Integrand size = 21, antiderivative size = 113 \[ \int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx=\frac {5 d^{7/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b}+\frac {5 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b}-\frac {5 d^3 \sqrt {d \cos (a+b x)}}{2 b}-\frac {d (d \cos (a+b x))^{5/2} \csc ^2(a+b x)}{2 b} \]
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Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2645, 294, 327, 335, 218, 212, 209} \[ \int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx=\frac {5 d^{7/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b}+\frac {5 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b}-\frac {5 d^3 \sqrt {d \cos (a+b x)}}{2 b}-\frac {d \csc ^2(a+b x) (d \cos (a+b x))^{5/2}}{2 b} \]
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Rule 209
Rule 212
Rule 218
Rule 294
Rule 327
Rule 335
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^{7/2}}{\left (1-\frac {x^2}{d^2}\right )^2} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {d (d \cos (a+b x))^{5/2} \csc ^2(a+b x)}{2 b}+\frac {(5 d) \text {Subst}\left (\int \frac {x^{3/2}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{4 b} \\ & = -\frac {5 d^3 \sqrt {d \cos (a+b x)}}{2 b}-\frac {d (d \cos (a+b x))^{5/2} \csc ^2(a+b x)}{2 b}+\frac {\left (5 d^3\right ) \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} \\ & = -\frac {5 d^3 \sqrt {d \cos (a+b x)}}{2 b}-\frac {d (d \cos (a+b x))^{5/2} \csc ^2(a+b x)}{2 b}+\frac {\left (5 d^3\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{d^2}} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{2 b} \\ & = -\frac {5 d^3 \sqrt {d \cos (a+b x)}}{2 b}-\frac {d (d \cos (a+b x))^{5/2} \csc ^2(a+b x)}{2 b}+\frac {\left (5 d^4\right ) \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b}+\frac {\left (5 d^4\right ) \text {Subst}\left (\int \frac {1}{d+x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b} \\ & = \frac {5 d^{7/2} \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b}+\frac {5 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b}-\frac {5 d^3 \sqrt {d \cos (a+b x)}}{2 b}-\frac {d (d \cos (a+b x))^{5/2} \csc ^2(a+b x)}{2 b} \\ \end{align*}
Time = 1.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx=\frac {(d \cos (a+b x))^{7/2} \left (5 \arctan \left (\sqrt {\cos (a+b x)}\right )+3 \text {arctanh}\left (\sqrt {\cos (a+b x)}\right )-8 \sqrt {\cos (a+b x)}-2 \sqrt {\cos (a+b x)} \csc ^2(a+b x)-\log \left (1-\sqrt {\cos (a+b x)}\right )+\log \left (1+\sqrt {\cos (a+b x)}\right )\right )}{4 b \cos ^{\frac {7}{2}}(a+b x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(309\) vs. \(2(89)=178\).
Time = 6.02 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.74
method | result | size |
default | \(\frac {-\frac {d^{3} \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d -d}}{8 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}-\frac {5 d^{4} \ln \left (\frac {-2 d +2 \sqrt {-d}\, \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d -d}}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}{4 \sqrt {-d}}-2 d^{3} \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}+\frac {5 d^{\frac {7}{2}} \ln \left (\frac {4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )-1}\right )}{8}+\frac {5 d^{\frac {7}{2}} \ln \left (\frac {-4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+2 \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\right )}{8}+\frac {d^{3} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}}{16 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-16}-\frac {d^{3} \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}}{16 \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}}{b}\) | \(310\) |
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (89) = 178\).
Time = 0.42 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.48 \[ \int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx=\left [-\frac {10 \, {\left (d^{3} \cos \left (b x + a\right )^{2} - d^{3}\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d}}{d \cos \left (b x + a\right ) + d}\right ) - 5 \, {\left (d^{3} \cos \left (b x + a\right )^{2} - d^{3}\right )} \sqrt {-d} \log \left (-\frac {d \cos \left (b x + a\right )^{2} + 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (4 \, d^{3} \cos \left (b x + a\right )^{2} - 5 \, d^{3}\right )} \sqrt {d \cos \left (b x + a\right )}}{16 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}}, -\frac {10 \, {\left (d^{3} \cos \left (b x + a\right )^{2} - d^{3}\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d}}{d \cos \left (b x + a\right ) - d}\right ) - 5 \, {\left (d^{3} \cos \left (b x + a\right )^{2} - d^{3}\right )} \sqrt {d} \log \left (-\frac {d \cos \left (b x + a\right )^{2} + 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (4 \, d^{3} \cos \left (b x + a\right )^{2} - 5 \, d^{3}\right )} \sqrt {d \cos \left (b x + a\right )}}{16 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}}\right ] \]
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Timed out. \[ \int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.04 \[ \int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx=\frac {\frac {4 \, \sqrt {d \cos \left (b x + a\right )} d^{6}}{d^{2} \cos \left (b x + a\right )^{2} - d^{2}} + 10 \, d^{\frac {9}{2}} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right ) - 5 \, d^{\frac {9}{2}} \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right ) - 16 \, \sqrt {d \cos \left (b x + a\right )} d^{4}}{8 \, b d} \]
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\[ \int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}} \csc \left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}}{{\sin \left (a+b\,x\right )}^3} \,d x \]
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