Integrand size = 18, antiderivative size = 396 \[ \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx=\frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3} \]
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Time = 1.15 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4290, 4276, 3404, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx=\frac {i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}-\frac {i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 \sqrt {b^2-a^2}}+\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}-\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 \sqrt {b^2-a^2}}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a d \sqrt {b^2-a^2}}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{2 a d \sqrt {b^2-a^2}}+\frac {x^6}{6 a} \]
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3404
Rule 4276
Rule 4290
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{a+b \csc (c+d x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {x^2}{a}-\frac {b x^2}{a (b+a \sin (c+d x))}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^6}{6 a}-\frac {b \text {Subst}\left (\int \frac {x^2}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{2 a} \\ & = \frac {x^6}{6 a}-\frac {b \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a} \\ & = \frac {x^6}{6 a}+\frac {(i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^2\right )}{\sqrt {-a^2+b^2}}-\frac {(i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^2\right )}{\sqrt {-a^2+b^2}} \\ & = \frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {(i b) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2} d}+\frac {(i b) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2} d} \\ & = \frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {b \text {Subst}\left (\int \operatorname {PolyLog}\left (2,\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2} d^2} \\ & = \frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a \sqrt {-a^2+b^2} d^3} \\ & = \frac {x^6}{6 a}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {b x^2 \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {i b \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3} \\ \end{align*}
Time = 1.43 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx=\frac {\sqrt {a^2-b^2} d^3 x^6-3 b d^2 x^4 \log \left (1-\frac {a e^{i \left (c+d x^2\right )}}{-i b+\sqrt {a^2-b^2}}\right )+3 b d^2 x^4 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )+6 i b d x^2 \operatorname {PolyLog}\left (2,\frac {a e^{i \left (c+d x^2\right )}}{-i b+\sqrt {a^2-b^2}}\right )-6 i b d x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )-6 b \operatorname {PolyLog}\left (3,\frac {a e^{i \left (c+d x^2\right )}}{-i b+\sqrt {a^2-b^2}}\right )+6 b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{6 a \sqrt {a^2-b^2} d^3} \]
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\[\int \frac {x^{5}}{a +b \csc \left (d \,x^{2}+c \right )}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (332) = 664\).
Time = 0.38 (sec) , antiderivative size = 1445, normalized size of antiderivative = 3.65 \[ \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx=\int \frac {x^{5}}{a + b \csc {\left (c + d x^{2} \right )}}\, dx \]
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\[ \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx=\int { \frac {x^{5}}{b \csc \left (d x^{2} + c\right ) + a} \,d x } \]
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\[ \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx=\int { \frac {x^{5}}{b \csc \left (d x^{2} + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {x^5}{a+b \csc \left (c+d x^2\right )} \, dx=\int \frac {x^5}{a+\frac {b}{\sin \left (d\,x^2+c\right )}} \,d x \]
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