Integrand size = 16, antiderivative size = 120 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {x^2}{2 a^2}+\frac {b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d x^2\right )\right )} \]
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Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4290, 3870, 4004, 3916, 2739, 632, 212} \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 d \left (a^2-b^2\right )^{3/2}}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )}+\frac {x^2}{2 a^2} \]
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Rule 212
Rule 632
Rule 2739
Rule 3870
Rule 3916
Rule 4004
Rule 4290
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b \csc (c+d x))^2} \, dx,x,x^2\right ) \\ & = -\frac {b^2 \cot \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d x^2\right )\right )}-\frac {\text {Subst}\left (\int \frac {-a^2+b^2+a b \csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,x^2\right )}{2 a \left (a^2-b^2\right )} \\ & = \frac {x^2}{2 a^2}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d x^2\right )\right )}-\frac {\left (b \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {\csc (c+d x)}{a+b \csc (c+d x)} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )} \\ & = \frac {x^2}{2 a^2}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d x^2\right )\right )}-\frac {\left (2 a^2-b^2\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a \sin (c+d x)}{b}} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )} \\ & = \frac {x^2}{2 a^2}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d x^2\right )\right )}-\frac {\left (2 a^2-b^2\right ) \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a^2 \left (a^2-b^2\right ) d} \\ & = \frac {x^2}{2 a^2}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d x^2\right )\right )}+\frac {\left (2 \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a^2 \left (a^2-b^2\right ) d} \\ & = \frac {x^2}{2 a^2}+\frac {b \left (2 a^2-b^2\right ) \text {arctanh}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2} d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \csc \left (c+d x^2\right )\right )} \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.32 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\frac {\csc \left (c+d x^2\right ) \left (\frac {a b^2 \cot \left (c+d x^2\right )}{(-a+b) (a+b)}+\left (c+d x^2\right ) \left (a+b \csc \left (c+d x^2\right )\right )-\frac {2 b \left (-2 a^2+b^2\right ) \arctan \left (\frac {a+b \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {-a^2+b^2}}\right ) \left (a+b \csc \left (c+d x^2\right )\right )}{\left (-a^2+b^2\right )^{3/2}}\right ) \left (b+a \sin \left (c+d x^2\right )\right )}{2 a^2 d \left (a+b \csc \left (c+d x^2\right )\right )^2} \]
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Time = 0.50 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.49
method | result | size |
derivativedivides | \(\frac {-\frac {2 b \left (\frac {\frac {a^{2} \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a b}{2 a^{2}-2 b^{2}}}{\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )^{2} b}{2}+a \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+\frac {b}{2}}+\frac {2 \left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 b \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (2 a^{2}-2 b^{2}\right ) \sqrt {-a^{2}+b^{2}}}\right )}{a^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{2 d}\) | \(179\) |
default | \(\frac {-\frac {2 b \left (\frac {\frac {a^{2} \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a b}{2 a^{2}-2 b^{2}}}{\frac {\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )^{2} b}{2}+a \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+\frac {b}{2}}+\frac {2 \left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 b \tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (2 a^{2}-2 b^{2}\right ) \sqrt {-a^{2}+b^{2}}}\right )}{a^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a^{2}}}{2 d}\) | \(179\) |
risch | \(\frac {x^{2}}{2 a^{2}}-\frac {i b^{2} \left (i a +b \,{\mathrm e}^{i \left (d \,x^{2}+c \right )}\right )}{a^{2} \left (-a^{2}+b^{2}\right ) d \left (2 b \,{\mathrm e}^{i \left (d \,x^{2}+c \right )}-i a \,{\mathrm e}^{2 i \left (d \,x^{2}+c \right )}+i a \right )}+\frac {b \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {b \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}\) | \(420\) |
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Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (111) = 222\).
Time = 0.29 (sec) , antiderivative size = 536, normalized size of antiderivative = 4.47 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sin \left (d x^{2} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2} + {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \sin \left (d x^{2} + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{2} + c\right )^{2} + 2 \, a b \sin \left (d x^{2} + c\right ) + a^{2} + b^{2} + 2 \, {\left (b \cos \left (d x^{2} + c\right ) \sin \left (d x^{2} + c\right ) + a \cos \left (d x^{2} + c\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (d x^{2} + c\right )^{2} - 2 \, a b \sin \left (d x^{2} + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x^{2} + c\right )}{4 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sin \left (d x^{2} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}}, \frac {{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sin \left (d x^{2} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2} + {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \sin \left (d x^{2} + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (d x^{2} + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (d x^{2} + c\right )}\right ) - {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sin \left (d x^{2} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}}\right ] \]
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\[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x}{\left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]
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Timed out. \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.45 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=-\frac {{\left (2 \, a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {d x^{2} + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} d - a^{2} b^{2} d\right )} \sqrt {-a^{2} + b^{2}}} - \frac {a b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + b^{2}}{{\left (a^{3} d - a b^{2} d\right )} {\left (b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + b\right )}} + \frac {d x^{2} + c}{2 \, a^{2} d} \]
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Time = 22.51 (sec) , antiderivative size = 2755, normalized size of antiderivative = 22.96 \[ \int \frac {x}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \]
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