Integrand size = 16, antiderivative size = 39 \[ \int \frac {1}{\sqrt {a+b \csc ^2(c+d x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \csc ^2(c+d x)}}\right )}{\sqrt {a} d} \]
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Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4213, 385, 209} \[ \int \frac {1}{\sqrt {a+b \csc ^2(c+d x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)+b}}\right )}{\sqrt {a} d} \]
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Rule 209
Rule 385
Rule 4213
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{d} \\ & = -\frac {\arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {a+b+b \cot ^2(c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(39)=78\).
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.51 \[ \int \frac {1}{\sqrt {a+b \csc ^2(c+d x)}} \, dx=-\frac {\sqrt {-a-2 b+a \cos (2 (c+d x))} \csc (c+d x) \log \left (\sqrt {2} \sqrt {a} \cos (c+d x)+\sqrt {-a-2 b+a \cos (2 (c+d x))}\right )}{\sqrt {2} \sqrt {a} d \sqrt {a+b \csc ^2(c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs. \(2(33)=66\).
Time = 2.42 (sec) , antiderivative size = 165, normalized size of antiderivative = 4.23
method | result | size |
default | \(-\frac {\sqrt {-\frac {a \cos \left (d x +c \right )^{2}-a -b}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \ln \left (4 \sqrt {-\frac {a \cos \left (d x +c \right )^{2}-a -b}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \cos \left (d x +c \right ) \sqrt {-a}+4 \sqrt {-a}\, \sqrt {-\frac {a \cos \left (d x +c \right )^{2}-a -b}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-4 \cos \left (d x +c \right ) a \right ) \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ) \sqrt {4}}{2 d \sqrt {a +b \csc \left (d x +c \right )^{2}}\, \sqrt {-a}}\) | \(165\) |
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Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (33) = 66\).
Time = 0.33 (sec) , antiderivative size = 414, normalized size of antiderivative = 10.62 \[ \int \frac {1}{\sqrt {a+b \csc ^2(c+d x)}} \, dx=\left [-\frac {\sqrt {-a} \log \left (128 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \, {\left (a^{4} + a^{3} b\right )} \cos \left (d x + c\right )^{6} + 160 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (16 \, a^{3} \cos \left (d x + c\right )^{7} - 24 \, {\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )\right )}{8 \, a d}, \frac {\arctan \left (\frac {{\left (8 \, a^{2} \cos \left (d x + c\right )^{4} - 8 \, {\left (a^{2} + a b\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a - b}{\cos \left (d x + c\right )^{2} - 1}} \sin \left (d x + c\right )}{4 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{5} - 3 \, {\left (a^{3} + a^{2} b\right )} \cos \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )\right )}}\right )}{4 \, \sqrt {a} d}\right ] \]
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\[ \int \frac {1}{\sqrt {a+b \csc ^2(c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \csc ^{2}{\left (c + d x \right )}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 987 vs. \(2 (33) = 66\).
Time = 0.46 (sec) , antiderivative size = 987, normalized size of antiderivative = 25.31 \[ \int \frac {1}{\sqrt {a+b \csc ^2(c+d x)}} \, dx=\text {Too large to display} \]
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Exception generated. \[ \int \frac {1}{\sqrt {a+b \csc ^2(c+d x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\sqrt {a+b \csc ^2(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {b}{{\sin \left (c+d\,x\right )}^2}}} \,d x \]
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