1+\cos (c+d x))^2}-\frac {9}{4 a^3 d (1+\cos (c+d x))}-\frac {7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac {57 \log (1+\cos (c+d x))}{64 a^3 d} \]
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Time = 0.12 (sec) , antiderivative size = 143, 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} \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {1}{32 a^3 d (1-\cos (c+d x))}-\frac {9}{4 a^3 d (\cos (c+d x)+1)}+\frac {39}{32 a^3 d (\cos (c+d x)+1)^2}-\frac {5}{12 a^3 d (\cos (c+d x)+1)^3}+\frac {1}{16 a^3 d (\cos (c+d x)+1)^4}-\frac {7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac {57 \log (\cos (c+d x)+1)}{64 a^3 d} \]
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Rule 90
Rule 3964
Rubi steps \begin{align*} \text {integral}& = -\frac {a^4 \text {Subst}\left (\int \frac {x^6}{(a-a x)^2 (a+a x)^5} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^4 \text {Subst}\left (\int \left (\frac {1}{32 a^7 (-1+x)^2}+\frac {7}{64 a^7 (-1+x)}+\frac {1}{4 a^7 (1+x)^5}-\frac {5}{4 a^7 (1+x)^4}+\frac {39}{16 a^7 (1+x)^3}-\frac {9}{4 a^7 (1+x)^2}+\frac {57}{64 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {1}{32 a^3 d (1-\cos (c+d x))}+\frac {1}{16 a^3 d (1+\cos (c+d x))^4}-\frac {5}{12 a^3 d (1+\cos (c+d x))^3}+\frac {39}{32 a^3 d (1+\cos (c+d x))^2}-\frac {9}{4 a^3 d (1+\cos (c+d x))}-\frac {7 \log (1-\cos (c+d x))}{64 a^3 d}-\frac {57 \log (1+\cos (c+d x))}{64 a^3 d} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (864+12 \cot ^2\left (\frac {1}{2} (c+d x)\right )+24 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (57 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+7 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-234 \sec ^2\left (\frac {1}{2} (c+d x)\right )+40 \sec ^4\left (\frac {1}{2} (c+d x)\right )-3 \sec ^6\left (\frac {1}{2} (c+d x)\right )\right ) \sec ^3(c+d x)}{96 a^3 d (1+\sec (c+d x))^3} \]
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Time = 0.75 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(\frac {\frac {1}{16 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {5}{12 \left (\cos \left (d x +c \right )+1\right )^{3}}+\frac {39}{32 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {9}{4 \left (\cos \left (d x +c \right )+1\right )}-\frac {57 \ln \left (\cos \left (d x +c \right )+1\right )}{64}+\frac {1}{32 \cos \left (d x +c \right )-32}-\frac {7 \ln \left (\cos \left (d x +c \right )-1\right )}{64}}{d \,a^{3}}\) | \(91\) |
default | \(\frac {\frac {1}{16 \left (\cos \left (d x +c \right )+1\right )^{4}}-\frac {5}{12 \left (\cos \left (d x +c \right )+1\right )^{3}}+\frac {39}{32 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {9}{4 \left (\cos \left (d x +c \right )+1\right )}-\frac {57 \ln \left (\cos \left (d x +c \right )+1\right )}{64}+\frac {1}{32 \cos \left (d x +c \right )-32}-\frac {7 \ln \left (\cos \left (d x +c \right )-1\right )}{64}}{d \,a^{3}}\) | \(91\) |
risch | \(\frac {i x}{a^{3}}+\frac {2 i c}{a^{3} d}-\frac {213 \,{\mathrm e}^{9 i \left (d x +c \right )}+606 \,{\mathrm e}^{8 i \left (d x +c \right )}+472 \,{\mathrm e}^{7 i \left (d x +c \right )}-846 \,{\mathrm e}^{6 i \left (d x +c \right )}-1658 \,{\mathrm e}^{5 i \left (d x +c \right )}-846 \,{\mathrm e}^{4 i \left (d x +c \right )}+472 \,{\mathrm e}^{3 i \left (d x +c \right )}+606 \,{\mathrm e}^{2 i \left (d x +c \right )}+213 \,{\mathrm e}^{i \left (d x +c \right )}}{48 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{32 a^{3} d}-\frac {57 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{32 a^{3} d}\) | \(193\) |
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Time = 0.29 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {426 \, \cos \left (d x + c\right )^{4} + 606 \, \cos \left (d x + c\right )^{3} - 190 \, \cos \left (d x + c\right )^{2} + 171 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 21 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 666 \, \cos \left (d x + c\right ) - 272}{192 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}} \]
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\[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
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Time = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (213 \, \cos \left (d x + c\right )^{4} + 303 \, \cos \left (d x + c\right )^{3} - 95 \, \cos \left (d x + c\right )^{2} - 333 \, \cos \left (d x + c\right ) - 136\right )}}{a^{3} \cos \left (d x + c\right )^{5} + 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} - 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}} + \frac {171 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {21 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{192 \, d} \]
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Time = 0.41 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {12 \, {\left (\frac {7 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {84 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {768 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\frac {504 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {132 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{12}}}{768 \, d} \]
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Time = 14.66 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {7\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{32}-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}+\frac {21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{256}}{a^3\,d} \]
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