Optimal. Leaf size=781 \[ -\frac {\sqrt {c^2 d x^2+d} \left (g-c^2 f x\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right ) (f+g x)^2}+\frac {\sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}-\frac {a \sqrt {c^2 d x^2+d}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{g^2 \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right )}-\frac {b c^2 f \sqrt {c^2 d x^2+d} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}+\frac {b c^2 f \sqrt {c^2 d x^2+d} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}-\frac {b c^2 f \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}+\frac {b c^2 f \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}+\frac {b c \sqrt {c^2 d x^2+d} \log (f+g x)}{g^2 \sqrt {c^2 x^2+1}}-\frac {b \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^2}{2 g^2 \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right )} \]
[Out]
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Rubi [A] time = 2.55, antiderivative size = 781, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 22, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {5835, 5823, 37, 5813, 12, 1651, 844, 215, 725, 206, 5859, 5857, 5675, 5831, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac {b c^2 f \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}+\frac {b c^2 f \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}-\frac {\sqrt {c^2 d x^2+d} \left (g-c^2 f x\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right ) (f+g x)^2}+\frac {\sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac {a c^3 f^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{g^2 \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right )}+\frac {a c^2 f \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}-\frac {a \sqrt {c^2 d x^2+d}}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^2}{2 g^2 \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right )}-\frac {b c^2 f \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}+\frac {b c^2 f \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}+\frac {b c \sqrt {c^2 d x^2+d} \log (f+g x)}{g^2 \sqrt {c^2 x^2+1}}-\frac {b \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{g (f+g x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 37
Rule 206
Rule 215
Rule 725
Rule 844
Rule 1651
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3322
Rule 3324
Rule 5675
Rule 5813
Rule 5823
Rule 5831
Rule 5835
Rule 5857
Rule 5859
Rubi steps
\begin {align*} \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(f+g x)^2} \, 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)^2} \, 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)^2}-\frac {\sqrt {d+c^2 d x^2} \int \frac {\left (-2 g+2 c^2 f x\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(f+g x)^3} \, dx}{2 b c \sqrt {1+c^2 x^2}}\\ &=-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {\sqrt {d+c^2 d x^2} \int \frac {\left (g-c^2 f x\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x)^2 \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {\sqrt {d+c^2 d x^2} \int \frac {\left (g-c^2 f x\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{(f+g x)^2 \sqrt {1+c^2 x^2}} \, dx}{\left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}\\ &=-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {\sqrt {d+c^2 d x^2} \int \left (\frac {a \left (-g+c^2 f x\right )^2}{(f+g x)^2 \sqrt {1+c^2 x^2}}+\frac {b \left (-g+c^2 f x\right )^2 \sinh ^{-1}(c x)}{(f+g x)^2 \sqrt {1+c^2 x^2}}\right ) \, dx}{\left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}\\ &=-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {\left (a \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (-g+c^2 f x\right )^2}{(f+g x)^2 \sqrt {1+c^2 x^2}} \, dx}{\left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {\left (b \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (-g+c^2 f x\right )^2 \sinh ^{-1}(c x)}{(f+g x)^2 \sqrt {1+c^2 x^2}} \, dx}{\left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}\\ &=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}-\frac {\left (a \sqrt {d+c^2 d x^2}\right ) \int \frac {c^2 f \left (c^2 f^2+g^2\right )-c^4 f^2 \left (\frac {c^2 f^2}{g}+g\right ) x}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{\left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2}}+\frac {\left (b \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {c^4 f^2 \sinh ^{-1}(c x)}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (c^2 f^2+g^2\right )^2 \sinh ^{-1}(c x)}{g^2 (f+g x)^2 \sqrt {1+c^2 x^2}}-\frac {2 c^2 f \left (c^2 f^2+g^2\right ) \sinh ^{-1}(c x)}{g^2 (f+g x) \sqrt {1+c^2 x^2}}\right ) \, dx}{\left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}\\ &=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}-\frac {\left (a c^2 f \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 (2 b c^2 f \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 {\left (a c^4 f^2 \left (\frac {c^2 f^2}{g}+g\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{g \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2}}+\frac {\left (b c^4 f^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{g^2 \left (c^2 f^2+g^2\right ) \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)^2 \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 (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {\left (a c^2 f \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 (2 b c^2 f \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 {\left (b c \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))^2} \, 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 (f+g x)}-\frac {b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {a c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {\left (b c^2 f \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 {\left (4 b c^2 f \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 {\left (b c \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{g \sqrt {1+c^2 x^2}}\\ &=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {a c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{c f+x} \, dx,x,c g x\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^2 f \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 {\left (4 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {\left (4 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\\ &=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {a c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {\left (2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {\left (2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\\ &=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {a c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1+c^2 x^2}}-\frac {\left (b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {\left (b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {\left (2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\\ &=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {a c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1+c^2 x^2}}-\frac {2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {2 b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {\left (b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {\left (b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\\ &=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \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)^2}+\frac {a c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1+c^2 x^2}}-\frac {b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c^2 f \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 {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 9.65, size = 1384, normalized size = 1.77 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \]
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 [B] time = 0.71, size = 1814, normalized size = 2.32 \[ \text {result too large to display} \]
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 \[ -{\left (\frac {c^{2} d f \operatorname {arsinh}\left (\frac {c f x}{g {\left | x + \frac {f}{g} \right |}} - \frac {1}{c {\left | x + \frac {f}{g} \right |}}\right )}{\sqrt {\frac {c^{2} d f^{2}}{g^{2}} + d} g^{3}} - \frac {c \sqrt {d} \operatorname {arsinh}\left (c x\right )}{g^{2}} + \frac {\sqrt {c^{2} d x^{2} + d}}{g^{2} x + f g}\right )} a + b \int \frac {\sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{g^{2} x^{2} + 2 \, f g x + f^{2}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d}}{{\left (f+g\,x\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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