Integrand size = 24, antiderivative size = 78 \[ \int \frac {\csc ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {5 \sec (c+d x)}{2 a^2 d}+\frac {5 \sec ^3(c+d x)}{6 a^2 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a^2 d} \]
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Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3254, 2702, 294, 308, 213} \[ \int \frac {\csc ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {5 \text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {5 \sec ^3(c+d x)}{6 a^2 d}+\frac {5 \sec (c+d x)}{2 a^2 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a^2 d} \]
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Rule 213
Rule 294
Rule 308
Rule 2702
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc ^3(c+d x) \sec ^4(c+d x) \, dx}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = -\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a^2 d}+\frac {5 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a^2 d} \\ & = -\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a^2 d}+\frac {5 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{2 a^2 d} \\ & = \frac {5 \sec (c+d x)}{2 a^2 d}+\frac {5 \sec ^3(c+d x)}{6 a^2 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a^2 d}+\frac {5 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a^2 d} \\ & = -\frac {5 \text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {5 \sec (c+d x)}{2 a^2 d}+\frac {5 \sec ^3(c+d x)}{6 a^2 d}-\frac {\csc ^2(c+d x) \sec ^3(c+d x)}{2 a^2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(208\) vs. \(2(78)=156\).
Time = 0.38 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.67 \[ \int \frac {\csc ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {2 \csc ^8(c+d x) \left (22-40 \cos (2 (c+d x))+13 \cos (3 (c+d x))-30 \cos (4 (c+d x))+13 \cos (5 (c+d x))+15 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+15 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-15 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-15 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\cos (c+d x) \left (-26-30 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+30 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{3 a^2 d \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Time = 0.89 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {1}{4+4 \cos \left (d x +c \right )}-\frac {5 \ln \left (1+\cos \left (d x +c \right )\right )}{4}+\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {2}{\cos \left (d x +c \right )}+\frac {1}{4 \cos \left (d x +c \right )-4}+\frac {5 \ln \left (\cos \left (d x +c \right )-1\right )}{4}}{d \,a^{2}}\) | \(75\) |
default | \(\frac {\frac {1}{4+4 \cos \left (d x +c \right )}-\frac {5 \ln \left (1+\cos \left (d x +c \right )\right )}{4}+\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {2}{\cos \left (d x +c \right )}+\frac {1}{4 \cos \left (d x +c \right )-4}+\frac {5 \ln \left (\cos \left (d x +c \right )-1\right )}{4}}{d \,a^{2}}\) | \(75\) |
parallelrisch | \(\frac {60 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-165 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+225 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-130}{24 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(127\) |
risch | \(\frac {15 \,{\mathrm e}^{9 i \left (d x +c \right )}+20 \,{\mathrm e}^{7 i \left (d x +c \right )}-22 \,{\mathrm e}^{5 i \left (d x +c \right )}+20 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{2}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{2}}\) | \(132\) |
norman | \(\frac {\frac {1}{8 a d}+\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {75 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {65 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {55 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}}{a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} d}\) | \(135\) |
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Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.51 \[ \int \frac {\csc ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {30 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 4}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
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\[ \int \frac {\csc ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} - 2 \sin ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10 \[ \int \frac {\csc ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{3}} - \frac {15 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {15 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (70) = 140\).
Time = 0.33 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.24 \[ \int \frac {\csc ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {\frac {3 \, {\left (\frac {10 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {30 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{a^{2} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {16 \, {\left (\frac {12 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 7\right )}}{a^{2} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{24 \, d} \]
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Time = 12.94 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int \frac {\csc ^3(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {-\frac {5\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {5\,{\cos \left (c+d\,x\right )}^2}{3}+\frac {1}{3}}{d\,\left (a^2\,{\cos \left (c+d\,x\right )}^3-a^2\,{\cos \left (c+d\,x\right )}^5\right )}-\frac {5\,\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{2\,a^2\,d} \]
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