Integrand size = 21, antiderivative size = 92 \[ \int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {a^2 \log (\cos (c+d x))}{d}-\frac {a (a+b) \log (1-\sec (c+d x))}{2 d}-\frac {a (a-b) \log (1+\sec (c+d x))}{2 d}-\frac {\cot ^2(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{2 d} \]
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Time = 0.17 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3970, 1819, 815} \[ \int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {\cot ^2(c+d x) \left (a^2+2 a b \sec (c+d x)+b^2\right )}{2 d}-\frac {a^2 \log (\cos (c+d x))}{d}-\frac {a (a+b) \log (1-\sec (c+d x))}{2 d}-\frac {a (a-b) \log (\sec (c+d x)+1)}{2 d} \]
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Rule 815
Rule 1819
Rule 3970
Rubi steps \begin{align*} \text {integral}& = \frac {b^4 \text {Subst}\left (\int \frac {(a+x)^2}{x \left (b^2-x^2\right )^2} \, dx,x,b \sec (c+d x)\right )}{d} \\ & = -\frac {\cot ^2(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{2 d}-\frac {b^2 \text {Subst}\left (\int \frac {-2 a^2-2 a x}{x \left (b^2-x^2\right )} \, dx,x,b \sec (c+d x)\right )}{2 d} \\ & = -\frac {\cot ^2(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{2 d}-\frac {b^2 \text {Subst}\left (\int \left (-\frac {a (a+b)}{b^2 (b-x)}-\frac {2 a^2}{b^2 x}+\frac {a (a-b)}{b^2 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{2 d} \\ & = -\frac {a^2 \log (\cos (c+d x))}{d}-\frac {a (a+b) \log (1-\sec (c+d x))}{2 d}-\frac {a (a-b) \log (1+\sec (c+d x))}{2 d}-\frac {\cot ^2(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{2 d} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.97 \[ \int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {4 a^2 \log (\cos (c+d x))+2 a (a+b) \log (1-\sec (c+d x))+2 a (a-b) \log (1+\sec (c+d x))+\frac {(a+b)^2}{-1+\sec (c+d x)}-\frac {(a-b)^2}{1+\sec (c+d x)}}{4 d} \]
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Time = 0.75 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{3}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )-\frac {b^{2}}{2 \sin \left (d x +c \right )^{2}}}{d}\) | \(92\) |
default | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{3}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )}{2}-\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )-\frac {b^{2}}{2 \sin \left (d x +c \right )^{2}}}{d}\) | \(92\) |
risch | \(i a^{2} x +\frac {2 i a^{2} c}{d}+\frac {2 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+2 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 a b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a b}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a b}{d}\) | \(165\) |
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.23 \[ \int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {2 \, a b \cos \left (d x + c\right ) + a^{2} + b^{2} - {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} - a^{2} + a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left ({\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} - a^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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\[ \int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \cot ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.78 \[ \int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {{\left (a^{2} - a b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) + {\left (a^{2} + a b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, a b \cos \left (d x + c\right ) + a^{2} + b^{2}}{\cos \left (d x + c\right )^{2} - 1}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (86) = 172\).
Time = 0.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.27 \[ \int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {8 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - 4 \, {\left (a^{2} + a b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) + \frac {{\left (a^{2} + 2 \, a b + b^{2} + \frac {4 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{8 \, d} \]
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Time = 14.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07 \[ \int \cot ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\left (a-b\right )}^2}{8\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2+b\,a\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{8}+\frac {a\,b}{4}+\frac {b^2}{8}\right )}{d} \]
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