Integrand size = 21, antiderivative size = 115 \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {5 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{3/2}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{3/2}}+\frac {5}{2 b d \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d \sqrt {d \cos (a+b x)}} \]
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Time = 0.05 (sec) , antiderivative size = 115, 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, 296, 331, 335, 304, 209, 212} \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {5 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{3/2}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{3/2}}+\frac {5}{2 b d \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d \sqrt {d \cos (a+b x)}} \]
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Rule 209
Rule 212
Rule 296
Rule 304
Rule 331
Rule 335
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^{3/2} \left (1-\frac {x^2}{d^2}\right )^2} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {\csc ^2(a+b x)}{2 b d \sqrt {d \cos (a+b x)}}-\frac {5 \text {Subst}\left (\int \frac {1}{x^{3/2} \left (1-\frac {x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{4 b d} \\ & = \frac {5}{2 b d \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d \sqrt {d \cos (a+b x)}}-\frac {5 \text {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{4 b d^3} \\ & = \frac {5}{2 b d \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d \sqrt {d \cos (a+b x)}}-\frac {5 \text {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{d^2}} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{2 b d^3} \\ & = \frac {5}{2 b d \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d \sqrt {d \cos (a+b x)}}-\frac {5 \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b d}+\frac {5 \text {Subst}\left (\int \frac {1}{d+x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b d} \\ & = \frac {5 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{3/2}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{3/2}}+\frac {5}{2 b d \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d \sqrt {d \cos (a+b x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.35 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.79 \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {-\left (-\cot ^2(a+b x)\right )^{3/4} \left (-4+\cot ^2(a+b x)\right )+5 \cot ^2(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\csc ^2(a+b x)\right )}{2 b d \sqrt {d \cos (a+b x)} \left (-\cot ^2(a+b x)\right )^{3/4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(688\) vs. \(2(91)=182\).
Time = 0.12 (sec) , antiderivative size = 689, normalized size of antiderivative = 5.99
method | result | size |
default | \(-\frac {\sqrt {-d}\, \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}-\left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (-20 d^{\frac {3}{2}} \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}\, \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 ) d -10 \sqrt {-d}\, \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 ) d \right )-5 \left (6 d^{\frac {3}{2}} \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 )-4 \sqrt {-d}\, \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}+3 \sqrt {-d}\, \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 ) d +3 \sqrt {-d}\, \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 ) d \right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+5 \left (2 d^{\frac {3}{2}} \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 )-4 \sqrt {-d}\, \sqrt {d}\, \sqrt {-2 d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+d}+\sqrt {-d}\, \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 ) d +\sqrt {-d}\, \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 ) d \right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 d^{\frac {5}{2}} \sqrt {-d}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-3 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right ) b}\) | \(689\) |
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Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (91) = 182\).
Time = 0.34 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.53 \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\left [\frac {10 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) + 1\right )}}{2 \, d \cos \left (b x + a\right )}\right ) - 5 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \sqrt {-d} \log \left (\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, \sqrt {d \cos \left (b x + a\right )} {\left (5 \, \cos \left (b x + a\right )^{2} - 4\right )}}{16 \, {\left (b d^{2} \cos \left (b x + a\right )^{3} - b d^{2} \cos \left (b x + a\right )\right )}}, \frac {10 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - 1\right )}}{2 \, \sqrt {d} \cos \left (b x + a\right )}\right ) + 5 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \sqrt {d} \log \left (\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, \sqrt {d \cos \left (b x + a\right )} {\left (5 \, \cos \left (b x + a\right )^{2} - 4\right )}}{16 \, {\left (b d^{2} \cos \left (b x + a\right )^{3} - b d^{2} \cos \left (b x + a\right )\right )}}\right ] \]
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\[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {\frac {10 \, \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right )}{\sqrt {d}} + \frac {5 \, \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right )}{\sqrt {d}} + \frac {4 \, {\left (5 \, d^{2} \cos \left (b x + a\right )^{2} - 4 \, d^{2}\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}} - \sqrt {d \cos \left (b x + a\right )} d^{2}}}{8 \, b d} \]
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\[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^3\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \]
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