Optimal. Leaf size=249 \[ -\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 b c^3 \sqrt{-c^2 x^2-1} \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right )}{11025 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right )}{11025 x^2 \sqrt{-c^2 x^2}}+\frac{b c d^2 \sqrt{-c^2 x^2-1}}{49 x^6 \sqrt{-c^2 x^2}}-\frac{2 b c d \sqrt{-c^2 x^2-1} \left (15 c^2 d-49 e\right )}{1225 x^4 \sqrt{-c^2 x^2}} \]
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Rubi [A] time = 0.196614, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {270, 6302, 12, 1265, 453, 271, 264} \[ -\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{2 b c^3 \sqrt{-c^2 x^2-1} \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right )}{11025 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right )}{11025 x^2 \sqrt{-c^2 x^2}}+\frac{b c d^2 \sqrt{-c^2 x^2-1}}{49 x^6 \sqrt{-c^2 x^2}}-\frac{2 b c d \sqrt{-c^2 x^2-1} \left (15 c^2 d-49 e\right )}{1225 x^4 \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 6302
Rule 12
Rule 1265
Rule 453
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2 \left (a+b \text{csch}^{-1}(c x)\right )}{x^8} \, dx &=-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{(b c x) \int \frac{-15 d^2-42 d e x^2-35 e^2 x^4}{105 x^8 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{(b c x) \int \frac{-15 d^2-42 d e x^2-35 e^2 x^4}{x^8 \sqrt{-1-c^2 x^2}} \, dx}{105 \sqrt{-c^2 x^2}}\\ &=\frac{b c d^2 \sqrt{-1-c^2 x^2}}{49 x^6 \sqrt{-c^2 x^2}}-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{(b c x) \int \frac{6 d \left (15 c^2 d-49 e\right )-245 e^2 x^2}{x^6 \sqrt{-1-c^2 x^2}} \, dx}{735 \sqrt{-c^2 x^2}}\\ &=\frac{b c d^2 \sqrt{-1-c^2 x^2}}{49 x^6 \sqrt{-c^2 x^2}}-\frac{2 b c d \left (15 c^2 d-49 e\right ) \sqrt{-1-c^2 x^2}}{1225 x^4 \sqrt{-c^2 x^2}}-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}-\frac{\left (b c \left (-360 c^4 d^2+1176 c^2 d e-1225 e^2\right ) x\right ) \int \frac{1}{x^4 \sqrt{-1-c^2 x^2}} \, dx}{3675 \sqrt{-c^2 x^2}}\\ &=\frac{b c d^2 \sqrt{-1-c^2 x^2}}{49 x^6 \sqrt{-c^2 x^2}}-\frac{2 b c d \left (15 c^2 d-49 e\right ) \sqrt{-1-c^2 x^2}}{1225 x^4 \sqrt{-c^2 x^2}}+\frac{b c \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt{-1-c^2 x^2}}{11025 x^2 \sqrt{-c^2 x^2}}-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}+\frac{\left (2 b c^3 \left (-360 c^4 d^2+1176 c^2 d e-1225 e^2\right ) x\right ) \int \frac{1}{x^2 \sqrt{-1-c^2 x^2}} \, dx}{11025 \sqrt{-c^2 x^2}}\\ &=-\frac{2 b c^3 \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt{-1-c^2 x^2}}{11025 \sqrt{-c^2 x^2}}+\frac{b c d^2 \sqrt{-1-c^2 x^2}}{49 x^6 \sqrt{-c^2 x^2}}-\frac{2 b c d \left (15 c^2 d-49 e\right ) \sqrt{-1-c^2 x^2}}{1225 x^4 \sqrt{-c^2 x^2}}+\frac{b c \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt{-1-c^2 x^2}}{11025 x^2 \sqrt{-c^2 x^2}}-\frac{d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{csch}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.259464, size = 152, normalized size = 0.61 \[ \frac{-105 a \left (15 d^2+42 d e x^2+35 e^2 x^4\right )+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (-45 d^2 \left (16 c^6 x^6-8 c^4 x^4+6 c^2 x^2-5\right )+294 d e x^2 \left (8 c^4 x^4-4 c^2 x^2+3\right )+1225 e^2 x^4 \left (1-2 c^2 x^2\right )\right )-105 b \text{csch}^{-1}(c x) \left (15 d^2+42 d e x^2+35 e^2 x^4\right )}{11025 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.207, size = 223, normalized size = 0.9 \begin{align*}{c}^{7} \left ({\frac{a}{{c}^{4}} \left ( -{\frac{{d}^{2}}{7\,{c}^{3}{x}^{7}}}-{\frac{2\,de}{5\,{c}^{3}{x}^{5}}}-{\frac{{e}^{2}}{3\,{c}^{3}{x}^{3}}} \right ) }+{\frac{b}{{c}^{4}} \left ( -{\frac{{\rm arccsch} \left (cx\right ){d}^{2}}{7\,{c}^{3}{x}^{7}}}-{\frac{2\,{\rm arccsch} \left (cx\right )de}{5\,{c}^{3}{x}^{5}}}-{\frac{{\rm arccsch} \left (cx\right ){e}^{2}}{3\,{c}^{3}{x}^{3}}}-{\frac{ \left ({c}^{2}{x}^{2}+1 \right ) \left ( 720\,{c}^{10}{d}^{2}{x}^{6}-2352\,{c}^{8}de{x}^{6}-360\,{c}^{8}{d}^{2}{x}^{4}+2450\,{c}^{6}{e}^{2}{x}^{6}+1176\,{c}^{6}de{x}^{4}+270\,{c}^{6}{d}^{2}{x}^{2}-1225\,{c}^{4}{e}^{2}{x}^{4}-882\,{c}^{4}de{x}^{2}-225\,{d}^{2}{c}^{4} \right ) }{11025\,{c}^{8}{x}^{8}}{\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 = 1.04044, size = 313, normalized size = 1.26 \begin{align*} \frac{1}{245} \, b d^{2}{\left (\frac{5 \, c^{8}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{7}{2}} - 21 \, c^{8}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + 35 \, c^{8}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 35 \, c^{8} \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c} - \frac{35 \, \operatorname{arcsch}\left (c x\right )}{x^{7}}\right )} + \frac{2}{75} \, b d e{\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{arcsch}\left (c x\right )}{x^{5}}\right )} + \frac{1}{9} \, b e^{2}{\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{arcsch}\left (c x\right )}{x^{3}}\right )} - \frac{a e^{2}}{3 \, x^{3}} - \frac{2 \, a d e}{5 \, x^{5}} - \frac{a d^{2}}{7 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55013, size = 481, normalized size = 1.93 \begin{align*} -\frac{3675 \, a e^{2} x^{4} + 4410 \, a d e x^{2} + 1575 \, a d^{2} + 105 \,{\left (35 \, b e^{2} x^{4} + 42 \, b d e x^{2} + 15 \, b d^{2}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (2 \,{\left (360 \, b c^{7} d^{2} - 1176 \, b c^{5} d e + 1225 \, b c^{3} e^{2}\right )} x^{7} -{\left (360 \, b c^{5} d^{2} - 1176 \, b c^{3} d e + 1225 \, b c e^{2}\right )} x^{5} - 225 \, b c d^{2} x + 18 \,{\left (15 \, b c^{3} d^{2} - 49 \, b c d e\right )} x^{3}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{11025 \, x^{7}} \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{acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{8}}\, 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 )}^{2}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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