Optimal. Leaf size=664 \[ \frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}\right )}{g^2 \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (\frac{c^2 f^2}{g^2}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{c^2 x^2+1} (f+g x)}+\frac{\sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{c x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{c^2 x^2+1}}-\frac{a \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{c^2 x^2+1}}+\frac{a \sqrt{c^2 d x^2+d}}{g}+\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \sinh ^{-1}(c x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right )}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \sinh ^{-1}(c x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt{c^2 x^2+1}}-\frac{b c x \sqrt{c^2 d x^2+d}}{g \sqrt{c^2 x^2+1}}+\frac{b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g} \]
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Rubi [A] time = 1.64957, antiderivative size = 664, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 20, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5835, 5823, 683, 5815, 6742, 261, 725, 206, 5859, 1654, 12, 5857, 5717, 8, 5831, 3322, 2264, 2190, 2279, 2391} \[ \frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}\right )}{g^2 \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (\frac{c^2 f^2}{g^2}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{c^2 x^2+1} (f+g x)}+\frac{\sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{c x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{c^2 x^2+1}}-\frac{a \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{c^2 x^2+1}}+\frac{a \sqrt{c^2 d x^2+d}}{g}+\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \sinh ^{-1}(c x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right )}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \sinh ^{-1}(c x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt{c^2 x^2+1}}-\frac{b c x \sqrt{c^2 d x^2+d}}{g \sqrt{c^2 x^2+1}}+\frac{b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g} \]
Antiderivative was successfully verified.
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Rule 5835
Rule 5823
Rule 683
Rule 5815
Rule 6742
Rule 261
Rule 725
Rule 206
Rule 5859
Rule 1654
Rule 12
Rule 5857
Rule 5717
Rule 8
Rule 5831
Rule 3322
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx &=\frac{\sqrt{d+c^2 d x^2} \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{\sqrt{d+c^2 d x^2} \int \frac{\left (-g+2 c^2 f x+c^2 g x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c \sqrt{1+c^2 x^2}}\\ &=-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\sqrt{d+c^2 d x^2} \int \frac{\left (\frac{c^2 x}{g}+\frac{1+\frac{c^2 f^2}{g^2}}{f+g x}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\sqrt{d+c^2 d x^2} \int \left (\frac{a \left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt{1+c^2 x^2}}+\frac{b \left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right ) \sinh ^{-1}(c x)}{g^2 (f+g x) \sqrt{1+c^2 x^2}}\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\left (a \sqrt{d+c^2 d x^2}\right ) \int \frac{c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right ) \sinh ^{-1}(c x)}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a \sqrt{d+c^2 d x^2}}{g}-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\left (a \sqrt{d+c^2 d x^2}\right ) \int \frac{c^2 g^2 \left (c^2 f^2+g^2\right )}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{c^2 g^4 \sqrt{1+c^2 x^2}}+\frac{\left (b \sqrt{d+c^2 d x^2}\right ) \int \left (\frac{c^2 g x \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}}+\frac{\left (c^2 f^2+g^2\right ) \sinh ^{-1}(c x)}{(f+g x) \sqrt{1+c^2 x^2}}\right ) \, dx}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a \sqrt{d+c^2 d x^2}}{g}-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}+\frac{\left (b c^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{g \sqrt{1+c^2 x^2}}+\frac{\left (a \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \int \frac{\sinh ^{-1}(c x)}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a \sqrt{d+c^2 d x^2}}{g}+\frac{b \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g}-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{\left (b c \sqrt{d+c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt{1+c^2 x^2}}-\frac{\left (a \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c^2 f^2+g^2-x^2} \, dx,x,\frac{g-c^2 f x}{\sqrt{1+c^2 x^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a \sqrt{d+c^2 d x^2}}{g}-\frac{b c x \sqrt{d+c^2 d x^2}}{g \sqrt{1+c^2 x^2}}+\frac{b \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g}-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (2 b \left (c^2 f^2+g^2\right ) \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c e^x f-g+e^{2 x} g} \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a \sqrt{d+c^2 d x^2}}{g}-\frac{b c x \sqrt{d+c^2 d x^2}}{g \sqrt{1+c^2 x^2}}+\frac{b \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g}-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (2 b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g-2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{g \sqrt{1+c^2 x^2}}-\frac{\left (2 b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g+2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{g \sqrt{1+c^2 x^2}}\\ &=\frac{a \sqrt{d+c^2 d x^2}}{g}-\frac{b c x \sqrt{d+c^2 d x^2}}{g \sqrt{1+c^2 x^2}}+\frac{b \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g}-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}-\frac{b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}-\frac{\left (b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f-2 \sqrt{c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f+2 \sqrt{c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a \sqrt{d+c^2 d x^2}}{g}-\frac{b c x \sqrt{d+c^2 d x^2}}{g \sqrt{1+c^2 x^2}}+\frac{b \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g}-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}-\frac{b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}-\frac{\left (b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f-2 \sqrt{c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f+2 \sqrt{c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{g^2 \sqrt{1+c^2 x^2}}\\ &=\frac{a \sqrt{d+c^2 d x^2}}{g}-\frac{b c x \sqrt{d+c^2 d x^2}}{g \sqrt{1+c^2 x^2}}+\frac{b \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g}-\frac{c x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt{1+c^2 x^2}}-\frac{\left (1+\frac{c^2 f^2}{g^2}\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x) \sqrt{1+c^2 x^2}}+\frac{\sqrt{1+c^2 x^2} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac{a \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{1+c^2 x^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}-\frac{b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}-\frac{b \sqrt{c^2 f^2+g^2} \sqrt{d+c^2 d x^2} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [C] time = 6.04509, size = 1353, normalized size = 2.04 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.247, size = 992, normalized size = 1.5 \begin{align*} \text{result 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 \operatorname{arsinh}\left (c x\right ) + a\right )}}{g x + f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )}{f + g x}\, dx \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{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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