Optimal. Leaf size=380 \[ -\frac {b c \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}} \]
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Rubi [A] time = 0.75, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5802, 5832, 3324, 3320, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac {b^2 c^3 d \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3320
Rule 3324
Rule 5802
Rule 5832
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {(b c) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b c^2\right ) \operatorname {Subst}\left (\int \frac {a+b x}{(c d+e \cosh (x))^2} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 d^2-e^2}+\frac {\left (b c^3 d\right ) \operatorname {Subst}\left (\int \frac {a+b x}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{c d+x} \, dx,x,c e x\right )}{e \left (c^2 d^2-e^2\right )}+\frac {\left (2 b c^3 d\right ) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{e+2 c d e^x+e e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {\left (2 b c^3 d\right ) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{2 c d-2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (2 b c^3 d\right ) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{2 c d+2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {\left (b^2 c^3 d\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (b^2 c^3 d\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {\left (b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 7.78, size = 1093, normalized size = 2.88 \[ -\frac {a^2}{2 e (d+e x)^2}+b c^2 \left (-\frac {\cosh ^{-1}(c x)}{e (c d+c e x)^2}+\frac {2 c d \tanh ^{-1}\left (\frac {\sqrt {c d-e} \sqrt {\frac {c x-1}{c x+1}}}{\sqrt {c d+e}}\right )}{(c d-e)^{3/2} e (c d+e)^{3/2}}-\frac {\sqrt {c x-1} \sqrt {c x+1}}{(c d-e) (c d+e) (c d+c e x)}\right ) a+b^2 c^2 \left (-\frac {\cosh ^{-1}(c x)^2}{2 e (c d+c e x)^2}-\frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{(c d-e) (c d+e) (c d+c e x)}+\frac {\log \left (\frac {e x}{d}+1\right )}{c^2 d^2 e-e^3}+\frac {c d \left (2 \cosh ^{-1}(c x) \tan ^{-1}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )-2 i \cos ^{-1}\left (-\frac {c d}{e}\right ) \tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )+\left (\cos ^{-1}\left (-\frac {c d}{e}\right )+2 \left (\tan ^{-1}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )+\tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )\right )\right ) \log \left (\frac {\sqrt {e^2-c^2 d^2} e^{-\frac {1}{2} \cosh ^{-1}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c (d+e x)}}\right )+\left (\cos ^{-1}\left (-\frac {c d}{e}\right )-2 \left (\tan ^{-1}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )+\tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )\right )\right ) \log \left (\frac {\sqrt {e^2-c^2 d^2} e^{\frac {1}{2} \cosh ^{-1}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c (d+e x)}}\right )-\left (\cos ^{-1}\left (-\frac {c d}{e}\right )+2 \tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )\right ) \log \left (\frac {(c d+e) \left (c d-e+i \sqrt {e^2-c^2 d^2}\right ) \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-1\right )}{e \left (c d+e+i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac {c d}{e}\right )-2 \tan ^{-1}\left (\frac {(e-c d) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {e^2-c^2 d^2}}\right )\right ) \log \left (\frac {(c d+e) \left (-c d+e+i \sqrt {e^2-c^2 d^2}\right ) \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )+1\right )}{e \left (c d+e+i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text {Li}_2\left (\frac {\left (c d-i \sqrt {e^2-c^2 d^2}\right ) \left (c d+e-i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text {Li}_2\left (\frac {\left (c d+i \sqrt {e^2-c^2 d^2}\right ) \left (c d+e-i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {e^2-c^2 d^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{e \left (e^2-c^2 d^2\right )^{3/2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 1170, normalized size = 3.08 \[ -\frac {c^{2} a^{2}}{2 \left (c x e +c d \right )^{2} e}-\frac {c^{4} b^{2} \mathrm {arccosh}\left (c x \right )^{2} d^{2}}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}-\frac {c^{3} b^{2} \mathrm {arccosh}\left (c x \right ) e \sqrt {c x +1}\, \sqrt {c x -1}\, x}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}-\frac {c^{3} b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, d}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}+\frac {c^{4} b^{2} \mathrm {arccosh}\left (c x \right ) e \,x^{2}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}+\frac {2 c^{4} b^{2} \mathrm {arccosh}\left (c x \right ) x d}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}+\frac {c^{4} b^{2} \mathrm {arccosh}\left (c x \right ) d^{2}}{e \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}+\frac {c^{2} b^{2} \mathrm {arccosh}\left (c x \right )^{2} e}{2 \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}+\frac {c^{3} b^{2} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{3} b^{2} \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{3} b^{2} \dilog \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}-\frac {c^{3} b^{2} \dilog \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right ) d}{\left (c^{2} d^{2}-e^{2}\right )^{\frac {3}{2}} e}+\frac {c^{2} b^{2} \ln \left (2 c d \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2} e +e \right )}{\left (c^{2} d^{2}-e^{2}\right ) e}-\frac {2 c^{2} b^{2} \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{\left (c^{2} d^{2}-e^{2}\right ) e}-\frac {c^{2} a b \,\mathrm {arccosh}\left (c x \right )}{\left (c x e +c d \right )^{2} e}-\frac {c^{4} a b \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) x d}{e \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{4} a b \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) d^{2}}{e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{2} a b \sqrt {c x +1}\, \sqrt {c x -1}}{\left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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