Optimal. Leaf size=774 \[ \frac {\text {Li}_2\left (-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-c a^2-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {Li}_2\left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-c a^2-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {Li}_2\left (-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {-c} a+\sqrt {-c a^2-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {Li}_2\left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {-c} a+\sqrt {-c a^2-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}+1\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}+1\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a x+1} \sqrt {a \sqrt {-c}-\sqrt {d}}}{\sqrt {a x-1} \sqrt {a \sqrt {-c}+\sqrt {d}}}\right )}{2 c \sqrt {d} \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a x+1} \sqrt {a \sqrt {-c}+\sqrt {d}}}{\sqrt {a x-1} \sqrt {a \sqrt {-c}-\sqrt {d}}}\right )}{2 c \sqrt {d} \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}}} \]
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Rubi [A] time = 1.10, antiderivative size = 774, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5707, 5802, 93, 208, 5800, 5562, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {PolyLog}\left (2,\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {PolyLog}\left (2,-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {PolyLog}\left (2,\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {a^2 (-c)-d}}+1\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}+a \sqrt {-c}}+1\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a x+1} \sqrt {a \sqrt {-c}-\sqrt {d}}}{\sqrt {a x-1} \sqrt {a \sqrt {-c}+\sqrt {d}}}\right )}{2 c \sqrt {d} \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a x+1} \sqrt {a \sqrt {-c}+\sqrt {d}}}{\sqrt {a x-1} \sqrt {a \sqrt {-c}-\sqrt {d}}}\right )}{2 c \sqrt {d} \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 208
Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5707
Rule 5800
Rule 5802
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx &=\int \left (-\frac {d \cosh ^{-1}(a x)}{4 c \left (\sqrt {-c} \sqrt {d}-d x\right )^2}-\frac {d \cosh ^{-1}(a x)}{4 c \left (\sqrt {-c} \sqrt {d}+d x\right )^2}-\frac {d \cosh ^{-1}(a x)}{2 c \left (-c d-d^2 x^2\right )}\right ) \, dx\\ &=-\frac {d \int \frac {\cosh ^{-1}(a x)}{\left (\sqrt {-c} \sqrt {d}-d x\right )^2} \, dx}{4 c}-\frac {d \int \frac {\cosh ^{-1}(a x)}{\left (\sqrt {-c} \sqrt {d}+d x\right )^2} \, dx}{4 c}-\frac {d \int \frac {\cosh ^{-1}(a x)}{-c d-d^2 x^2} \, dx}{2 c}\\ &=-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x} \left (\sqrt {-c} \sqrt {d}-d x\right )} \, dx}{4 c}-\frac {a \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x} \left (\sqrt {-c} \sqrt {d}+d x\right )} \, dx}{4 c}-\frac {d \int \left (-\frac {\sqrt {-c} \cosh ^{-1}(a x)}{2 c d \left (\sqrt {-c}-\sqrt {d} x\right )}-\frac {\sqrt {-c} \cosh ^{-1}(a x)}{2 c d \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx}{2 c}\\ &=-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {\int \frac {\cosh ^{-1}(a x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{4 (-c)^{3/2}}+\frac {\int \frac {\cosh ^{-1}(a x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{4 (-c)^{3/2}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{a \sqrt {-c} \sqrt {d}+d-\left (a \sqrt {-c} \sqrt {d}-d\right ) x^2} \, dx,x,\frac {\sqrt {1+a x}}{\sqrt {-1+a x}}\right )}{2 c}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a \sqrt {-c} \sqrt {d}-d-\left (a \sqrt {-c} \sqrt {d}+d\right ) x^2} \, dx,x,\frac {\sqrt {1+a x}}{\sqrt {-1+a x}}\right )}{2 c}\\ &=-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {x \sinh (x)}{a \sqrt {-c}-\sqrt {d} \cosh (x)} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {x \sinh (x)}{a \sqrt {-c}+\sqrt {d} \cosh (x)} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}\\ &=-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {e^x x}{a \sqrt {-c}-\sqrt {-a^2 c-d}-\sqrt {d} e^x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {e^x x}{a \sqrt {-c}+\sqrt {-a^2 c-d}-\sqrt {d} e^x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {e^x x}{a \sqrt {-c}-\sqrt {-a^2 c-d}+\sqrt {d} e^x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {e^x x}{a \sqrt {-c}+\sqrt {-a^2 c-d}+\sqrt {d} e^x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2}}\\ &=-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {d} e^x}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {d} e^x}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {d} e^x}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {d} e^x}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt {d}}\\ &=-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {d} x}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {d} x}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {d} x}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {d} x}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}\\ &=-\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cosh ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {1+a x}}{\sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {-1+a x}}\right )}{2 c \sqrt {a \sqrt {-c}-\sqrt {d}} \sqrt {a \sqrt {-c}+\sqrt {d}} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cosh ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cosh ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {Li}_2\left (-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {Li}_2\left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\text {Li}_2\left (-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\text {Li}_2\left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{4 (-c)^{3/2} \sqrt {d}}\\ \end {align*}
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Mathematica [C] time = 1.38, size = 687, normalized size = 0.89 \[ \frac {i \left (2 \text {Li}_2\left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {-c a^2-d}-i a \sqrt {c}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{i \sqrt {c} a+\sqrt {-c a^2-d}}\right )+\cosh ^{-1}(a x) \left (-\cosh ^{-1}(a x)+2 \left (\log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{-\sqrt {a^2 (-c)-d}+i a \sqrt {c}}\right )+\log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}+i a \sqrt {c}}\right )\right )\right )\right )-i \left (2 \text {Li}_2\left (-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {-c a^2-d}-i a \sqrt {c}}\right )+2 \text {Li}_2\left (\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{i \sqrt {c} a+\sqrt {-c a^2-d}}\right )+\cosh ^{-1}(a x) \left (-\cosh ^{-1}(a x)+2 \left (\log \left (1+\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}-i a \sqrt {c}}\right )+\log \left (1-\frac {\sqrt {d} e^{\cosh ^{-1}(a x)}}{\sqrt {a^2 (-c)-d}+i a \sqrt {c}}\right )\right )\right )\right )+2 \sqrt {c} \left (\frac {a \log \left (\frac {2 d \left (-i \sqrt {a x-1} \sqrt {a x+1} \sqrt {a^2 (-c)-d}+a^2 \sqrt {c} x+i \sqrt {d}\right )}{a \sqrt {a^2 (-c)-d} \left (\sqrt {c}+i \sqrt {d} x\right )}\right )}{\sqrt {a^2 (-c)-d}}+\frac {\cosh ^{-1}(a x)}{\sqrt {d} x-i \sqrt {c}}\right )-2 \sqrt {c} \left (-\frac {a \log \left (\frac {2 d \left (\sqrt {a x-1} \sqrt {a x+1} \sqrt {a^2 (-c)-d}-i a^2 \sqrt {c} x-\sqrt {d}\right )}{a \sqrt {a^2 (-c)-d} \left (\sqrt {d} x+i \sqrt {c}\right )}\right )}{\sqrt {a^2 (-c)-d}}-\frac {\cosh ^{-1}(a x)}{\sqrt {d} x+i \sqrt {c}}\right )}{8 c^{3/2} \sqrt {d}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.40, size = 1632, normalized size = 2.11 \[ \text {result too large to display} \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (a x \right )}}{\left (c + d x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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