Integrand size = 21, antiderivative size = 93 \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=-\frac {3 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d} \]
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Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2645, 296, 335, 218, 212, 209} \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=-\frac {3 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {\csc ^2(a+b x) \sqrt {d \cos (a+b x)}}{2 b d} \]
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Rule 209
Rule 212
Rule 218
Rule 296
Rule 335
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{d^2}\right )^2} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{4 b d} \\ & = -\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d}-\frac {3 \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{d^2}} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{2 b d} \\ & = -\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d}-\frac {3 \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b}-\frac {3 \text {Subst}\left (\int \frac {1}{d+x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b} \\ & = -\frac {3 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.74 \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {d \left (-\cot ^2(a+b x)\right )^{3/4} \left (\sqrt [4]{-\cot ^2(a+b x)}-\operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{4},\frac {7}{4},\csc ^2(a+b x)\right )\right )}{2 b (d \cos (a+b x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs. \(2(73)=146\).
Time = 0.09 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.04
method | result | size |
default | \(\frac {-\frac {\sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d -d}}{8 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}+\frac {3 \ln \left (\frac {-2 d +2 \sqrt {-d}\, \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d -d}}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}{4 \sqrt {-d}}-\frac {3 \ln \left (\frac {4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )-1}\right )}{8 \sqrt {d}}-\frac {3 \ln \left (\frac {-4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\right )}{8 \sqrt {d}}+\frac {\sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}}{16 d \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}-\frac {\sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}}{16 d \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}}{b}\) | \(283\) |
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Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (73) = 146\).
Time = 0.36 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.59 \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\left [\frac {6 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) + 1\right )}}{2 \, d \cos \left (b x + a\right )}\right ) - 3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {-d} \log \left (\frac {d \cos \left (b x + a\right )^{2} + 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, \sqrt {d \cos \left (b x + a\right )}}{16 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\right )}}, -\frac {6 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - 1\right )}}{2 \, \sqrt {d} \cos \left (b x + a\right )}\right ) - 3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {d} \log \left (\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) - 8 \, \sqrt {d \cos \left (b x + a\right )}}{16 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\right )}}\right ] \]
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\[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )}}{\sqrt {d \cos {\left (a + b x \right )}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.11 \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {\frac {4 \, \sqrt {d \cos \left (b x + a\right )} d^{2}}{d^{2} \cos \left (b x + a\right )^{2} - d^{2}} - 6 \, \sqrt {d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right ) + 3 \, \sqrt {d} \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right )}{8 \, b d} \]
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Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {d^{3} {\left (\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{{\left (d^{2} \cos \left (b x + a\right )^{2} - d^{2}\right )} d^{2}} + \frac {3 \, \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {-d}}\right )}{\sqrt {-d} d^{3}} - \frac {3 \, \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right )}{d^{\frac {7}{2}}}\right )}}{4 \, b} \]
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Timed out. \[ \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^3\,\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \]
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