Optimal. Leaf size=506 \[ -\frac{8 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{8 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{16 c^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{32 b c^3 \sqrt{c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{32 b c^3 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (c^2 d x^2+d\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2}{3 d^2 x \sqrt{c^2 d x^2+d}} \]
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Rubi [A] time = 1.09419, antiderivative size = 506, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 15, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.536, Rules used = {5747, 5690, 5687, 5714, 3718, 2190, 2279, 2391, 5717, 191, 5755, 5720, 5461, 4182, 271} \[ -\frac{8 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{8 b^2 c^3 \sqrt{c^2 x^2+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{16 c^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{32 b c^3 \sqrt{c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{32 b c^3 \sqrt{c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (c^2 d x^2+d\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2}{3 d^2 x \sqrt{c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5690
Rule 5687
Rule 5714
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 5717
Rule 191
Rule 5755
Rule 5720
Rule 5461
Rule 4182
Rule 271
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}-\left (2 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx+\frac{\left (2 b c \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\left (8 c^4\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx+\frac{\left (b^2 c^2 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d+c^2 d x^2}}-\frac{8 b c^3 \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{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{\left (16 c^4\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 d}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (2 b^2 c^4 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (16 b c^5 \sqrt{1+c^2 x^2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d+c^2 d x^2}}+\frac{2 b^2 c^4 x}{d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (4 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 b^2 c^4 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (32 b c^5 \sqrt{1+c^2 x^2}\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d+c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (8 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (32 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d+c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{16 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{32 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (64 b c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (4 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (4 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (4 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 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 x \sqrt{d+c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{16 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{32 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{32 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (2 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (2 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (2 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}-\frac{\left (2 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (32 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d+c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{16 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{32 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{32 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{8 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{\left (16 b^2 c^3 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 x \sqrt{d+c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac{2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac{8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac{16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{16 c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d+c^2 d x^2}}+\frac{32 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{32 b c^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{1+c^2 x^2} \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d+c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 3.02378, size = 417, normalized size = 0.82 \[ \frac{b^2 c^3 \left (c^2 x^2+1\right )^{3/2} \left (8 \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+8 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )-\frac{\sqrt{c^2 x^2+1}}{c x}-\frac{c x}{\sqrt{c^2 x^2+1}}+\frac{8 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2}{c x}-\frac{\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2}{c^3 x^3}+\frac{8 c x \sinh ^{-1}(c x)^2}{\sqrt{c^2 x^2+1}}+\frac{c x \sinh ^{-1}(c x)^2}{\left (c^2 x^2+1\right )^{3/2}}+\frac{\sinh ^{-1}(c x)}{c^2 x^2+1}-\frac{\sinh ^{-1}(c x)}{c^2 x^2}-16 \sinh ^{-1}(c x)^2-16 \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-16 \sinh ^{-1}(c x) \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )\right )+\frac{a^2 \left (16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right )}{x^3}-\frac{a b \left (c x \sqrt{c^2 x^2+1} \left (16 \left (c^4 x^4+c^2 x^2\right ) \log (c x)+8 \left (c^4 x^4+c^2 x^2\right ) \log \left (c^2 x^2+1\right )+1\right )-2 \left (16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right ) \sinh ^{-1}(c x)\right )}{x^3}}{3 d \left (c^2 d x^2+d\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.33, size = 4955, normalized size = 9.8 \begin{align*} \text{output too large to display} \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^{10} + 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} + d^{3} x^{4}}, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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