Optimal. Leaf size=158 \[ -\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}+\frac{5 b c^4 \sqrt{1-c x}}{96 x^2 \sqrt{\frac{1}{c x+1}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{c x+1}}}+\frac{5}{96} b c^6 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right )+\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{c x+1}}} \]
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Rubi [A] time = 0.0728699, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6283, 103, 12, 92, 208} \[ -\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}+\frac{5 b c^4 \sqrt{1-c x}}{96 x^2 \sqrt{\frac{1}{c x+1}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{c x+1}}}+\frac{5}{96} b c^6 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right )+\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{c x+1}}} \]
Antiderivative was successfully verified.
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Rule 6283
Rule 103
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{x^7} \, dx &=-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}-\frac{1}{6} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^7 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}+\frac{1}{36} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{5 c^2}{x^5 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}-\frac{1}{36} \left (5 b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^5 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}+\frac{1}{144} \left (5 b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{3 c^2}{x^3 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}-\frac{1}{48} \left (5 b c^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^3 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^4 \sqrt{1-c x}}{96 x^2 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}-\frac{1}{96} \left (5 b c^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{c^2}{x \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^4 \sqrt{1-c x}}{96 x^2 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}-\frac{1}{96} \left (5 b c^6 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^4 \sqrt{1-c x}}{96 x^2 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}+\frac{1}{96} \left (5 b c^7 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{c-c x^2} \, dx,x,\sqrt{1-c x} \sqrt{1+c x}\right )\\ &=\frac{b \sqrt{1-c x}}{36 x^6 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^2 \sqrt{1-c x}}{144 x^4 \sqrt{\frac{1}{1+c x}}}+\frac{5 b c^4 \sqrt{1-c x}}{96 x^2 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{6 x^6}+\frac{5}{96} b c^6 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{1+c x}\right )\\ \end{align*}
Mathematica [A] time = 0.142301, size = 157, normalized size = 0.99 \[ -\frac{a}{6 x^6}+b \left (\frac{5 c^4}{96 x^2}+\frac{5 c^3}{144 x^3}+\frac{5 c^2}{144 x^4}+\frac{5 c^5}{96 x}+\frac{c}{36 x^5}+\frac{1}{36 x^6}\right ) \sqrt{\frac{1-c x}{c x+1}}-\frac{5}{96} b c^6 \log (x)+\frac{5}{96} b c^6 \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )-\frac{b \text{sech}^{-1}(c x)}{6 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.189, size = 155, normalized size = 1. \begin{align*}{c}^{6} \left ( -{\frac{a}{6\,{c}^{6}{x}^{6}}}+b \left ( -{\frac{{\rm arcsech} \left (cx\right )}{6\,{c}^{6}{x}^{6}}}+{\frac{1}{288\,{c}^{5}{x}^{5}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( 15\,{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ){c}^{6}{x}^{6}+15\,{c}^{4}{x}^{4}\sqrt{-{c}^{2}{x}^{2}+1}+10\,\sqrt{-{c}^{2}{x}^{2}+1}{c}^{2}{x}^{2}+8\,\sqrt{-{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01455, size = 250, normalized size = 1.58 \begin{align*} \frac{1}{576} \, b{\left (\frac{15 \, c^{7} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} - 1} + 1\right ) - 15 \, c^{7} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} - 1} - 1\right ) - \frac{2 \,{\left (15 \, c^{12} x^{5}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{5}{2}} - 40 \, c^{10} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} + 33 \, c^{8} x \sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )}}{c^{6} x^{6}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{3} - 3 \, c^{4} x^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} - 1}}{c} - \frac{96 \, \operatorname{arsech}\left (c x\right )}{x^{6}}\right )} - \frac{a}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83268, size = 227, normalized size = 1.44 \begin{align*} \frac{3 \,{\left (5 \, b c^{6} x^{6} - 16 \, b\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (15 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 8 \, b c x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 48 \, a}{288 \, x^{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 \frac{a + b \operatorname{asech}{\left (c x \right )}}{x^{7}}\, 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{b \operatorname{arsech}\left (c x\right ) + a}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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