Optimal. Leaf size=508 \[ \frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{4 e \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{4}{9} n x \left (\frac{2 e}{a^2}+9 d\right )+\frac{2 n \sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x)}{9 a^3}-\frac{4 n \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 e n x}{27 a^2}+\frac{2 e n (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{27 a^3}+\frac{4 e n \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a^3}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 d \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 e x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}-d n x \cosh ^{-1}(a x)^2+\frac{2 d n \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2+\frac{2 e n x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a}+2 d x \log \left (c x^n\right )+\frac{2}{27} e x^3 \log \left (c x^n\right )-2 d n x-\frac{2}{27} e n x^3 \]
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Rubi [A] time = 1.54606, antiderivative size = 508, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5707, 5654, 5718, 8, 5662, 5759, 30, 2387, 6, 5743, 5761, 4180, 2279, 2391} \[ \frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{4 e \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{4}{9} n x \left (\frac{2 e}{a^2}+9 d\right )+\frac{2 n \sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x)}{9 a^3}-\frac{4 n \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 e n x}{27 a^2}+\frac{2 e n (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{27 a^3}+\frac{4 e n \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a^3}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 d \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 e x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}-d n x \cosh ^{-1}(a x)^2+\frac{2 d n \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2+\frac{2 e n x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a}+2 d x \log \left (c x^n\right )+\frac{2}{27} e x^3 \log \left (c x^n\right )-2 d n x-\frac{2}{27} e n x^3 \]
Antiderivative was successfully verified.
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Rule 5707
Rule 5654
Rule 5718
Rule 8
Rule 5662
Rule 5759
Rule 30
Rule 2387
Rule 6
Rule 5743
Rule 5761
Rule 4180
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \cosh ^{-1}(a x)^2 \log \left (c x^n\right ) \, dx &=2 d x \log \left (c x^n\right )+\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2}{27} e x^3 \log \left (c x^n\right )-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-n \int \left (2 d+\frac{4 e}{9 a^2}+\frac{2 e x^2}{27}-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a x}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3 x}-\frac{2 e x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a}+d \cosh ^{-1}(a x)^2+\frac{1}{3} e x^2 \cosh ^{-1}(a x)^2\right ) \, dx\\ &=2 d x \log \left (c x^n\right )+\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2}{27} e x^3 \log \left (c x^n\right )-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-n \int \left (2 d+\frac{4 e}{9 a^2}+\frac{2 e x^2}{27}+\frac{\left (-\frac{2 d}{a}-\frac{4 e}{9 a^3}\right ) \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{x}-\frac{2 e x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a}+d \cosh ^{-1}(a x)^2+\frac{1}{3} e x^2 \cosh ^{-1}(a x)^2\right ) \, dx\\ &=-\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x-\frac{2}{81} e n x^3+2 d x \log \left (c x^n\right )+\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2}{27} e x^3 \log \left (c x^n\right )-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-(d n) \int \cosh ^{-1}(a x)^2 \, dx-\frac{1}{3} (e n) \int x^2 \cosh ^{-1}(a x)^2 \, dx+\frac{(2 e n) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \, dx}{9 a}+\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int \frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{x} \, dx}{9 a^3}\\ &=-\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x-\frac{2}{81} e n x^3+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2+2 d x \log \left (c x^n\right )+\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2}{27} e x^3 \log \left (c x^n\right )-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+(2 a d n) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{(2 e n) \int \left (-1+a^2 x^2\right ) \, dx}{27 a^2}+\frac{1}{9} (2 a e n) \int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int \frac{\cosh ^{-1}(a x)}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{9 a^3}-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int 1 \, dx}{9 a^2}\\ &=\frac{2 e n x}{27 a^2}-\frac{2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}-\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x-\frac{4}{81} e n x^3+\frac{2 d n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2+2 d x \log \left (c x^n\right )+\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2}{27} e x^3 \log \left (c x^n\right )-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-(2 d n) \int 1 \, dx-\frac{1}{27} (2 e n) \int x^2 \, dx+\frac{(4 e n) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a}-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{9 a^3}\\ &=-2 d n x+\frac{2 e n x}{27 a^2}-\frac{2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}-\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x-\frac{2}{27} e