Optimal. Leaf size=195 \[ \frac{e^3 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^2 d}+\frac{e^3 \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^2 d}-\frac{e^3 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^2 d}-\frac{e^3 \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^2 d}-\frac{e^3 \sqrt{c+d x-1} (c+d x)^3 \sqrt{c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]
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Rubi [A] time = 0.2984, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5866, 12, 5666, 3303, 3298, 3301} \[ \frac{e^3 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac{e^3 \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac{e^3 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac{e^3 \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac{e^3 \sqrt{c+d x-1} (c+d x)^3 \sqrt{c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5666
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^3}{\left (a+b \cosh ^{-1}(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^3 x^3}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \frac{x^3}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e^3 \operatorname{Subst}\left (\int \left (-\frac{\cosh (2 x)}{2 (a+b x)}-\frac{\cosh (4 x)}{2 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^3 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b d}+\frac{e^3 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b d}\\ &=-\frac{e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\left (e^3 \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,\cosh ^{-1}(c+d x)\right )}{2 b d}+\frac{\left (e^3 \cosh \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b d}-\frac{\left (e^3 \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,\cosh ^{-1}(c+d x)\right )}{2 b d}-\frac{\left (e^3 \sinh \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b d}\\ &=-\frac{e^3 \sqrt{-1+c+d x} (c+d x)^3 \sqrt{1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^3 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac{e^3 \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac{e^3 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac{e^3 \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c+d x)\right )}{2 b^2 d}\\ \end{align*}
Mathematica [A] time = 2.16606, size = 230, normalized size = 1.18 \[ \frac{e^3 \left (-3 \left (\cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\log \left (a+b \cosh ^{-1}(c+d x)\right )\right )+4 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-4 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\frac{2 b \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) (c+d x)^3}{a+b \cosh ^{-1}(c+d x)}+3 \log \left (a+b \cosh ^{-1}(c+d x)\right )\right )}{2 b^2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.115, size = 418, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ({\frac{{e}^{3}}{ \left ( 16\,a+16\,b{\rm arccosh} \left (dx+c\right ) \right ) b} \left ( -8\, \left ( dx+c \right ) ^{3}\sqrt{dx+c-1}\sqrt{dx+c+1}+4\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) +8\, \left ( dx+c \right ) ^{4}-8\, \left ( dx+c \right ) ^{2}+1 \right ) }-{\frac{{e}^{3}}{4\,{b}^{2}}{{\rm e}^{4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,4\,{\rm arccosh} \left (dx+c\right )+4\,{\frac{a}{b}} \right ) }+{\frac{{e}^{3}}{ \left ( 8\,a+8\,b{\rm arccosh} \left (dx+c\right ) \right ) b} \left ( -2\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) +2\, \left ( dx+c \right ) ^{2}-1 \right ) }-{\frac{{e}^{3}}{4\,{b}^{2}}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\rm arccosh} \left (dx+c\right )+2\,{\frac{a}{b}} \right ) }-{\frac{{e}^{3}}{8\,b \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}^{3}}{4\,{b}^{2}}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\rm arccosh} \left (dx+c\right )-2\,{\frac{a}{b}} \right ) }-{\frac{{e}^{3}}{16\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( 8\, \left ( dx+c \right ) ^{4}-8\, \left ( dx+c \right ) ^{2}+8\, \left ( dx+c \right ) ^{3}\sqrt{dx+c-1}\sqrt{dx+c+1}-4\,\sqrt{dx+c+1}\sqrt{dx+c-1} \left ( dx+c \right ) +1 \right ) }-{\frac{{e}^{3}}{4\,{b}^{2}}{{\rm e}^{-4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-4\,{\rm arccosh} \left (dx+c\right )-4\,{\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*} \text{result too large to display} \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{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}{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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{3} \left (\int \frac{c^{3}}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac{d^{3} x^{3}}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac{3 c d^{2} x^{2}}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac{3 c^{2} d x}{a^{2} + 2 a b \operatorname{acosh}{\left (c + d x \right )} + b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \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{{\left (d e x + c e\right )}^{3}}{{\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.
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