Optimal. Leaf size=126 \[ -\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}+\frac{3 b c^2 \sqrt{1-c x}}{32 x^2 \sqrt{\frac{1}{c x+1}}}+\frac{3}{32} b c^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right )+\frac{b \sqrt{1-c x}}{16 x^4 \sqrt{\frac{1}{c x+1}}} \]
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Rubi [A] time = 0.054912, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}+\frac{3 b c^2 \sqrt{1-c x}}{32 x^2 \sqrt{\frac{1}{c x+1}}}+\frac{3}{32} b c^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right )+\frac{b \sqrt{1-c x}}{16 x^4 \sqrt{\frac{1}{c x+1}}} \]
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^5} \, dx &=-\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}-\frac{1}{4} \left (b \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}}{16 x^4 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}+\frac{1}{16} \left (b \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}}{16 x^4 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}-\frac{1}{16} \left (3 b c^2 \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}}{16 x^4 \sqrt{\frac{1}{1+c x}}}+\frac{3 b c^2 \sqrt{1-c x}}{32 x^2 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}-\frac{1}{32} \left (3 b c^2 \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}}{16 x^4 \sqrt{\frac{1}{1+c x}}}+\frac{3 b c^2 \sqrt{1-c x}}{32 x^2 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}-\frac{1}{32} \left (3 b c^4 \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}}{16 x^4 \sqrt{\frac{1}{1+c x}}}+\frac{3 b c^2 \sqrt{1-c x}}{32 x^2 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}+\frac{1}{32} \left (3 b c^5 \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}}{16 x^4 \sqrt{\frac{1}{1+c x}}}+\frac{3 b c^2 \sqrt{1-c x}}{32 x^2 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{4 x^4}+\frac{3}{32} b c^4 \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 = 0.0949213, size = 137, normalized size = 1.09 \[ -\frac{a}{4 x^4}+b \left (\frac{3 c^2}{32 x^2}+\frac{3 c^3}{32 x}+\frac{c}{16 x^3}+\frac{1}{16 x^4}\right ) \sqrt{\frac{1-c x}{c x+1}}-\frac{3}{32} b c^4 \log (x)+\frac{3}{32} b c^4 \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )-\frac{b \text{sech}^{-1}(c x)}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.182, size = 135, normalized size = 1.1 \begin{align*}{c}^{4} \left ( -{\frac{a}{4\,{c}^{4}{x}^{4}}}+b \left ( -{\frac{{\rm arcsech} \left (cx\right )}{4\,{c}^{4}{x}^{4}}}+{\frac{1}{32\,{c}^{3}{x}^{3}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ){c}^{4}{x}^{4}+3\,\sqrt{-{c}^{2}{x}^{2}+1}{c}^{2}{x}^{2}+2\,\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 = 0.988312, size = 198, normalized size = 1.57 \begin{align*} \frac{1}{64} \, b{\left (\frac{3 \, c^{5} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} - 1} + 1\right ) - 3 \, c^{5} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} - 1} - 1\right ) - \frac{2 \,{\left (3 \, c^{8} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} - 5 \, c^{6} x \sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )}}{c^{4} x^{4}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{2} - 2 \, c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + 1}}{c} - \frac{16 \, \operatorname{arsech}\left (c x\right )}{x^{4}}\right )} - \frac{a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90141, size = 198, normalized size = 1.57 \begin{align*} \frac{{\left (3 \, b c^{4} x^{4} - 8 \, b\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (3 \, b c^{3} x^{3} + 2 \, b c x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 8 \, a}{32 \, x^{4}} \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^{5}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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