n x^3+\frac{2 d n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+\frac{4 e n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2-\frac{4 \left (9 a^2 d+2 e\right ) n \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+2 d x \log \left (c x^n\right )+\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2}{27} e x^3 \log \left (c x^n\right )-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{(4 e n) \int 1 \, dx}{27 a^2}+\frac{\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{9 a^3}-\frac{\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{9 a^3}\\ &=-2 d n x-\frac{2 e n x}{27 a^2}-\frac{2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}-\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x-\frac{2}{27} e n x^3+\frac{2 d n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+\frac{4 e n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2-\frac{4 \left (9 a^2 d+2 e\right ) n \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+2 d x \log \left (c x^n\right )+\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2}{27} e x^3 \log \left (c x^n\right )-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{9 a^3}\\ &=-2 d n x-\frac{2 e n x}{27 a^2}-\frac{2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}-\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x-\frac{2}{27} e n x^3+\frac{2 d n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+\frac{4 e n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2-\frac{4 \left (9 a^2 d+2 e\right ) n \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+2 d x \log \left (c x^n\right )+\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2}{27} e x^3 \log \left (c x^n\right )-\frac{2 d \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{2 i \left (9 a^2 d+2 e\right ) n \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 i \left (9 a^2 d+2 e\right ) n \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}\\ \end{align*}
Mathematica [A] time = 3.5644, size = 770, normalized size = 1.52 \[ \frac{4 e n \sqrt{a x-1} \sqrt{a x+1} \left (\frac{i \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}-\frac{a x}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}+\cosh ^{-1}(a x)+\frac{i \cosh ^{-1}(a x) \left (\log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}\right )}{9 a^3}+\frac{2 d n \sqrt{a x-1} \sqrt{a x+1} \left (\frac{i \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}-\frac{a x}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}+\cosh ^{-1}(a x)+\frac{i \cosh ^{-1}(a x) \left (\log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}\right )}{a}+\frac{e \left (27 a x \left (\cosh ^{-1}(a x)^2+2\right )+\left (9 \cosh ^{-1}(a x)^2+2\right ) \cosh \left (3 \cosh ^{-1}(a x)\right )-6 \cosh ^{-1}(a x) \left (9 \sqrt{\frac{a x-1}{a x+1}} (a x+1)+\sinh \left (3 \cosh ^{-1}(a x)\right )\right )\right ) \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )}{324 a^3}+\frac{e n \log (x) \left (2 a^3 x^3+9 a^3 x^3 \cosh ^{-1}(a x)^2-6 a^2 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)+12 a x-12 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)\right )}{27 a^3}-\frac{4 e n x}{9 a^2}-\frac{e n \sqrt{a x-1} \left (-9 a x-12 \left (\frac{a x-1}{a x+1}\right )^{3/2} (a x+1)^3 \cosh ^{-1}(a x)+\cosh \left (3 \cosh ^{-1}(a x)\right )\right )}{162 a^3 \sqrt{\frac{a x-1}{a x+1}} \sqrt{a x+1}}+\frac{d \left (a x \left (\cosh ^{-1}(a x)^2+2\right )-2 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)\right ) \left (\log \left (c x^n\right )+n (-\log (x))-n\right )}{a}+\frac{d n \log (x) \left (2 a x+a x \cosh ^{-1}(a x)^2-2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)\right )}{a}-2 d n x-\frac{2}{81} e n x^3 \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.849, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\ln \left ( c{x}^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{9} \,{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} x^{3} + 9 \,{\left (d n - d \log \left (c\right )\right )} x - 3 \,{\left (e x^{3} + 3 \, d x\right )} \log \left (x^{n}\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{2} - \int -\frac{2 \,{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} a^{3} x^{5} +{\left (9 \,{\left (d n - d \log \left (c\right )\right )} a^{3} -{\left (e n - 3 \, e \log \left (c\right )\right )} a\right )} x^{3} - 9 \,{\left (d n - d \log \left (c\right )\right )} a x +{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} a^{2} x^{4} + 9 \,{\left (d n - d \log \left (c\right )\right )} a^{2} x^{2} - 3 \,{\left (a^{2} e x^{4} + 3 \, a^{2} d x^{2}\right )} \log \left (x^{n}\right )\right )} \sqrt{a x + 1} \sqrt{a x - 1} - 3 \,{\left (a^{3} e x^{5} +{\left (3 \, a^{3} d - a e\right )} x^{3} - 3 \, a d x\right )} \log \left (x^{n}\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}{9 \,{\left (a^{3} x^{3} +{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{a x - 1} - a x\right )}}\,{d x} \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 ({\left (e x^{2} + d\right )} \operatorname{arcosh}\left (a x\right )^{2} \log \left (c x^{n}\right ), 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{\left (e x^{2} + d\right )} \operatorname{arcosh}\left (a x\right )^{2} \log \left (c x^{n}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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