Integrand size = 16, antiderivative size = 129 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {b \operatorname {PolyLog}\left (3,-e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {b \operatorname {PolyLog}\left (3,e^{i \left (c+d x^2\right )}\right )}{d^3} \]
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Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {14, 4290, 4268, 2611, 2320, 6724} \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b \operatorname {PolyLog}\left (3,-e^{i \left (d x^2+c\right )}\right )}{d^3}+\frac {b \operatorname {PolyLog}\left (3,e^{i \left (d x^2+c\right )}\right )}{d^3}+\frac {i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (d x^2+c\right )}\right )}{d^2}-\frac {i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (d x^2+c\right )}\right )}{d^2} \]
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Rule 14
Rule 2320
Rule 2611
Rule 4268
Rule 4290
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^5+b x^5 \csc \left (c+d x^2\right )\right ) \, dx \\ & = \frac {a x^6}{6}+b \int x^5 \csc \left (c+d x^2\right ) \, dx \\ & = \frac {a x^6}{6}+\frac {1}{2} b \text {Subst}\left (\int x^2 \csc (c+d x) \, dx,x,x^2\right ) \\ & = \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b \text {Subst}\left (\int x \log \left (1-e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d}+\frac {b \text {Subst}\left (\int x \log \left (1+e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d} \\ & = \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {(i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d^2}+\frac {(i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d^2} \\ & = \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {b \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {b \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{d^3} \\ & = \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {b \operatorname {PolyLog}\left (3,-e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {b \operatorname {PolyLog}\left (3,e^{i \left (c+d x^2\right )}\right )}{d^3} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.23 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}-\frac {b \left (d^2 x^4 \text {arctanh}\left (\cos \left (c+d x^2\right )+i \sin \left (c+d x^2\right )\right )-i d x^2 \operatorname {PolyLog}\left (2,-\cos \left (c+d x^2\right )-i \sin \left (c+d x^2\right )\right )+i d x^2 \operatorname {PolyLog}\left (2,\cos \left (c+d x^2\right )+i \sin \left (c+d x^2\right )\right )+\operatorname {PolyLog}\left (3,-\cos \left (c+d x^2\right )-i \sin \left (c+d x^2\right )\right )-\operatorname {PolyLog}\left (3,\cos \left (c+d x^2\right )+i \sin \left (c+d x^2\right )\right )\right )}{d^3} \]
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\[\int x^{5} \left (a +b \csc \left (d \,x^{2}+c \right )\right )d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (111) = 222\).
Time = 0.29 (sec) , antiderivative size = 425, normalized size of antiderivative = 3.29 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {2 \, a d^{3} x^{6} - 3 \, b d^{2} x^{4} \log \left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) - 3 \, b d^{2} x^{4} \log \left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) - 6 i \, b d x^{2} {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + 6 i \, b d x^{2} {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) - 6 i \, b d x^{2} {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + 6 i \, b d x^{2} {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) + 3 \, b c^{2} \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) + 3 \, b c^{2} \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (b d^{2} x^{4} - b c^{2}\right )} \log \left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) + 3 \, {\left (b d^{2} x^{4} - b c^{2}\right )} \log \left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) + 6 \, b {\rm polylog}\left (3, \cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + 6 \, b {\rm polylog}\left (3, \cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) - 6 \, b {\rm polylog}\left (3, -\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) - 6 \, b {\rm polylog}\left (3, -\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right )}{12 \, d^{3}} \]
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\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^{5} \left (a + b \csc {\left (c + d x^{2} \right )}\right )\, dx \]
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\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \]
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\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \]
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Timed out. \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^5\,\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right ) \,d x \]
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