Optimal. Leaf size=664 \[ -\frac {\sqrt {c^2 d x^2+d} \left (\frac {c^2 f^2}{g^2}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {c^2 x^2+1} (f+g x)}+\frac {\sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)}-\frac {c x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g \sqrt {c^2 x^2+1}}-\frac {a \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1}}+\frac {a \sqrt {c^2 d x^2+d}}{g}+\frac {b \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1}}-\frac {b \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1}}+\frac {b \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \sinh ^{-1}(c x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^2 \sqrt {c^2 x^2+1}}-\frac {b \sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2} \sinh ^{-1}(c x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt {c^2 x^2+1}}-\frac {b c x \sqrt {c^2 d x^2+d}}{g \sqrt {c^2 x^2+1}}+\frac {b \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{g} \]
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Rubi [A] time = 1.65, antiderivative size = 664, normalized size of antiderivative = 1.00, 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 8
Rule 12
Rule 206
Rule 261
Rule 683
Rule 725
Rule 1654
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3322
Rule 5717
Rule 5815
Rule 5823
Rule 5831
Rule 5835
Rule 5857
Rule 5859
Rule 6742
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*}
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Mathematica [C] time = 6.37, size = 1353, normalized size = 2.04 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 992, normalized size = 1.49 \[ \frac {a \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{g}-\frac {a \,c^{2} d f \ln \left (\frac {-\frac {c^{2} d f}{g}+c^{2} d \left (x +\frac {f}{g}\right )}{\sqrt {c^{2} d}}+\sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\right )}{g^{2} \sqrt {c^{2} d}}-\frac {a d \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right ) c^{2} f^{2}}{g^{3} \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}-\frac {a d \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \arcsinh \left (c x \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}\, g^{2}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2} c^{2}}{\left (c^{2} x^{2}+1\right ) g}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c x}{\sqrt {c^{2} x^{2}+1}\, g}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{\left (c^{2} x^{2}+1\right ) g}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \arcsinh \left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )}{\sqrt {c^{2} x^{2}+1}\, g^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \arcsinh \left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )}{\sqrt {c^{2} x^{2}+1}\, g^{2}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \dilog \left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )}{\sqrt {c^{2} x^{2}+1}\, g^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \dilog \left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )}{\sqrt {c^{2} x^{2}+1}\, g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -{\left (\frac {c \sqrt {d} f \operatorname {arsinh}\left (c x\right )}{g^{2}} - \frac {\sqrt {\frac {c^{2} d f^{2}}{g^{2}} + d} \operatorname {arsinh}\left (\frac {c f x}{{\left | g x + f \right |}} - \frac {g}{c {\left | g x + f \right |}}\right )}{g} - \frac {\sqrt {c^{2} d x^{2} + d}}{g}\right )} a + b \int \frac {\sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{g x + f}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d}}{f+g\,x} \,d x \]
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 \[ \int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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