Optimal. Leaf size=687 \[ \frac{5 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 d x^2+d}}-\frac{5 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 d x^2+d}}-\frac{13 i b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{13 i b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{5 b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{c^2 d x^2+d}}+\frac{5 b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{c^2 d x^2+d}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{c^2 d x^2+d}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{26 b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \sqrt{c^2 d x^2+d}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (c^2 d x^2+d\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}+\frac{b^2 c^2}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{d^2 \sqrt{c^2 d x^2+d}} \]
[Out]
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Rubi [A] time = 1.259, antiderivative size = 687, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 18, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5747, 5755, 5764, 5760, 4182, 2531, 2282, 6589, 5693, 4180, 2279, 2391, 5690, 261, 266, 51, 63, 208} \[ \frac{5 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 d x^2+d}}-\frac{5 b c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt{c^2 d x^2+d}}-\frac{13 i b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{13 i b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{5 b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{c^2 d x^2+d}}+\frac{5 b^2 c^2 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{c^2 d x^2+d}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{c^2 d x^2+d}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{26 b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \sqrt{c^2 d x^2+d}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (c^2 d x^2+d\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}+\frac{b^2 c^2}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{d^2 \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5747
Rule 5755
Rule 5764
Rule 5760
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 5693
Rule 4180
Rule 2279
Rule 2391
Rule 5690
Rule 261
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{1}{2} \left (5 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx+\frac{\left (b c \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (5 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx}{2 d}+\frac{\left (b^2 c^2 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (3 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt{d+c^2 d x^2}} \, dx}{2 d^2}+\frac{\left (b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{6 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (3 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 b^2 c^4 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (3 b^2 c^4 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 c^2 \sqrt{1+c^2 x^2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt{1+c^2 x^2}} \, dx}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{6 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (3 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{2 d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{26 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (b^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (3 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (3 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{26 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{5 c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1+c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 b c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{6 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (3 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (3 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{26 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{5 c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1+c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{5 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{5 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{26 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{5 c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1+c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{5 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{5 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ &=\frac{b^2 c^2}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac{5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d+c^2 d x^2}}+\frac{26 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{5 c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1+c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{5 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1+c^2 x^2} \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{5 b c^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{5 b^2 c^2 \sqrt{1+c^2 x^2} \text{Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{5 b^2 c^2 \sqrt{1+c^2 x^2} \text{Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 7.97946, size = 983, normalized size = 1.43 \[ -\frac{5 a^2 \log (x) c^2}{2 d^{5/2}}+\frac{5 a^2 \log \left (d+\sqrt{d \left (c^2 x^2+1\right )} \sqrt{d}\right ) c^2}{2 d^{5/2}}+\frac{a b \left (-3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\frac{8 \sinh ^{-1}(c x)}{c^2 x^2+1}-48 \sinh ^{-1}(c x)+104 \sqrt{c^2 x^2+1} \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )-6 \sqrt{c^2 x^2+1} \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-60 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+60 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )-60 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )+60 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+6 \sqrt{c^2 x^2+1} \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+\frac{4 c x}{\sqrt{c^2 x^2+1}}\right ) c^2}{12 d^2 \sqrt{d \left (c^2 x^2+1\right )}}+\frac{b^2 \left (-3 \sqrt{c^2 x^2+1} \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)^2-3 \sqrt{c^2 x^2+1} \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)^2-60 \sqrt{c^2 x^2+1} \log \left (1-e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)^2+60 \sqrt{c^2 x^2+1} \log \left (1+e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)^2-\frac{8 \sinh ^{-1}(c x)^2}{c^2 x^2+1}-48 \sinh ^{-1}(c x)^2-12 \sqrt{c^2 x^2+1} \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)-104 i \sqrt{c^2 x^2+1} \log \left (1-i e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+104 i \sqrt{c^2 x^2+1} \log \left (1+i e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)-120 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+120 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+12 \sqrt{c^2 x^2+1} \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\frac{8 c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+24 \sqrt{c^2 x^2+1} \log \left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )-104 i \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+104 i \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )-120 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(c x)}\right )+120 \sqrt{c^2 x^2+1} \text{PolyLog}\left (3,e^{-\sinh ^{-1}(c x)}\right )+8\right ) c^2}{24 d^2 \sqrt{d \left (c^2 x^2+1\right )}}+\sqrt{d \left (c^2 x^2+1\right )} \left (-\frac{2 c^2 a^2}{d^3 \left (c^2 x^2+1\right )}-\frac{a^2}{2 d^3 x^2}-\frac{c^2 a^2}{3 d^3 \left (c^2 x^2+1\right )^2}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.388, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{{x}^{3}} \left ({c}^{2}d{x}^{2}+d \right ) ^{-{\frac{5}{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{\sqrt{c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{9} + 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} + d^{3} x^{3}}, 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 \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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