Optimal. Leaf size=727 \[ -\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {1-a^2 x^2} \sqrt {a^2 c+d}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \tanh ^{-1}\left (\frac {a^2 \sqrt {-c} x+\sqrt {d}}{\sqrt {1-a^2 x^2} \sqrt {a^2 c+d}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )} \]
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Rubi [A] time = 1.07, antiderivative size = 727, 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 = {4668, 4744, 725, 206, 4742, 4522, 2190, 2279, 2391} \[ -\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {1-a^2 x^2} \sqrt {a^2 c+d}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \tanh ^{-1}\left (\frac {a^2 \sqrt {-c} x+\sqrt {d}}{\sqrt {1-a^2 x^2} \sqrt {a^2 c+d}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 2190
Rule 2279
Rule 2391
Rule 4522
Rule 4668
Rule 4742
Rule 4744
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)}{\left (c+d x^2\right )^2} \, dx &=\int \left (-\frac {d \cos ^{-1}(a x)}{4 c \left (\sqrt {-c} \sqrt {d}-d x\right )^2}-\frac {d \cos ^{-1}(a x)}{4 c \left (\sqrt {-c} \sqrt {d}+d x\right )^2}-\frac {d \cos ^{-1}(a x)}{2 c \left (-c d-d^2 x^2\right )}\right ) \, dx\\ &=-\frac {d \int \frac {\cos ^{-1}(a x)}{\left (\sqrt {-c} \sqrt {d}-d x\right )^2} \, dx}{4 c}-\frac {d \int \frac {\cos ^{-1}(a x)}{\left (\sqrt {-c} \sqrt {d}+d x\right )^2} \, dx}{4 c}-\frac {d \int \frac {\cos ^{-1}(a x)}{-c d-d^2 x^2} \, dx}{2 c}\\ &=-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \int \frac {1}{\left (\sqrt {-c} \sqrt {d}-d x\right ) \sqrt {1-a^2 x^2}} \, dx}{4 c}+\frac {a \int \frac {1}{\left (\sqrt {-c} \sqrt {d}+d x\right ) \sqrt {1-a^2 x^2}} \, dx}{4 c}-\frac {d \int \left (-\frac {\sqrt {-c} \cos ^{-1}(a x)}{2 c d \left (\sqrt {-c}-\sqrt {d} x\right )}-\frac {\sqrt {-c} \cos ^{-1}(a x)}{2 c d \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx}{2 c}\\ &=-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}+\frac {\int \frac {\cos ^{-1}(a x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{4 (-c)^{3/2}}+\frac {\int \frac {\cos ^{-1}(a x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{4 (-c)^{3/2}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{a^2 c d+d^2-x^2} \, dx,x,\frac {-d+a^2 \sqrt {-c} \sqrt {d} x}{\sqrt {1-a^2 x^2}}\right )}{4 c}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a^2 c d+d^2-x^2} \, dx,x,\frac {d+a^2 \sqrt {-c} \sqrt {d} x}{\sqrt {1-a^2 x^2}}\right )}{4 c}\\ &=-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\operatorname {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}-\sqrt {d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}+\sqrt {d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}\\ &=-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\operatorname {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}-i \sqrt {d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}-i \sqrt {d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}+i \sqrt {d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}+i \sqrt {d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2}}\\ &=-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \log \left (1-\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \log \left (1+\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\operatorname {Subst}\left (\int \log \left (1-\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\operatorname {Subst}\left (\int \log \left (1+\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{4 (-c)^{3/2} \sqrt {d}}\\ &=-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {d} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {d} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {d} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {d} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{4 (-c)^{3/2} \sqrt {d}}\\ &=-\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\cos ^{-1}(a