Optimal. Leaf size=153 \[ \frac{1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+4 e\right )}{36 c^5 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left (3 c^2 d+2 e\right )}{12 c^5 \sqrt{c^2 x^2}}+\frac{b e x \left (c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.121809, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 5239, 12, 446, 77} \[ \frac{1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+4 e\right )}{36 c^5 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left (3 c^2 d+2 e\right )}{12 c^5 \sqrt{c^2 x^2}}+\frac{b e x \left (c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 5239
Rule 12
Rule 446
Rule 77
Rubi steps
\begin{align*} \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{x^3 \left (3 d+2 e x^2\right )}{12 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=\frac{1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \int \frac{x^3 \left (3 d+2 e x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{12 \sqrt{c^2 x^2}}\\ &=\frac{1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \operatorname{Subst}\left (\int \frac{x (3 d+2 e x)}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{24 \sqrt{c^2 x^2}}\\ &=\frac{1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac{(b c x) \operatorname{Subst}\left (\int \left (\frac{3 c^2 d+2 e}{c^4 \sqrt{-1+c^2 x}}+\frac{\left (3 c^2 d+4 e\right ) \sqrt{-1+c^2 x}}{c^4}+\frac{2 e \left (-1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{24 \sqrt{c^2 x^2}}\\ &=\frac{b \left (3 c^2 d+2 e\right ) x \sqrt{-1+c^2 x^2}}{12 c^5 \sqrt{c^2 x^2}}+\frac{b \left (3 c^2 d+4 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{36 c^5 \sqrt{c^2 x^2}}+\frac{b e x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt{c^2 x^2}}+\frac{1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.205599, size = 97, normalized size = 0.63 \[ \frac{1}{180} x \left (15 a x^3 \left (3 d+2 e x^2\right )+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left (3 c^4 \left (5 d x^2+2 e x^4\right )+c^2 \left (30 d+8 e x^2\right )+16 e\right )}{c^5}+15 b x^3 \csc ^{-1}(c x) \left (3 d+2 e x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.183, size = 134, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{4}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{6}{x}^{6}}{6}}+{\frac{{x}^{4}{c}^{6}d}{4}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccsc} \left (cx\right )e{c}^{6}{x}^{6}}{6}}+{\frac{{\rm arccsc} \left (cx\right ){c}^{6}{x}^{4}d}{4}}+{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 6\,{c}^{4}e{x}^{4}+15\,{c}^{4}d{x}^{2}+8\,{c}^{2}e{x}^{2}+30\,{c}^{2}d+16\,e \right ) }{180\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991776, size = 192, normalized size = 1.25 \begin{align*} \frac{1}{6} \, a e x^{6} + \frac{1}{4} \, a d x^{4} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arccsc}\left (c x\right ) + \frac{c^{2} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 3 \, x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d + \frac{1}{90} \,{\left (15 \, x^{6} \operatorname{arccsc}\left (c x\right ) + \frac{3 \, c^{4} x^{5}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 10 \, c^{2} x^{3}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.91554, size = 247, normalized size = 1.61 \begin{align*} \frac{30 \, a c^{6} e x^{6} + 45 \, a c^{6} d x^{4} + 15 \,{\left (2 \, b c^{6} e x^{6} + 3 \, b c^{6} d x^{4}\right )} \operatorname{arccsc}\left (c x\right ) +{\left (6 \, b c^{4} e x^{4} + 30 \, b c^{2} d +{\left (15 \, b c^{4} d + 8 \, b c^{2} e\right )} x^{2} + 16 \, b e\right )} \sqrt{c^{2} x^{2} - 1}}{180 \, c^{6}} \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 x^{3} \left (a + b \operatorname{acsc}{\left (c x \right )}\right ) \left (d + e 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{\left (e x^{2} + d\right )}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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