Optimal. Leaf size=183 \[ -\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\frac{2 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (12 c^2 d+25 e\right )}{225 x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (12 c^2 d+25 e\right )}{225 x^3}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{25 x^5} \]
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Rubi [A] time = 0.0994742, antiderivative size = 183, 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, 6301, 12, 453, 271, 264} \[ -\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\frac{2 b c^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (12 c^2 d+25 e\right )}{225 x}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (12 c^2 d+25 e\right )}{225 x^3}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{25 x^5} \]
Antiderivative was successfully verified.
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Rule 14
Rule 6301
Rule 12
Rule 453
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{x^6} \, dx &=-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e \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{-3 d-5 e x^2}{15 x^6 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\frac{1}{15} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-3 d-5 e x^2}{x^6 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{25 x^5}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\frac{1}{75} \left (b \left (-12 c^2 d-25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^4 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{25 x^5}+\frac{b \left (12 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{225 x^3}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}+\frac{1}{225} \left (2 b c^2 \left (-12 c^2 d-25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{b d \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{25 x^5}+\frac{b \left (12 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{225 x^3}+\frac{2 b c^2 \left (12 c^2 d+25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{225 x}-\frac{d \left (a+b \text{sech}^{-1}(c x)\right )}{5 x^5}-\frac{e \left (a+b \text{sech}^{-1}(c x)\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.164759, size = 101, normalized size = 0.55 \[ \frac{-15 a \left (3 d+5 e x^2\right )+b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (3 d \left (8 c^4 x^4+4 c^2 x^2+3\right )+25 e x^2 \left (2 c^2 x^2+1\right )\right )-15 b \text{sech}^{-1}(c x) \left (3 d+5 e x^2\right )}{225 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 142, normalized size = 0.8 \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 arcsech} \left (cx\right )d}{5\,{c}^{3}{x}^{5}}}-{\frac{{\rm arcsech} \left (cx\right )e}{3\,{c}^{3}{x}^{3}}}+{\frac{24\,{c}^{6}d{x}^{4}+50\,{c}^{4}e{x}^{4}+12\,{c}^{4}d{x}^{2}+25\,{c}^{2}{x}^{2}e+9\,{c}^{2}d}{225\,{c}^{4}{x}^{4}}\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 = 1.00069, size = 178, normalized size = 0.97 \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{arsech}\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{arsech}\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 = 1.89353, size = 297, normalized size = 1.62 \begin{align*} -\frac{75 \, a e x^{2} + 45 \, a d + 15 \,{\left (5 \, b e x^{2} + 3 \, b d\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (2 \,{\left (12 \, b c^{5} d + 25 \, b c^{3} e\right )} x^{5} + 9 \, b c d x +{\left (12 \, b c^{3} d + 25 \, b c e\right )} x^{3}\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{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{asech}{\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{arsech}\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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