Optimal. Leaf size=281 \[ -\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\frac{2 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right )}{11025 x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right )}{11025 x^3}+\frac{b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{2 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (15 c^2 d+49 e\right )}{1225 x^5} \]
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Rubi [A] time = 0.200737, antiderivative size = 281, 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, 6301, 12, 1265, 453, 271, 264} \[ -\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\frac{2 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right )}{11025 x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right )}{11025 x^3}+\frac{b d^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{2 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (15 c^2 d+49 e\right )}{1225 x^5} \]
Antiderivative was successfully verified.
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Rule 270
Rule 6301
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{sech}^{-1}(c x)\right )}{x^8} \, dx &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \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\\ &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\frac{1}{105} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-15 d^2-42 d e x^2-35 e^2 x^4}{x^8 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b d^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{49 x^7}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}-\frac{1}{735} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \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\\ &=\frac{b d^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{2 b d \left (15 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{1225 x^5}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}-\frac{\left (b \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^4 \sqrt{1-c^2 x^2}} \, dx}{3675}\\ &=\frac{b d^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{2 b d \left (15 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{1225 x^5}+\frac{b \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{11025 x^3}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}-\frac{\left (2 b c^2 \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2}} \, dx}{11025}\\ &=\frac{b d^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{49 x^7}+\frac{2 b d \left (15 c^2 d+49 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{1225 x^5}+\frac{b \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{11025 x^3}+\frac{2 b c^2 \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{11025 x}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{7 x^7}-\frac{2 d e \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e^2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.332848, size = 160, normalized size = 0.57 \[ \frac{-105 a \left (15 d^2+42 d e x^2+35 e^2 x^4\right )+b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (45 d^2 \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )+294 d e x^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )+1225 e^2 x^4 \left (2 c^2 x^2+1\right )\right )-105 b \text{sech}^{-1}(c x) \left (15 d^2+42 d e x^2+35 e^2 x^4\right )}{11025 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.189, size = 225, normalized size = 0.8 \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 arcsech} \left (cx\right ){d}^{2}}{7\,{c}^{3}{x}^{7}}}-{\frac{2\,{\rm arcsech} \left (cx\right )de}{5\,{c}^{3}{x}^{5}}}-{\frac{{\rm arcsech} \left (cx\right ){e}^{2}}{3\,{c}^{3}{x}^{3}}}+{\frac{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}}{11025\,{c}^{6}{x}^{6}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01764, size = 313, normalized size = 1.11 \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{arsech}\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{arsech}\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{arsech}\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 = 1.87113, size = 483, normalized size = 1.72 \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{asech}{\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{arsech}\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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