Integrand size = 19, antiderivative size = 99 \[ \int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx=-\frac {d^{7/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}-\frac {d^{7/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}+\frac {2 d^3 \sqrt {d \cos (a+b x)}}{b}+\frac {2 d (d \cos (a+b x))^{5/2}}{5 b} \]
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Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2645, 327, 335, 218, 212, 209} \[ \int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx=-\frac {d^{7/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}-\frac {d^{7/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}+\frac {2 d^3 \sqrt {d \cos (a+b x)}}{b}+\frac {2 d (d \cos (a+b x))^{5/2}}{5 b} \]
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Rule 209
Rule 212
Rule 218
Rule 327
Rule 335
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^{7/2}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = \frac {2 d (d \cos (a+b x))^{5/2}}{5 b}-\frac {d \text {Subst}\left (\int \frac {x^{3/2}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b} \\ & = \frac {2 d^3 \sqrt {d \cos (a+b x)}}{b}+\frac {2 d (d \cos (a+b x))^{5/2}}{5 b}-\frac {d^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 )}{b} \\ & = \frac {2 d^3 \sqrt {d \cos (a+b x)}}{b}+\frac {2 d (d \cos (a+b x))^{5/2}}{5 b}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{d^2}} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{b} \\ & = \frac {2 d^3 \sqrt {d \cos (a+b x)}}{b}+\frac {2 d (d \cos (a+b x))^{5/2}}{5 b}-\frac {d^4 \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{b}-\frac {d^4 \text {Subst}\left (\int \frac {1}{d+x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{b} \\ & = -\frac {d^{7/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}-\frac {d^{7/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{b}+\frac {2 d^3 \sqrt {d \cos (a+b x)}}{b}+\frac {2 d (d \cos (a+b x))^{5/2}}{5 b} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.81 \[ \int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx=\frac {d^3 \sqrt {d \cos (a+b x)} \left (-5 \arctan \left (\sqrt {\cos (a+b x)}\right )-5 \text {arctanh}\left (\sqrt {\cos (a+b x)}\right )+\sqrt {\cos (a+b x)} (11+\cos (2 (a+b x)))\right )}{5 b \sqrt {\cos (a+b x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(291\) vs. \(2(81)=162\).
Time = 0.37 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.95
method | result | size |
default | \(-\frac {5 d^{\frac {7}{2}} \ln \left (\frac {4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )-1}\right ) \sqrt {-d}+5 d^{\frac {7}{2}} \ln \left (-\frac {2 \left (2 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-\sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}+d \right )}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\right ) \sqrt {-d}-16 d^{3} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}\, \sqrt {-d}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+16 d^{3} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}\, \sqrt {-d}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-24 d^{3} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}\, \sqrt {-d}-10 d^{4} \ln \left (\frac {2 \sqrt {-d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}{10 \sqrt {-d}\, b}\) | \(292\) |
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Time = 0.40 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.02 \[ \int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx=\left [\frac {10 \, \sqrt {-d} d^{3} \arctan \left (\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d}}{d \cos \left (b x + a\right ) + d}\right ) + 5 \, \sqrt {-d} d^{3} \log \left (-\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (d^{3} \cos \left (b x + a\right )^{2} + 5 \, d^{3}\right )} \sqrt {d \cos \left (b x + a\right )}}{20 \, b}, \frac {10 \, d^{\frac {7}{2}} \arctan \left (\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d}}{d \cos \left (b x + a\right ) - d}\right ) + 5 \, d^{\frac {7}{2}} \log \left (-\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (d^{3} \cos \left (b x + a\right )^{2} + 5 \, d^{3}\right )} \sqrt {d \cos \left (b x + a\right )}}{20 \, b}\right ] \]
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Timed out. \[ \int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.99 \[ \int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx=-\frac {10 \, d^{\frac {9}{2}} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right ) - 5 \, d^{\frac {9}{2}} \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right ) - 4 \, \left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}} d^{2} - 20 \, \sqrt {d \cos \left (b x + a\right )} d^{4}}{10 \, b d} \]
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\[ \int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} \csc \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}}{\sin \left (a+b\,x\right )} \,d x \]
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