Optimal. Leaf size=773 \[ -\frac {g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2} (c f-g)}+\frac {(c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2} (c f+g)}-\frac {b g^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b g^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} (c f+g)}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{c x+1}\right )}{2 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} (c f-g)}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{c x+1}\right )}{2 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} (c f+g)} \]
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Rubi [A] time = 1.86, antiderivative size = 773, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 17, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {5836, 5834, 37, 5848, 12, 6719, 260, 266, 36, 31, 29, 5832, 3320, 2264, 2190, 2279, 2391} \[ -\frac {b g^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {b g^2 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac {g^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{d \sqrt {d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2} (c f-g)}+\frac {(c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt {d-c^2 d x^2} (c f+g)}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} (c f+g)}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{c x+1}\right )}{2 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} (c f-g)}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{c x+1}\right )}{2 d \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \sqrt {d-c^2 d x^2} (c f+g)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 37
Rule 260
Rule 266
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3320
Rule 5832
Rule 5834
Rule 5836
Rule 5848
Rule 6719
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{(-1+c x)^{3/2} (1+c x)^{3/2} (f+g x)} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (-\frac {c \left (a+b \cosh ^{-1}(c x)\right )}{2 (c f-g) \sqrt {-1+c x} (1+c x)^{3/2}}+\frac {c \left (a+b \cosh ^{-1}(c x)\right )}{2 (c f+g) (-1+c x)^{3/2} \sqrt {1+c x}}+\frac {g^2 \left (a+b \cosh ^{-1}(c x)\right )}{(c f-g) (c f+g) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\left (c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{(-1+c x)^{3/2} \sqrt {1+c x}} \, dx}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{d \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {a+b x}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (b c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {\frac {-1+c x}{1+c x}}}{c \sqrt {1-c^2 x^2}} \, dx}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (b c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{c \sqrt {\frac {-1+c x}{1+c x}} \sqrt {1-c^2 x^2}} \, dx}{2 d (c f+g) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 g^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{2 c e^x f+g+e^{2 x} g} \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {\frac {-1+c x}{1+c x}}}{\sqrt {1-c^2 x^2}} \, dx}{2 d (c f-g) \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {\frac {-1+c x}{1+c x}} \sqrt {1-c^2 x^2}} \, dx}{2 d (c f+g) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {\left (2 g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{2 c f+2 e^x g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (2 g^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{2 c f+2 e^x g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sqrt {-\frac {x^2}{\left (-1+x^2\right )^2}} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-\frac {x^2}{\left (-1+x^2\right )^2}}}{x^2} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{d (c f+g) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {\left (b g^2 \sqrt {-1+c x} \sqrt {1+c x}\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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (b g^2 \sqrt {-1+c x} \sqrt {1+c x}\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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{-1+x^2} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{d (c f-g) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-1+x^2\right )} \, dx,x,\sqrt {\frac {-1+c x}{1+c x}}\right )}{d (c f+g) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 d (c f-g) \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (b g^2 \sqrt {-1+c x} \sqrt {1+c x}\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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (b g^2 \sqrt {-1+c x} \sqrt {1+c x}\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 )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 d (c f+g) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 d (c f-g) \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b g^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b g^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 d (c f+g) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-(-1+c x) (1+c x)} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {-1+c x}{1+c x}\right )}{2 d (c f+g) \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}\\ &=-\frac {(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (-\frac {1-c x}{1+c x}\right )}{2 d (c f+g) \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 d (c f-g) \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log (1+c x)}{2 d (c f+g) \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b g^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b g^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [C] time = 9.73, size = 1203, normalized size = 1.56 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{c^{4} d^{2} g x^{5} + c^{4} d^{2} f x^{4} - 2 \, c^{2} d^{2} g x^{3} - 2 \, c^{2} d^{2} f x^{2} + d^{2} g x + d^{2} f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.63, size = 1926, normalized size = 2.49 \[ \text {result too large to display} \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{\left (f+g\,x\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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