Integrand size = 21, antiderivative size = 42 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {a+2 a \cos (c+d x)}{6 d (1-\cos (c+d x)) (a+a \cos (c+d x))^3} \]
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Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 82} \[ \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 a \cos (c+d x)+a}{6 d (1-\cos (c+d x)) (a \cos (c+d x)+a)^3} \]
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Rule 12
Rule 82
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(-a-a \cos (c+d x))^2} \, dx \\ & = \frac {a^3 \text {Subst}\left (\int \frac {x^2}{a^2 (-a-x)^2 (-a+x)^4} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \frac {x^2}{(-a-x)^2 (-a+x)^4} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a+2 a \cos (c+d x)}{6 d (1-\cos (c+d x)) (a+a \cos (c+d x))^3} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {(1+2 \cos (c+d x)) \csc ^2(c+d x)}{6 a^2 d (1+\cos (c+d x))^2} \]
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Time = 0.53 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10
method | result | size |
parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-3-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{96 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\) | \(46\) |
derivativedivides | \(\frac {\frac {1}{12 \left (\cos \left (d x +c \right )+1\right )^{3}}-\frac {1}{8 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {1}{16 \left (\cos \left (d x +c \right )+1\right )}+\frac {1}{16 \cos \left (d x +c \right )-16}}{d \,a^{2}}\) | \(57\) |
default | \(\frac {\frac {1}{12 \left (\cos \left (d x +c \right )+1\right )^{3}}-\frac {1}{8 \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {1}{16 \left (\cos \left (d x +c \right )+1\right )}+\frac {1}{16 \cos \left (d x +c \right )-16}}{d \,a^{2}}\) | \(57\) |
norman | \(\frac {-\frac {1}{32 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{96 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}\) | \(63\) |
risch | \(\frac {\frac {8 \,{\mathrm e}^{5 i \left (d x +c \right )}}{3}+\frac {8 \,{\mathrm e}^{4 i \left (d x +c \right )}}{3}+\frac {8 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}}{a^{2} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}\) | \(63\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.43 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {2 \, \cos \left (d x + c\right ) + 1}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )}} \]
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\[ \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.40 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {2 \, \cos \left (d x + c\right ) + 1}{6 \, {\left (a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right ) - a^{2}\right )} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (38) = 76\).
Time = 0.35 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.95 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}} + \frac {\frac {6 \, a^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{6}}}{96 \, d} \]
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Time = 13.81 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {\cos \left (c+d\,x\right )}{3}+\frac {1}{6}}{d\,\left (-a^2\,{\cos \left (c+d\,x\right )}^4-2\,a^2\,{\cos \left (c+d\,x\right )}^3+2\,a^2\,\cos \left (c+d\,x\right )+a^2\right )} \]
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