Integrand size = 16, antiderivative size = 45 \[ \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^2}{2}-\frac {a b \text {arctanh}\left (\cos \left (c+d x^2\right )\right )}{d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 d} \]
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Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4290, 3858, 3855, 3852, 8} \[ \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {a^2 x^2}{2}-\frac {a b \text {arctanh}\left (\cos \left (c+d x^2\right )\right )}{d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3858
Rule 4290
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int (a+b \csc (c+d x))^2 \, dx,x,x^2\right ) \\ & = \frac {a^2 x^2}{2}+(a b) \text {Subst}\left (\int \csc (c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \text {Subst}\left (\int \csc ^2(c+d x) \, dx,x,x^2\right ) \\ & = \frac {a^2 x^2}{2}-\frac {a b \text {arctanh}\left (\cos \left (c+d x^2\right )\right )}{d}-\frac {b^2 \text {Subst}\left (\int 1 \, dx,x,\cot \left (c+d x^2\right )\right )}{2 d} \\ & = \frac {a^2 x^2}{2}-\frac {a b \text {arctanh}\left (\cos \left (c+d x^2\right )\right )}{d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 d} \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.91 \[ \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {-b^2 \cot \left (\frac {1}{2} \left (c+d x^2\right )\right )+2 a \left (a c+a d x^2-2 b \log \left (\cos \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )+2 b \log \left (\sin \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )\right )+b^2 \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{4 d} \]
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Time = 0.63 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.13
method | result | size |
parts | \(\frac {a^{2} x^{2}}{2}-\frac {b^{2} \cot \left (d \,x^{2}+c \right )}{2 d}-\frac {a b \ln \left (\csc \left (d \,x^{2}+c \right )+\cot \left (d \,x^{2}+c \right )\right )}{d}\) | \(51\) |
derivativedivides | \(\frac {a^{2} \left (d \,x^{2}+c \right )+2 a b \ln \left (\csc \left (d \,x^{2}+c \right )-\cot \left (d \,x^{2}+c \right )\right )-b^{2} \cot \left (d \,x^{2}+c \right )}{2 d}\) | \(55\) |
default | \(\frac {a^{2} \left (d \,x^{2}+c \right )+2 a b \ln \left (\csc \left (d \,x^{2}+c \right )-\cot \left (d \,x^{2}+c \right )\right )-b^{2} \cot \left (d \,x^{2}+c \right )}{2 d}\) | \(55\) |
parallelrisch | \(\frac {2 a^{2} x^{2} d -\cot \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right ) b^{2}+\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right ) b^{2}+4 \ln \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right ) a b}{4 d}\) | \(63\) |
risch | \(\frac {a^{2} x^{2}}{2}-\frac {i b^{2}}{d \left ({\mathrm e}^{2 i \left (d \,x^{2}+c \right )}-1\right )}+\frac {a b \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}-1\right )}{d}-\frac {a b \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+1\right )}{d}\) | \(75\) |
norman | \(\frac {-\frac {b^{2}}{4 d}+\frac {a^{2} x^{2} \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{2}+\frac {b^{2} \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )^{2}}{4 d}}{\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}+\frac {a b \ln \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{d}\) | \(83\) |
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (41) = 82\).
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.09 \[ \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {a^{2} d x^{2} \sin \left (d x^{2} + c\right ) - a b \log \left (\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{2} + c\right ) + a b \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{2} + c\right ) - b^{2} \cos \left (d x^{2} + c\right )}{2 \, d \sin \left (d x^{2} + c\right )} \]
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\[ \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x \left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (41) = 82\).
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.18 \[ \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {1}{2} \, a^{2} x^{2} - \frac {a b \log \left (\cot \left (d x^{2} + c\right ) + \csc \left (d x^{2} + c\right )\right )}{d} - \frac {b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )}{d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (41) = 82\).
Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.87 \[ \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {2 \, {\left (d x^{2} + c\right )} a^{2} + 4 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) \right |}\right ) + b^{2} \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) - \frac {4 \, a b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + b^{2}}{\tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )}}{4 \, d} \]
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Time = 18.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.27 \[ \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\frac {a^2\,x^2}{2}-\frac {b^2\,1{}\mathrm {i}}{d\,\left ({\mathrm {e}}^{2{}\mathrm {i}\,d\,x^2+c\,2{}\mathrm {i}}-1\right )}-\frac {a\,b\,\ln \left (-a\,b\,x\,4{}\mathrm {i}-a\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,4{}\mathrm {i}\right )}{d}+\frac {a\,b\,\ln \left (a\,b\,x\,4{}\mathrm {i}-a\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,4{}\mathrm {i}\right )}{d} \]
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