Optimal. Leaf size=46 \[ \frac{c \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.110401, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6285, 3303, 3298, 3301} \[ \frac{c \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6285
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b \text{sech}^{-1}(c x)\right )} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\text{sech}^{-1}(c x)\right )\right )\\ &=-\left (\left (c \cosh \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 )\right )+\left (c \sinh \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 )\\ &=\frac{c \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{b}-\frac{c \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0747712, size = 43, normalized size = 0.93 \[ \frac{c \left (\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )-\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\text{sech}^{-1}(c x)\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.211, size = 54, normalized size = 1.2 \begin{align*} c \left ( -{\frac{1}{2\,b}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\frac{a}{b}}+{\rm arcsech} \left (cx\right ) \right ) }+{\frac{1}{2\,b}{{\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*} \int \frac{1}{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{2}}\,{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 x^{2} \operatorname{arsech}\left (c x\right ) + a 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 )}\, 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 )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]