Optimal. Leaf size=496 \[ -\frac{5 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{16 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 (1-c x) (c x+1)}{3 d x \sqrt{d-c^2 d x^2}} \]
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Rubi [A] time = 1.46136, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {5798, 5748, 5688, 5715, 3716, 2190, 2279, 2391, 5721, 5461, 4182, 5746, 95} \[ -\frac{5 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{16 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 (1-c x) (c x+1)}{3 d x \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5748
Rule 5688
Rule 5715
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rule 5721
Rule 5461
Rule 4182
Rule 5746
Rule 95
Rubi steps
\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )} \, dx}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (4 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (8 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (16 b c^5 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (16 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (16 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (8 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (16 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{5 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{5 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b^2 c^2 (1-c x) (1+c x)}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt{d-c^2 d x^2}}-\frac{4 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{16 b c^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{5 b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 2.51462, size = 529, normalized size = 1.07 \[ \frac{b^2 \left (5 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+3 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )-c^4 x^4+c^2 x^2+3 c^4 x^4 \cosh ^{-1}(c x)^2-8 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2+5 c^2 x^2 (c x-1) (c x+1) \cosh ^{-1}(c x)^2-6 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-10 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )+c x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)+(c x-1) (c x+1) \cosh ^{-1}(c x)^2\right )+a^2 \left (8 c^4 x^4-4 c^2 x^2-1\right )+a b \left (6 c^4 x^4 \cosh ^{-1}(c x)+2 c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (5 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left (5 \log (c x)+3 \log \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right )\right )\right )+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (c x+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)\right )\right )}{3 d x^3 \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.385, size = 2868, normalized size = 5.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, 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{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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