Optimal. Leaf size=453 \[ -\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b^2 c d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
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Rubi [A]
time = 0.42, antiderivative size = 453, normalized size of antiderivative = 1.00, number
of steps used = 15, number of rules used = 14, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules
used = {5928, 5895, 5893, 5883, 92, 54, 5912, 5919, 5882, 3799, 2221, 2317, 2438, 38}
\begin {gather*} -\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt {c x-1} \sqrt {c x+1}}+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac {2 b c d \sqrt {d-c^2 d x^2} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 c d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 54
Rule 92
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5883
Rule 5893
Rule 5895
Rule 5912
Rule 5919
Rule 5928
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^2} \, 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^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{2} b^2 c^2 d x \sqrt {d-c^2 d x^2}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {b^2 c d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \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 ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c d \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 2.67, size = 433, normalized size = 0.96 \begin {gather*} \frac {-12 a^2 d \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (2+c^2 x^2\right ) \sqrt {d-c^2 d x^2}+36 a^2 c d^{3/2} x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-24 a b d \sqrt {d-c^2 d x^2} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-c x \left (\cosh ^{-1}(c x)^2+2 \log (c x)\right )\right )-8 b^2 d \sqrt {d-c^2 d x^2} \left (\cosh ^{-1}(c x) \left (3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-c x \left (\cosh ^{-1}(c x) \left (3+\cosh ^{-1}(c x)\right )+6 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )\right )+3 c x \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right )+6 a b c d x \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+b^2 c d x \sqrt {d-c^2 d x^2} \left (4 \cosh ^{-1}(c x)^3+6 \cosh ^{-1}(c x) \cosh \left (2 \cosh ^{-1}(c x)\right )-3 \left (1+2 \cosh ^{-1}(c x)^2\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{24 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(941\) vs.
\(2(421)=842\).
time = 2.72, size = 942, normalized size = 2.08
method | result | size |
default | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {3 a^{2} c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{2}-\frac {3 a^{2} c^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d \,x^{3}}{4 \left (c x +1\right ) \left (c x -1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} d x}{4 \left (c x +1\right ) \left (c x -1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d \mathrm {arccosh}\left (c x \right )^{2} x^{3}}{2 \left (c x +1\right ) \left (c x -1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{3} c d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} d \mathrm {arccosh}\left (c x \right )^{2} x}{2 \left (c x +1\right ) \left (c x -1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d}{\left (c x +1\right ) \left (c x -1\right ) x}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c d \mathrm {arccosh}\left (c x \right )^{2}}{\sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c d \,\mathrm {arccosh}\left (c x \right )}{4 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c d}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c d}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} d \,\mathrm {arccosh}\left (c x \right ) x^{2}}{2 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} c d}{2 \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d \,\mathrm {arccosh}\left (c x \right ) x^{3}}{\left (c x +1\right ) \left (c x -1\right )}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} d \,x^{2}}{2 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c d \,\mathrm {arccosh}\left (c x \right )}{\sqrt {c x +1}\, \sqrt {c x -1}}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} d \,\mathrm {arccosh}\left (c x \right ) x}{\left (c x +1\right ) \left (c x -1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c d}{4 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) d}{\left (c x +1\right ) \left (c x -1\right ) x}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) c d}{\sqrt {c x -1}\, \sqrt {c x +1}}\) | \(942\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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