Optimal. Leaf size=573 \[ \frac{2 i b d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 i b^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{3} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2}{27} b^2 c^2 d x^2 \sqrt{d-c^2 d x^2}+\frac{68}{27} b^2 d \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 1.2549, antiderivative size = 585, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {5798, 5745, 5743, 5761, 4180, 2531, 2282, 6589, 5654, 74, 5680, 12, 460} \[ \frac{2 i b d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 i b^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{c x-1} \sqrt{c x+1}}+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{3} d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2}{27} b^2 c^2 d x^2 \sqrt{d-c^2 d x^2}+\frac{68}{27} b^2 d \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5745
Rule 5743
Rule 5761
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5654
Rule 74
Rule 5680
Rule 12
Rule 460
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{3} d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{3} d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-3+c^2 x^2\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{3} d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c d \sqrt{d-c^2 d x^2}\right ) \int \cosh ^{-1}(c x) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (-3+c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{2}{27} b^2 c^2 d x^2 \sqrt{d-c^2 d x^2}-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b^2 c d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{3} d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 i b d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 i b d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (14 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{68}{27} b^2 d \sqrt{d-c^2 d x^2}-\frac{2}{27} b^2 c^2 d x^2 \sqrt{d-c^2 d x^2}-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b^2 c d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{3} d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 i b d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 i b d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 i b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 i b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{68}{27} b^2 d \sqrt{d-c^2 d x^2}-\frac{2}{27} b^2 c^2 d x^2 \sqrt{d-c^2 d x^2}-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b^2 c d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{3} d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 i b d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 i b d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 i b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 i b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{68}{27} b^2 d \sqrt{d-c^2 d x^2}-\frac{2}{27} b^2 c^2 d x^2 \sqrt{d-c^2 d x^2}-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b^2 c d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt{-1+c x} \sqrt{1+c x}}+d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{1}{3} d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 i b d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 i b d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 i b^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 i b^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 2.70997, size = 650, normalized size = 1.13 \[ \frac{2 a b d \sqrt{d-c^2 d x^2} \left (i \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-i \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-i \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+b^2 d \sqrt{d-c^2 d x^2} \left (\frac{i \left (2 \cosh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-2 \cosh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right )-2 \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(c x)}\right )+\cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)^2+\frac{2 c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{1-c x}+2\right )-a^2 d^{3/2} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )-\frac{1}{3} a^2 d \left (c^2 x^2-4\right ) \sqrt{d-c^2 d x^2}+a^2 d^{3/2} \log (c x)-\frac{a b d \sqrt{d-c^2 d x^2} \left (9 c x+12 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )}{18 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{1}{54} b^2 d \sqrt{d-c^2 d x^2} \left (9 \left (\cosh \left (2 \cosh ^{-1}(c x)\right )-1\right ) \cosh ^{-1}(c x)^2+\frac{3 \sqrt{\frac{c x-1}{c x+1}} \left (9 c x-\cosh \left (3 \cosh ^{-1}(c x)\right )\right ) \cosh ^{-1}(c x)}{c x-1}+2 \left (\cosh \left (2 \cosh ^{-1}(c x)\right )-13\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.333, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{x} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \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{{\left (a^{2} c^{2} d x^{2} - a^{2} d +{\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \operatorname{arcosh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} - a b d\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{x}, 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{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{x}\, 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{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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