Optimal. Leaf size=476 \[ -\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+3 d^2 e \log (x) \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b e^2 \sqrt {c^2 x^2-1} \left (8 c^2 d+e\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{32 c^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b e^2 x \left (1-c^2 x^2\right ) \left (8 c^2 d+e\right )}{32 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 1.76, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 19, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {266, 43, 5790, 12, 6742, 1610, 1807, 1584, 459, 321, 217, 206, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}+3 d^2 e \log (x) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b e^2 x \left (1-c^2 x^2\right ) \left (8 c^2 d+e\right )}{32 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b e^2 \sqrt {c^2 x^2-1} \left (8 c^2 d+e\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{32 c^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 206
Rule 217
Rule 266
Rule 321
Rule 459
Rule 1584
Rule 1610
Rule 1807
Rule 2190
Rule 2279
Rule 2326
Rule 2328
Rule 2391
Rule 3717
Rule 4625
Rule 5790
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{4 x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \left (\frac {-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {12 d^2 e \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {1}{4} (b c) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx-\left (3 b c d^2 e\right ) \int \frac {\log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {\left (3 b c d^2 e \sqrt {1-c^2 x^2}\right ) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d^2 e \sqrt {1-c^2 x^2}\right ) \int \frac {\sin ^{-1}(c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {6 d e^2 x^3+e^3 x^5}{x \sqrt {-1+c^2 x^2}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d^2 e \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x^2 \left (6 d e^2+e^3 x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (6 i b d^2 e \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}--\frac {\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{16 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b d^2 e \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}--\frac {\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 i b d^2 e \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}--\frac {\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b e^2 \left (8 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{32 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac {3 b d^2 e \sqrt {1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b d^2 e \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 267, normalized size = 0.56 \[ \frac {1}{4} \left (-\frac {2 a d^3}{x^2}+12 a d^2 e \log (x)+6 a d e^2 x^2+a e^3 x^4-\frac {3 b d e^2 \left (c x \sqrt {c x-1} \sqrt {c x+1}+2 \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )\right )}{c^2}-\frac {b e^3 \left (c x \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 x^2+3\right )+6 \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )\right )}{8 c^4}+\frac {2 b d^3 \left (c x \sqrt {c x-1} \sqrt {c x+1}-\cosh ^{-1}(c x)\right )}{x^2}-6 b d^2 e \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+6 b d^2 e \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+6 b d e^2 x^2 \cosh ^{-1}(c x)+b e^3 x^4 \cosh ^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \operatorname {arcosh}\left (c x\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 296, normalized size = 0.62 \[ \frac {a \,x^{4} e^{3}}{4}+\frac {3 a \,x^{2} d \,e^{2}}{2}+3 a \,d^{2} e \ln \left (c x \right )-\frac {d^{3} a}{2 x^{2}}-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right )}{2 x^{2}}-\frac {3 b \,d^{2} e \mathrm {arccosh}\left (c x \right )^{2}}{2}+3 b \,d^{2} e \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) e^{3}}{32 c^{4}}+\frac {c \,d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 x}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x^{3}}{16 c}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{3} x}{32 c^{3}}+\frac {3 b \,d^{2} e \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {b \,\mathrm {arccosh}\left (c x \right ) x^{4} e^{3}}{4}-\frac {d^{3} b \,c^{2}}{2}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) x^{2} d \,e^{2}}{2}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, x d \,e^{2}}{4 c}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) d \,e^{2}}{4 c^{2}} \]
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 \[ \frac {1}{4} \, a e^{3} x^{4} + \frac {3}{2} \, a d e^{2} x^{2} + \frac {1}{2} \, b d^{3} {\left (\frac {\sqrt {c^{2} x^{2} - 1} c}{x} - \frac {\operatorname {arcosh}\left (c x\right )}{x^{2}}\right )} + 3 \, a d^{2} e \log \relax (x) - \frac {a d^{3}}{2 \, x^{2}} + \int b e^{3} x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 3 \, b d e^{2} x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \frac {3 \, b d^{2} e \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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