Integrand size = 22, antiderivative size = 47 \[ \int \frac {\csc (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d} \]
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Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3254, 2702, 308, 213} \[ \int \frac {\csc (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}+\frac {\sec (c+d x)}{a^2 d} \]
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Rule 213
Rule 308
Rule 2702
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc (c+d x) \sec ^4(c+d x) \, dx}{a^2} \\ & = \frac {\text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {\text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {\sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}+\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a^2 d}+\frac {\sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.30 \[ \int \frac {\csc (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {-\frac {\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {\sec (c+d x)}{d}+\frac {\sec ^3(c+d x)}{3 d}}{a^2} \]
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Time = 0.76 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2}}{d \,a^{2}}\) | \(49\) |
default | \(\frac {\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2}}{d \,a^{2}}\) | \(49\) |
norman | \(\frac {\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {8}{3 a d}-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d}\) | \(85\) |
parallelrisch | \(\frac {\left (9 \cos \left (d x +c \right )+3 \cos \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \cos \left (d x +c \right )+6 \cos \left (2 d x +2 c \right )+4 \cos \left (3 d x +3 c \right )+10}{3 a^{2} d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(92\) |
risch | \(\frac {2 \,{\mathrm e}^{5 i \left (d x +c \right )}+\frac {20 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}+2 \,{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{2}}\) | \(96\) |
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Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.49 \[ \int \frac {\csc (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {3 \, \cos \left (d x + c\right )^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, \cos \left (d x + c\right )^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 6 \, \cos \left (d x + c\right )^{2} - 2}{6 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
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\[ \int \frac {\csc (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\int \frac {\csc {\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} - 2 \sin ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \frac {\csc (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=-\frac {\frac {3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}} - \frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{a^{2} \cos \left (d x + c\right )^{3}}}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (45) = 90\).
Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.28 \[ \int \frac {\csc (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {\frac {3 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac {8 \, {\left (\frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{6 \, d} \]
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Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \frac {\csc (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx=\frac {{\cos \left (c+d\,x\right )}^2+\frac {1}{3}}{a^2\,d\,{\cos \left (c+d\,x\right )}^3}-\frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{a^2\,d} \]
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