Optimal. Leaf size=1270 \[ -\frac {b c^2 d (c f-g) \sqrt {d-c^2 d x^2} x^2}{4 g^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {c d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x}{2 b g^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {c d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{2 g^2}+\frac {b c d \left (4 c^2 x^2-9 c x-12\right ) \sqrt {d-c^2 d x^2} x}{36 g \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} x}{g^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)^2}{4 g \sqrt {c x-1} \sqrt {c x+1}}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {c x-1} \sqrt {c x+1} (f+g x)}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {c x-1} \sqrt {c x+1} (f+g x)}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac {b d \left (-2 c^2 x^2+3 c x+2\right ) \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{6 g}-\frac {a d \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{2 g \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 a d (c f-g)^{3/2} (c f+g)^{3/2} \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (\frac {\sqrt {c f+g} \sqrt {c x+1}}{\sqrt {c f-g} \sqrt {c x-1}}\right )}{g^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b d (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )}{g^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b d (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d (c f-g) (c f+g) \sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}+\frac {a d \left (-2 c^2 x^2+3 c x+2\right ) \sqrt {d-c^2 d x^2}}{6 g} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 3.85, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {c (c f-g) \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g^2}+\frac {c (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g}+\frac {(c f-g) (c f+g) \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g^2 (f+g x)}\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (c d (c f-g) \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, dx}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (c d (c f-g) \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c^2 d (c f-g) \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (g+2 c^2 f x+c^2 g x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (\frac {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) \left (-a-b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \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 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cosh ^{-1}(c x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}\right ) \, dx}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (a d (c f-g) (c f+g) \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}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d (c f-g) (c f+g) \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 ) \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^2 g x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 f^2-g^2\right ) \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)}\right ) \, dx}{g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (a d (c f-g) (c f+g) \sqrt {-1+c^2 x^2} \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^4 (-1+c x) (1+c x)}\\ &=-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c^2 d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int \frac {x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (a d (c f-g) (c f+g) \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {c^4 f^2 g^2-c^2 g^4}{(f+g x) \sqrt {-1+c^2 x^2}} \, dx}{c^2 g^6 (-1+c x) (1+c x)}\\ &=-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{g^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (a d (c f-g)^2 (c f+g)^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {-1+c^2 x^2}} \, dx}{g^4 (-1+c x) (1+c x)}\\ &=\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b d (c f-g)^2 (c f+g)^2 \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,\cosh ^{-1}(c x)\right )}{g^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (a d (c f-g)^2 (c f+g)^2 \sqrt {-1+c^2 x^2} \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^4 (-1+c x) (1+c x)}\\ &=\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {-1+c^2 x^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^4 \sqrt {c^2 f^2-g^2} (1-c x) (1+c x)}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b d (c f-g)^2 (c f+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,\cosh ^{-1}(c x)\right )}{g^3 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b d (c f-g)^2 (c f+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,\cosh ^{-1}(c x)\right )}{g^3 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {-1+c^2 x^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^4 \sqrt {c^2 f^2-g^2} (1-c x) (1+c x)}-\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d (c f-g)^2 (c f+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,\cosh ^{-1}(c x)\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d (c f-g)^2 (c f+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,\cosh ^{-1}(c x)\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {-1+c^2 x^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^4 \sqrt {c^2 f^2-g^2} (1-c x) (1+c x)}-\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d (c f-g)^2 (c f+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^{\cosh ^{-1}(c x)}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d (c f-g)^2 (c f+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^{\cosh ^{-1}(c x)}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 d (c f-g) x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac {c d (c f-g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac {d (c f-g) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}+\frac {a d (c f-g)^2 (c f+g)^2 \sqrt {-1+c^2 x^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^4 \sqrt {c^2 f^2-g^2} (1-c x) (1+c x)}-\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d (c f-g)^2 (c f+g)^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [C] time = 11.69, size = 3068, normalized size = 2.42 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a c^{2} d x^{2} - a d + {\left (b c^{2} d x^{2} - b d\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{g x + f}, x\right ) \]
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 = 0.64, size = 1965, normalized size = 1.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \]
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 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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