Optimal. Leaf size=86 \[ -\frac{c \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b^2}+\frac{c \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b^2}+\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{b x \left (a+b \text{sech}^{-1}(c x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.144917, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {6285, 3297, 3303, 3298, 3301} \[ -\frac{c \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b^2}+\frac{c \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b^2}+\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{b x \left (a+b \text{sech}^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6285
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b \text{sech}^{-1}(c x)\right )^2} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sinh (x)}{(a+b x)^2} \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=\frac{\sqrt{\frac{1-c x}{1+c x}} (1+c x)}{b x \left (a+b \text{sech}^{-1}(c x)\right )}-\frac{c \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )}{b}\\ &=\frac{\sqrt{\frac{1-c x}{1+c x}} (1+c x)}{b x \left (a+b \text{sech}^{-1}(c x)\right )}-\frac{\left (c \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )}{b}+\frac{\left (c \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )}{b}\\ &=\frac{\sqrt{\frac{1-c x}{1+c x}} (1+c x)}{b x \left (a+b \text{sech}^{-1}(c x)\right )}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b^2}+\frac{c \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.333992, size = 82, normalized size = 0.95 \[ \frac{-c \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )+c \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )+\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x \left (a+b \text{sech}^{-1}(c x)\right )}}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.258, size = 164, normalized size = 1.9 \begin{align*} c \left ({\frac{1}{2\,xbc \left ( a+b{\rm arcsech} \left (cx\right ) \right ) } \left ( \sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}cx-1 \right ) }+{\frac{1}{2\,{b}^{2}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\frac{a}{b}}+{\rm arcsech} \left (cx\right ) \right ) }+{\frac{1}{2\,xbc \left ( a+b{\rm arcsech} \left (cx\right ) \right ) } \left ( \sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}cx+1 \right ) }+{\frac{1}{2\,{b}^{2}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arcsech} \left (cx\right )-{\frac{a}{b}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2} \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^{2} -{\left (b^{2} x^{2} \log \left (x\right ) +{\left (b^{2} \log \left (c\right ) - a b\right )} x^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} +{\left (\sqrt{c x + 1} \sqrt{-c x + 1} b^{2} x^{2} -{\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{2}\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right )} + \int -\frac{c^{4} x^{4} - 2 \, c^{2} x^{2} -{\left (c^{2} x^{2} + 1\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} -{\left (c^{2} x^{2} - 2\right )} \sqrt{c x + 1} \sqrt{-c x + 1} + 1}{{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x^{2} \log \left (x\right ) -{\left (b^{2} x^{2} \log \left (x\right ) +{\left (b^{2} \log \left (c\right ) - a b\right )} x^{2}\right )}{\left (c x + 1\right )}{\left (c x - 1\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^{2} - 2 \,{\left ({\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{2} \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^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} +{\left ({\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} x^{2} + 2 \,{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} x^{2} -{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x^{2}\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x^{2} \operatorname{arsech}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arsech}\left (c x\right ) + a^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (a + b \operatorname{asech}{\left (c x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]