x)}{4 c \sqrt {d} \left (\sqrt {-c}+\sqrt {d} x\right )}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}-a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d}+a^2 \sqrt {-c} x}{\sqrt {a^2 c+d} \sqrt {1-a^2 x^2}}\right )}{4 c \sqrt {d} \sqrt {a^2 c+d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}-\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}+\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{4 (-c)^{3/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 2.47, size = 1065, normalized size = 1.46 \[ \frac {4 \sin ^{-1}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) \tan \left (\frac {1}{2} \cos ^{-1}(a x)\right )}{\sqrt {c a^2+d}}\right )-4 \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c} a}{\sqrt {d}}+1}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {\left (\sqrt {c} a+i \sqrt {d}\right ) \tan \left (\frac {1}{2} \cos ^{-1}(a x)\right )}{\sqrt {c a^2+d}}\right )+i \cos ^{-1}(a x) \log \left (1-\frac {i \left (\sqrt {c a^2+d}-a \sqrt {c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+2 i \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c} a}{\sqrt {d}}+1}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {c a^2+d}-a \sqrt {c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )-i \cos ^{-1}(a x) \log \left (\frac {i e^{i \cos ^{-1}(a x)} \left (\sqrt {c a^2+d}-a \sqrt {c}\right )}{\sqrt {d}}+1\right )-2 i \sin ^{-1}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (\frac {i e^{i \cos ^{-1}(a x)} \left (\sqrt {c a^2+d}-a \sqrt {c}\right )}{\sqrt {d}}+1\right )-i \cos ^{-1}(a x) \log \left (1-\frac {i \left (\sqrt {c} a+\sqrt {c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+2 i \sin ^{-1}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {c} a+\sqrt {c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+i \cos ^{-1}(a x) \log \left (\frac {i e^{i \cos ^{-1}(a x)} \left (\sqrt {c} a+\sqrt {c a^2+d}\right )}{\sqrt {d}}+1\right )-2 i \sin ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c} a}{\sqrt {d}}+1}}{\sqrt {2}}\right ) \log \left (\frac {i e^{i \cos ^{-1}(a x)} \left (\sqrt {c} a+\sqrt {c a^2+d}\right )}{\sqrt {d}}+1\right )+\sqrt {c} \left (\frac {\cos ^{-1}(a x)}{\sqrt {d} x-i \sqrt {c}}-\frac {a \log \left (\frac {2 d \left (-i \sqrt {c} x a^2+\sqrt {d}+\sqrt {c a^2+d} \sqrt {1-a^2 x^2}\right )}{a \sqrt {c a^2+d} \left (\sqrt {d} x-i \sqrt {c}\right )}\right )}{\sqrt {c a^2+d}}\right )+\sqrt {c} \left (\frac {\cos ^{-1}(a x)}{\sqrt {d} x+i \sqrt {c}}-\frac {a \log \left (-\frac {2 d \left (i \sqrt {c} x a^2+\sqrt {d}+\sqrt {c a^2+d} \sqrt {1-a^2 x^2}\right )}{a \sqrt {c a^2+d} \left (\sqrt {d} x+i \sqrt {c}\right )}\right )}{\sqrt {c a^2+d}}\right )-\text {Li}_2\left (-\frac {i \left (\sqrt {c a^2+d}-a \sqrt {c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+\text {Li}_2\left (\frac {i \left (\sqrt {c a^2+d}-a \sqrt {c}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )+\text {Li}_2\left (-\frac {i \left (\sqrt {c} a+\sqrt {c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )-\text {Li}_2\left (\frac {i \left (\sqrt {c} a+\sqrt {c a^2+d}\right ) e^{i \cos ^{-1}(a x)}}{\sqrt {d}}\right )}{4 c^{3/2} \sqrt {d}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (a x\right )}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]
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 \[ \int \frac {\arccos \left (a x\right )}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.83, size = 1654, normalized size = 2.28 \[ \text {result too large to display} \]
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 \[ \int \frac {\arccos \left (a x\right )}{{\left (d x^{2} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^2} \,d x \]
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 \[ \int \frac {\operatorname {acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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