Optimal. Leaf size=358 \[ \frac{\sqrt{\pi } e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{\sqrt{3 \pi } e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{\sqrt{\pi } e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{\sqrt{3 \pi } e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{\sqrt{\pi } d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{2 d \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]
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Rubi [A] time = 0.831471, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5707, 5656, 5781, 3307, 2180, 2204, 2205, 5666} \[ \frac{\sqrt{\pi } e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{\sqrt{3 \pi } e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{\sqrt{\pi } e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{\sqrt{3 \pi } e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{\sqrt{\pi } d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}-\frac{2 d \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 5707
Rule 5656
Rule 5781
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5666
Rubi steps
\begin{align*} \int \frac{d+e x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=\int \left (\frac{d}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac{e x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}}\right ) \, dx\\ &=d \int \frac{1}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx+e \int \frac{x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{(2 c d) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x} \sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b}-\frac{(2 e) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{a+b x}}-\frac{3 \cosh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{(2 d) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}+\frac{e \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3}+\frac{(3 e) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{d \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}+\frac{d \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}+\frac{e \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{e \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{(3 e) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac{(3 e) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{(2 d) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^2 c}+\frac{(2 d) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{b^2 c}+\frac{e \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{2 b^2 c^3}+\frac{e \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{2 b^2 c^3}+\frac{(3 e) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{2 b^2 c^3}+\frac{(3 e) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{2 b^2 c^3}\\ &=-\frac{2 d \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}+\frac{e e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{e e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{b^{3/2} c}+\frac{e e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}+\frac{e e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^3}\\ \end{align*}
Mathematica [A] time = 1.91394, size = 268, normalized size = 0.75 \[ \frac{e^{-\frac{3 a}{b}} \left (e^{\frac{4 a}{b}} \left (-\left (4 c^2 d+e\right )\right ) \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )+e^{\frac{2 a}{b}} \left (4 c^2 d+e\right ) \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )-e^{\frac{3 a}{b}} \left (\sqrt{3} e e^{\frac{3 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (4 c^2 d+e\right )+2 e \sinh \left (3 \cosh ^{-1}(c x)\right )\right )+\sqrt{3} e \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{4 b c^3 \sqrt{a+b \cosh ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.118, size = 0, normalized size = 0. \begin{align*} \int{(e{x}^{2}+d) \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \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*} \int \frac{e x^{2} + d}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 )^{\frac{3}{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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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