Integrand size = 18, antiderivative size = 186 \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=-\frac {2 x^3 \text {arctanh}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \operatorname {PolyLog}\left (3,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \operatorname {PolyLog}\left (3,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 i \operatorname {PolyLog}\left (4,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 i \operatorname {PolyLog}\left (4,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}} \]
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Time = 0.68 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6852, 4268, 2611, 6744, 2320, 6724} \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=-\frac {2 x^3 \text {arctanh}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \operatorname {PolyLog}\left (3,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \operatorname {PolyLog}\left (3,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 i \operatorname {PolyLog}\left (4,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 i \operatorname {PolyLog}\left (4,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}} \]
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Rule 2320
Rule 2611
Rule 4268
Rule 6724
Rule 6744
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \frac {\sec (x) \int x^3 \csc (x) \, dx}{\sqrt {a \sec ^2(x)}} \\ & = -\frac {2 x^3 \text {arctanh}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(3 \sec (x)) \int x^2 \log \left (1-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}+\frac {(3 \sec (x)) \int x^2 \log \left (1+e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}} \\ & = -\frac {2 x^3 \text {arctanh}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(6 i \sec (x)) \int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}+\frac {(6 i \sec (x)) \int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}} \\ & = -\frac {2 x^3 \text {arctanh}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \operatorname {PolyLog}\left (3,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \operatorname {PolyLog}\left (3,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {(6 \sec (x)) \int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}}-\frac {(6 \sec (x)) \int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx}{\sqrt {a \sec ^2(x)}} \\ & = -\frac {2 x^3 \text {arctanh}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \operatorname {PolyLog}\left (3,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \operatorname {PolyLog}\left (3,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {(6 i \sec (x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i x}\right )}{\sqrt {a \sec ^2(x)}}+\frac {(6 i \sec (x)) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i x}\right )}{\sqrt {a \sec ^2(x)}} \\ & = -\frac {2 x^3 \text {arctanh}\left (e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 x \operatorname {PolyLog}\left (3,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 x \operatorname {PolyLog}\left (3,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {6 i \operatorname {PolyLog}\left (4,-e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}}+\frac {6 i \operatorname {PolyLog}\left (4,e^{i x}\right ) \sec (x)}{\sqrt {a \sec ^2(x)}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=-\frac {i \left (\pi ^4-2 x^4+8 i x^3 \log \left (1-e^{-i x}\right )-8 i x^3 \log \left (1+e^{i x}\right )-24 x^2 \operatorname {PolyLog}\left (2,e^{-i x}\right )-24 x^2 \operatorname {PolyLog}\left (2,-e^{i x}\right )+48 i x \operatorname {PolyLog}\left (3,e^{-i x}\right )-48 i x \operatorname {PolyLog}\left (3,-e^{i x}\right )+48 \operatorname {PolyLog}\left (4,e^{-i x}\right )+48 \operatorname {PolyLog}\left (4,-e^{i x}\right )\right ) \sec (x)}{8 \sqrt {a \sec ^2(x)}} \]
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Time = 0.94 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {2 i {\mathrm e}^{i x} \left (\frac {i x^{3} \ln \left ({\mathrm e}^{i x}+1\right )}{2}+\frac {3 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right )}{2}+3 i x \operatorname {polylog}\left (3, -{\mathrm e}^{i x}\right )-3 \operatorname {polylog}\left (4, -{\mathrm e}^{i x}\right )-\frac {i x^{3} \ln \left (1-{\mathrm e}^{i x}\right )}{2}-\frac {3 x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right )}{2}-3 i x \operatorname {polylog}\left (3, {\mathrm e}^{i x}\right )+3 \operatorname {polylog}\left (4, {\mathrm e}^{i x}\right )\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}\) | \(137\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (141) = 282\).
Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.76 \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\frac {6 \, x \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 6 \, x \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) - 6 \, x \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 6 \, x \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 6 i \, \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (4, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 6 i \, \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (4, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 6 i \, \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (4, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 6 i \, \sqrt {\frac {a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) {\rm polylog}\left (4, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - {\left (x^{3} \cos \left (x\right ) \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + x^{3} \cos \left (x\right ) \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - x^{3} \cos \left (x\right ) \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - x^{3} \cos \left (x\right ) \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + 3 i \, x^{2} \cos \left (x\right ) {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 3 i \, x^{2} \cos \left (x\right ) {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 3 i \, x^{2} \cos \left (x\right ) {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 3 i \, x^{2} \cos \left (x\right ) {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{2}}}}{2 \, a} \]
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\[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\int \frac {x^{3} \csc {\left (x \right )} \sec {\left (x \right )}}{\sqrt {a \sec ^{2}{\left (x \right )}}}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.70 \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=-\frac {2 i \, x^{3} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + 2 i \, x^{3} \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + x^{3} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - x^{3} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 6 i \, x^{2} {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + 6 i \, x^{2} {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) + 12 \, x {\rm Li}_{3}(-e^{\left (i \, x\right )}) - 12 \, x {\rm Li}_{3}(e^{\left (i \, x\right )}) + 12 i \, {\rm Li}_{4}(-e^{\left (i \, x\right )}) - 12 i \, {\rm Li}_{4}(e^{\left (i \, x\right )})}{2 \, \sqrt {a}} \]
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\[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\int { \frac {x^{3} \csc \left (x\right ) \sec \left (x\right )}{\sqrt {a \sec \left (x\right )^{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \, dx=\int \frac {x^3}{\cos \left (x\right )\,\sin \left (x\right )\,\sqrt {\frac {a}{{\cos \left (x\right )}^2}}} \,d x \]
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