Optimal. Leaf size=85 \[ -\frac{c^2 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right )}{b^2}+\frac{c^2 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right )}{b^2}+\frac{c^2 \sinh \left (2 \text{sech}^{-1}(c x)\right )}{2 b \left (a+b \text{sech}^{-1}(c x)\right )} \]
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Rubi [A] time = 0.164571, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6285, 5448, 12, 3297, 3303, 3298, 3301} \[ -\frac{c^2 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right )}{b^2}+\frac{c^2 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right )}{b^2}+\frac{c^2 \sinh \left (2 \text{sech}^{-1}(c x)\right )}{2 b \left (a+b \text{sech}^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 6285
Rule 5448
Rule 12
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx &=-\left (c^2 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{(a+b x)^2} \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\left (c^2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 (a+b x)^2} \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\left (\frac{1}{2} c^2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{(a+b x)^2} \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=\frac{c^2 \sinh \left (2 \text{sech}^{-1}(c x)\right )}{2 b \left (a+b \text{sech}^{-1}(c x)\right )}-\frac{c^2 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )}{b}\\ &=\frac{c^2 \sinh \left (2 \text{sech}^{-1}(c x)\right )}{2 b \left (a+b \text{sech}^{-1}(c x)\right )}-\frac{\left (c^2 \cosh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )}{b}+\frac{\left (c^2 \sinh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )}{b}\\ &=-\frac{c^2 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right )}{b^2}+\frac{c^2 \sinh \left (2 \text{sech}^{-1}(c x)\right )}{2 b \left (a+b \text{sech}^{-1}(c x)\right )}+\frac{c^2 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \text{sech}^{-1}(c x)\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.335204, size = 92, normalized size = 1.08 \[ \frac{c^2 \left (-\cosh \left (\frac{2 a}{b}\right )\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )\right )+c^2 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )\right )+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x^2 \left (a+b \text{sech}^{-1}(c x)\right )}}{b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.237, size = 186, normalized size = 2.2 \begin{align*}{c}^{2} \left ({\frac{1}{4\,{c}^{2}{x}^{2} \left ( a+b{\rm arcsech} \left (cx\right ) \right ) b} \left ( 2\,\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}cx+{c}^{2}{x}^{2}-2 \right ) }+{\frac{1}{2\,{b}^{2}}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\frac{a}{b}}+2\,{\rm arcsech} \left (cx\right ) \right ) }-{\frac{1}{4\,{c}^{2}{x}^{2} \left ( a+b{\rm arcsech} \left (cx\right ) \right ) b} \left ({c}^{2}{x}^{2}-2-2\,\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}cx \right ) }+{\frac{1}{2\,{b}^{2}}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\rm arcsech} \left (cx\right )-2\,{\frac{a}{b}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c^{2} x^{3} +{\left (c^{2} x^{3} - x\right )} \sqrt{c x + 1} \sqrt{-c x + 1} - x}{{\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{3} \log \left (x\right ) +{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} - b^{2} \log \left (c\right ) + a b\right )} x^{3} -{\left (b^{2} x^{3} \log \left (x\right ) +{\left (b^{2} \log \left (c\right ) - a b\right )} x^{3}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} +{\left (\sqrt{c x + 1} \sqrt{-c x + 1} b^{2} x^{3} -{\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{3}\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right )} + \int -\frac{2 \, c^{4} x^{4} - 4 \, c^{2} x^{2} - 2 \,{\left (c x + 1\right )}{\left (c x - 1\right )} +{\left (c^{4} x^{4} - 4 \, c^{2} x^{2} + 4\right )} \sqrt{c x + 1} \sqrt{-c x + 1} + 2}{{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x^{3} \log \left (x\right ) +{\left ({\left (b^{2} c^{4} \log \left (c\right ) - a b c^{4}\right )} x^{4} - 2 \,{\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} + b^{2} \log \left (c\right ) - a b\right )} x^{3} -{\left (b^{2} x^{3} \log \left (x\right ) +{\left (b^{2} \log \left (c\right ) - a b\right )} x^{3}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} - 2 \,{\left ({\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{3} \log \left (x\right ) +{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} - b^{2} \log \left (c\right ) + a b\right )} x^{3}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} +{\left ({\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} x^{3} + 2 \,{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} x^{3} -{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x^{3}\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x^{3} \operatorname{arsech}\left (c x\right )^{2} + 2 \, a b x^{3} \operatorname{arsech}\left (c x\right ) + a^{2} x^{3}}, x\right ) \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{1}{x^{3} \left (a + b \operatorname{asech}{\left (c x \right )}\right )^{2}}\, 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{1}{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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