Optimal. Leaf size=98 \[ \frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^2 d}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^2 d}-\frac{\sqrt{c+d x-1} \sqrt{c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.252003, antiderivative size = 94, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5864, 5656, 5781, 3303, 3298, 3301} \[ \frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac{\sqrt{c+d x-1} \sqrt{c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5864
Rule 5656
Rule 5781
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \cosh ^{-1}(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{\sqrt{-1+c+d x} \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )} \, dx,x,c+d x\right )}{b d}\\ &=-\frac{\sqrt{-1+c+d x} \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{\sqrt{-1+c+d x} \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\cosh \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}-\frac{\sinh \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{\sqrt{-1+c+d x} \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ \end{align*}
Mathematica [A] time = 0.990587, size = 143, normalized size = 1.46 \[ \frac{\sqrt{\frac{c+d x-1}{c+d x+1}} \coth \left (\frac{1}{2} \cosh ^{-1}(c+d x)\right ) \left (\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-\log \left (a+b \cosh ^{-1}(c+d x)\right )\right )-\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1}}{a+b \cosh ^{-1}(c+d x)}+\log \left (\frac{b \cosh ^{-1}(c+d x)}{a}+1\right )}{b^2 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 139, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({\frac{1}{2\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( -\sqrt{dx+c-1}\sqrt{dx+c+1}+dx+c \right ) }-{\frac{1}{2\,{b}^{2}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (dx+c\right )+{\frac{a}{b}} \right ) }-{\frac{1}{2\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{1}{2\,{b}^{2}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (dx+c\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*} \text{result too large to display} \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} \operatorname{arcosh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcosh}\left (d x + c\right ) + a^{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}{\left (a + b \operatorname{acosh}{\left (c + d 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{arcosh}\left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]