Optimal. Leaf size=396 \[ \frac{b x^2 \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 \sqrt{b^2-a^2}}-\frac{b x^2 \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 \sqrt{b^2-a^2}}+\frac{i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^3 \sqrt{b^2-a^2}}-\frac{i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^3 \sqrt{b^2-a^2}}+\frac{i b x^4 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a d \sqrt{b^2-a^2}}-\frac{i b x^4 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a d \sqrt{b^2-a^2}}+\frac{x^6}{6 a} \]
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Rubi [A] time = 0.945527, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4205, 4191, 3323, 2264, 2190, 2531, 2282, 6589} \[ \frac{b x^2 \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^2 \sqrt{b^2-a^2}}-\frac{b x^2 \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^2 \sqrt{b^2-a^2}}+\frac{i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{a d^3 \sqrt{b^2-a^2}}-\frac{i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{a d^3 \sqrt{b^2-a^2}}+\frac{i b x^4 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{b-\sqrt{b^2-a^2}}\right )}{2 a d \sqrt{b^2-a^2}}-\frac{i b x^4 \log \left (1-\frac{i a e^{i \left (c+d x^2\right )}}{\sqrt{b^2-a^2}+b}\right )}{2 a d \sqrt{b^2-a^2}}+\frac{x^6}{6 a} \]
Antiderivative was successfully verified.
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Rule 4205
Rule 4191
Rule 3323
Rule 2264
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^5}{a+b \csc \left (c+d x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{a+b \csc (c+d x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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 \operatorname{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 \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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 \text{Li}_2\left (\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 \text{Li}_2\left (\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 \operatorname{Subst}\left (\int \text{Li}_2\left (\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 \operatorname{Subst}\left (\int \text{Li}_2\left (\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 \text{Li}_2\left (\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 \text{Li}_2\left (\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{Subst}\left (\int \frac{\text{Li}_2\left (\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) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\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 \text{Li}_2\left (\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 \text{Li}_2\left (\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{Li}_3\left (\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 \text{Li}_3\left (\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*}
Mathematica [A] time = 1.15507, size = 488, normalized size = 1.23 \[ \frac{6 i b e^{i c} d x^2 \text{PolyLog}\left (2,\frac{i a e^{i \left (2 c+d x^2\right )}}{i \sqrt{e^{2 i c} \left (a^2-b^2\right )}+b e^{i c}}\right )-6 i b e^{i c} d x^2 \text{PolyLog}\left (2,-\frac{a e^{i \left (2 c+d x^2\right )}}{\sqrt{e^{2 i c} \left (a^2-b^2\right )}+i b e^{i c}}\right )-6 b e^{i c} \text{PolyLog}\left (3,\frac{i a e^{i \left (2 c+d x^2\right )}}{i \sqrt{e^{2 i c} \left (a^2-b^2\right )}+b e^{i c}}\right )+6 b e^{i c} \text{PolyLog}\left (3,-\frac{a e^{i \left (2 c+d x^2\right )}}{\sqrt{e^{2 i c} \left (a^2-b^2\right )}+i b e^{i c}}\right )+d^3 x^6 \sqrt{e^{2 i c} \left (a^2-b^2\right )}-3 b e^{i c} d^2 x^4 \log \left (1+\frac{a e^{i \left (2 c+d x^2\right )}}{i b e^{i c}-\sqrt{e^{2 i c} \left (a^2-b^2\right )}}\right )+3 b e^{i c} d^2 x^4 \log \left (1+\frac{a e^{i \left (2 c+d x^2\right )}}{\sqrt{e^{2 i c} \left (a^2-b^2\right )}+i b e^{i c}}\right )}{6 a d^3 \sqrt{e^{2 i c} \left (a^2-b^2\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.319, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{a+b\csc \left ( d{x}^{2}+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.795814, size = 3459, normalized size = 8.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{a + b \csc{\left (c + d x^{2} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{b \csc \left (d x^{2} + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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