Optimal. Leaf size=163 \[ -\frac{e \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}+\frac{e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}-\frac{e (c+d x)^2}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.50852, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {5866, 12, 5668, 5775, 5670, 5448, 3303, 3298, 3301, 5676} \[ -\frac{e \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{b^3 d}+\frac{e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{b^3 d}-\frac{e (c+d x)^2}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5866
Rule 12
Rule 5668
Rule 5775
Rule 5670
Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rule 5676
Rubi steps
\begin{align*} \int \frac{c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e x}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{x}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{e \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}+\frac{e \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e (c+d x)^2}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{x}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e (c+d x)^2}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e (c+d x)^2}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e (c+d x)^2}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e (c+d x)^2}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\left (e \cosh \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,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac{\left (e \sinh \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,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac{e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e (c+d x)^2}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac{2 a}{b}\right )}{b^3 d}+\frac{e \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{b^3 d}\\ \end{align*}
Mathematica [A] time = 0.325081, size = 127, normalized size = 0.78 \[ \frac{e \left (-\frac{b^2 \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1}}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}-2 \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )+2 \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\frac{b \left (1-2 (c+d x)^2\right )}{a+b \cosh ^{-1}(c+d x)}\right )}{2 b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 254, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ( -{\frac{e \left ( 2\,b{\rm arccosh} \left (dx+c\right )+2\,a-b \right ) }{8\,{b}^{2} \left ({b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}+2\,ab{\rm arccosh} \left (dx+c\right )+{a}^{2} \right ) } \left ( -2\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) +2\, \left ( dx+c \right ) ^{2}-1 \right ) }+{\frac{e}{2\,{b}^{3}}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\rm arccosh} \left (dx+c\right )+2\,{\frac{a}{b}} \right ) }-{\frac{e}{8\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{2}} \left ( 2\, \left ( dx+c \right ) ^{2}-1+2\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) \right ) }-{\frac{e}{4\,{b}^{2} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( 2\, \left ( dx+c \right ) ^{2}-1+2\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) \right ) }-{\frac{e}{2\,{b}^{3}}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\rm arccosh} \left (dx+c\right )-2\,{\frac{a}{b}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{d e x + c e}{b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (d x + c\right ) + a^{3}}, 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*} e \left (\int \frac{c}{a^{3} + 3 a^{2} b \operatorname{acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac{d x}{a^{3} + 3 a^{2} b \operatorname{acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \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{d e x + c e}{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]