Optimal. Leaf size=152 \[ -\frac{d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac{2 b c^3 \sqrt{c^2 x^2-1} \left (12 c^2 d+25 e\right )}{225 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (12 c^2 d+25 e\right )}{225 x^2 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{c^2 x^2-1}}{25 x^4 \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.0921755, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 5239, 12, 453, 271, 264} \[ -\frac{d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac{2 b c^3 \sqrt{c^2 x^2-1} \left (12 c^2 d+25 e\right )}{225 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (12 c^2 d+25 e\right )}{225 x^2 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{c^2 x^2-1}}{25 x^4 \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 5239
Rule 12
Rule 453
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx &=-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}+\frac{(b c x) \int \frac{-3 d-5 e x^2}{15 x^6 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}+\frac{(b c x) \int \frac{-3 d-5 e x^2}{x^6 \sqrt{-1+c^2 x^2}} \, dx}{15 \sqrt{c^2 x^2}}\\ &=-\frac{b c d \sqrt{-1+c^2 x^2}}{25 x^4 \sqrt{c^2 x^2}}-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}+\frac{\left (b c \left (-12 c^2 d-25 e\right ) x\right ) \int \frac{1}{x^4 \sqrt{-1+c^2 x^2}} \, dx}{75 \sqrt{c^2 x^2}}\\ &=-\frac{b c d \sqrt{-1+c^2 x^2}}{25 x^4 \sqrt{c^2 x^2}}-\frac{b c \left (12 c^2 d+25 e\right ) \sqrt{-1+c^2 x^2}}{225 x^2 \sqrt{c^2 x^2}}-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}+\frac{\left (2 b c^3 \left (-12 c^2 d-25 e\right ) x\right ) \int \frac{1}{x^2 \sqrt{-1+c^2 x^2}} \, dx}{225 \sqrt{c^2 x^2}}\\ &=-\frac{2 b c^3 \left (12 c^2 d+25 e\right ) \sqrt{-1+c^2 x^2}}{225 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{-1+c^2 x^2}}{25 x^4 \sqrt{c^2 x^2}}-\frac{b c \left (12 c^2 d+25 e\right ) \sqrt{-1+c^2 x^2}}{225 x^2 \sqrt{c^2 x^2}}-\frac{d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.159635, size = 94, normalized size = 0.62 \[ -\frac{15 a \left (3 d+5 e x^2\right )+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (3 d \left (8 c^4 x^4+4 c^2 x^2+3\right )+25 e x^2 \left (2 c^2 x^2+1\right )\right )+15 b \csc ^{-1}(c x) \left (3 d+5 e x^2\right )}{225 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.185, size = 140, normalized size = 0.9 \begin{align*}{c}^{5} \left ({\frac{a}{{c}^{2}} \left ( -{\frac{d}{5\,{c}^{3}{x}^{5}}}-{\frac{e}{3\,{c}^{3}{x}^{3}}} \right ) }+{\frac{b}{{c}^{2}} \left ( -{\frac{{\rm arccsc} \left (cx\right )d}{5\,{c}^{3}{x}^{5}}}-{\frac{{\rm arccsc} \left (cx\right )e}{3\,{c}^{3}{x}^{3}}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 24\,{c}^{6}d{x}^{4}+50\,{c}^{4}e{x}^{4}+12\,{c}^{4}d{x}^{2}+25\,{c}^{2}e{x}^{2}+9\,{c}^{2}d \right ) }{225\,{c}^{6}{x}^{6}}{\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 = 0.972544, size = 185, normalized size = 1.22 \begin{align*} -\frac{1}{75} \, b d{\left (\frac{3 \, c^{6}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 10 \, c^{6}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, c^{6} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} + \frac{15 \, \operatorname{arccsc}\left (c x\right )}{x^{5}}\right )} + \frac{1}{9} \, b e{\left (\frac{c^{4}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, c^{4} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} - \frac{3 \, \operatorname{arccsc}\left (c x\right )}{x^{3}}\right )} - \frac{a e}{3 \, x^{3}} - \frac{a d}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62581, size = 217, normalized size = 1.43 \begin{align*} -\frac{75 \, a e x^{2} + 45 \, a d + 15 \,{\left (5 \, b e x^{2} + 3 \, b d\right )} \operatorname{arccsc}\left (c x\right ) +{\left (2 \,{\left (12 \, b c^{4} d + 25 \, b c^{2} e\right )} x^{4} +{\left (12 \, b c^{2} d + 25 \, b e\right )} x^{2} + 9 \, b d\right )} \sqrt{c^{2} x^{2} - 1}}{225 \, x^{5}} \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{\left (a + b \operatorname{acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{6}}\, 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{{\left (e x^{2} + d\right )}{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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