Optimal. Leaf size=257 \[ \frac{e \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}+\frac{3 e \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac{e \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^3}-\frac{3 e \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d \sqrt{c x-1} \sqrt{c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac{e x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.595345, antiderivative size = 249, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5707, 5656, 5781, 3303, 3298, 3301, 5666} \[ \frac{e \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^3}+\frac{3 e \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{e \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{3 e \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^3}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}-\frac{d \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}-\frac{d \sqrt{c x-1} \sqrt{c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac{e x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5707
Rule 5656
Rule 5781
Rule 3303
Rule 3298
Rule 3301
Rule 5666
Rubi steps
\begin{align*} \int \frac{d+e x^2}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac{d}{\left (a+b \cosh ^{-1}(c x)\right )^2}+\frac{e x^2}{\left (a+b \cosh ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d \int \frac{1}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx+e \int \frac{x^2}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx\\ &=-\frac{d \sqrt{-1+c x} \sqrt{1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac{e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac{(c d) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b}-\frac{e \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 (a+b x)}-\frac{3 \cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{d \sqrt{-1+c x} \sqrt{1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac{e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac{d \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}+\frac{e \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{(3 e) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac{d \sqrt{-1+c x} \sqrt{1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac{e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (d \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}+\frac{\left (e \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{\left (3 e \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac{\left (d \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}-\frac{\left (e \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac{\left (3 e \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac{d \sqrt{-1+c x} \sqrt{1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac{e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac{d \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}+\frac{e \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^3}+\frac{3 e \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{d \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}-\frac{e \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{3 e \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^3}\\ \end{align*}
Mathematica [A] time = 1.52872, size = 225, normalized size = 0.88 \[ \frac{4 c^2 d \left (\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-\frac{4 b c^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (d+e x^2\right )}{a+b \cosh ^{-1}(c x)}+3 e \left (3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+\cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-\sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )\right )-8 e \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+8 e \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.144, size = 465, normalized size = 1.8 \begin{align*}{\frac{1}{c} \left ({\frac{e}{8\,b{c}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( -4\,\sqrt{cx+1}\sqrt{cx-1}{x}^{2}{c}^{2}+\sqrt{cx-1}\sqrt{cx+1}+4\,{c}^{3}{x}^{3}-3\,cx \right ) }-{\frac{3\,e}{8\,{c}^{2}{b}^{2}}{{\rm e}^{3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,3\,{\rm arccosh} \left (cx\right )+3\,{\frac{a}{b}} \right ) }-{\frac{e}{8\,b{c}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( 4\,{c}^{3}{x}^{3}-3\,cx+4\,\sqrt{cx+1}\sqrt{cx-1}{x}^{2}{c}^{2}-\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{3\,e}{8\,{c}^{2}{b}^{2}}{{\rm e}^{-3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ) }+{\frac{d}{2\,b \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( -\sqrt{cx-1}\sqrt{cx+1}+cx \right ) }+{\frac{e}{8\,b{c}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( -\sqrt{cx-1}\sqrt{cx+1}+cx \right ) }-{\frac{d}{2\,{b}^{2}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }-{\frac{e}{8\,{c}^{2}{b}^{2}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }-{\frac{d}{2\,b \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{e}{8\,b{c}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{d}{2\,{b}^{2}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) }-{\frac{e}{8\,{c}^{2}{b}^{2}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\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*} -\frac{c^{3} e x^{5} +{\left (c^{3} d - c e\right )} x^{3} - c d x +{\left (c^{2} e x^{4} +{\left (c^{2} d - e\right )} x^{2} - d\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{a b c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x - a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} + \int \frac{3 \, c^{5} e x^{6} +{\left (c^{5} d - 6 \, c^{3} e\right )} x^{4} +{\left (3 \, c^{3} e x^{4} +{\left (c^{3} d - c e\right )} x^{2} + c d\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} -{\left (2 \, c^{3} d - 3 \, c e\right )} x^{2} +{\left (6 \, c^{4} e x^{5} +{\left (2 \, c^{4} d - 7 \, c^{2} e\right )} x^{3} -{\left (c^{2} d - 2 \, e\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} + c d}{a b c^{5} x^{4} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{2} - 2 \, a b c^{3} x^{2} + a b c + 2 \,{\left (a b c^{4} x^{3} - a b c^{2} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{4} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{2} - 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \,{\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}\,{d x} \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{e x^{2} + d}{b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\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*} \int \frac{d + e x^{2}}{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}\, dx \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{e x^{2} + d}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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