Optimal. Leaf size=634 \[ \frac{b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{d^3}+\frac{b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{d^3}+\frac{b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{d^3}+\frac{b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{d^3}+\frac{b e \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{d^3}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{2 e \left (a+b \cosh ^{-1}(c x)\right )^2}{b d^3}-\frac{2 e \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^3}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}+\frac{b c e \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{2 d^{5/2} \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d^2 x} \]
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Rubi [A] time = 1.08718, antiderivative size = 616, normalized size of antiderivative = 0.97, number of steps used = 31, number of rules used = 14, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5792, 5662, 95, 5660, 3718, 2190, 2279, 2391, 5788, 519, 377, 208, 5800, 5562} \[ \frac{b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{d^3}+\frac{b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{d^3}+\frac{b e \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{d^3}+\frac{b e \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{d^3}-\frac{b e \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{d^3}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}-\frac{2 e \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^3}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}+\frac{b c e \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{2 d^{5/2} \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d^2 x} \]
Warning: Unable to verify antiderivative.
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Rule 5792
Rule 5662
Rule 95
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 5788
Rule 519
Rule 377
Rule 208
Rule 5800
Rule 5562
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx &=\int \left (\frac{a+b \cosh ^{-1}(c x)}{d^2 x^3}-\frac{2 e \left (a+b \cosh ^{-1}(c x)\right )}{d^3 x}+\frac{e^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )^2}+\frac{2 e^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \cosh ^{-1}(c x)}{x^3} \, dx}{d^2}-\frac{(2 e) \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx}{d^3}+\frac{\left (2 e^2\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^3}+\frac{e^2 \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{d^2}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}+\frac{(b c) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 d^2}-\frac{(2 e) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}+\frac{(b c e) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )} \, dx}{2 d^2}+\frac{\left (2 e^2\right ) \int \left (-\frac{a+b \cosh ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{d^3}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )^2}{b d^3}-\frac{(4 e) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{e^{3/2} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{d^3}+\frac{e^{3/2} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{d^3}+\frac{\left (b c e \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{2 d^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )^2}{b d^3}-\frac{2 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{(2 b e) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}+\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}+\frac{\left (b c e \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{2 d^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}+\frac{b c e \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt{c^2 d+e} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^3}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}+\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}+\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}+\frac{b c e \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt{c^2 d+e} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{d^3}-\frac{2 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^3}-\frac{b e \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^3}-\frac{(b e) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{(b e) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{(b e) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{(b e) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}+\frac{b c e \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt{c^2 d+e} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{d^3}-\frac{2 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^3}-\frac{b e \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^3}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{2 d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (d+e x^2\right )}+\frac{b c e \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e} x}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{2 d^{5/2} \sqrt{c^2 d+e} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{d^3}-\frac{2 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{b e \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{b e \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{b e \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{d^3}+\frac{b e \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{d^3}-\frac{b e \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^3}\\ \end{align*}
Mathematica [F] time = 5.71047, size = 0, normalized size = 0. \[ \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left (d+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 0.24, size = 723, normalized size = 1.1 \begin{align*} \text{result too large to display} \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{1}{2} \, a{\left (\frac{2 \, e x^{2} + d}{d^{2} e x^{4} + d^{3} x^{2}} - \frac{2 \, e \log \left (e x^{2} + d\right )}{d^{3}} + \frac{4 \, e \log \left (x\right )}{d^{3}}\right )} + b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{e^{2} x^{7} + 2 \, d e x^{5} + d^{2} x^{3}}\,{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{b \operatorname{arcosh}\left (c x\right ) + a}{e^{2} x^{7} + 2 \, d e x^{5} + d^{2} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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