Optimal. Leaf size=765 \[ -\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}} \]
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Rubi [A] time = 1.25474, antiderivative size = 765, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {5241, 4667, 4743, 725, 206, 4741, 4519, 2190, 2279, 2391} \[ -\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}} \]
Antiderivative was successfully verified.
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Rule 5241
Rule 4667
Rule 4743
Rule 725
Rule 206
Rule 4741
Rule 4519
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}-d x\right )^2}-\frac{d \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}+d x\right )^2}-\frac{d \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{d \operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}-d x\right )^2} \, dx,x,\frac{1}{x}\right )}{4 e}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}+d x\right )^2} \, dx,x,\frac{1}{x}\right )}{4 e}+\frac{d \operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac{1}{x}\right )}{2 e}\\ &=\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-d} \sqrt{e}-d x\right ) \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c e}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-d} \sqrt{e}+d x\right ) \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c e}+\frac{d \operatorname{Subst}\left (\int \left (-\frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}-\frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{2 e}\\ &=\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 e^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 e^{3/2}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{d^2+\frac{d e}{c^2}-x^2} \, dx,x,\frac{-d+\frac{\sqrt{-d} \sqrt{e}}{c^2 x}}{\sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 c e}-\frac{b \operatorname{Subst}\left (\int \frac{1}{d^2+\frac{d e}{c^2}-x^2} \, dx,x,\frac{d+\frac{\sqrt{-d} \sqrt{e}}{c^2 x}}{\sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 c e}\\ &=\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{4 e^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \cos (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \sin (x)} \, dx,x,\csc ^{-1}(c x)\right )}{4 e^{3/2}}\\ &=\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}-i \sqrt{-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{4 e^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}-i \sqrt{-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{4 e^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}+i \sqrt{-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{4 e^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}+i \sqrt{-d} e^{i x}} \, dx,x,\csc ^{-1}(c x)\right )}{4 e^{3/2}}\\ &=\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{i \sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{4 \sqrt{-d} e^{3/2}}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{i \sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{4 \sqrt{-d} e^{3/2}}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{i \sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{4 \sqrt{-d} e^{3/2}}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{i \sqrt{-d} e^{i x}}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{4 \sqrt{-d} e^{3/2}}\\ &=\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i \sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i \sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{4 \sqrt{-d} e^{3/2}}\\ &=\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{a+b \csc ^{-1}(c x)}{4 e \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left (-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left (\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left (-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left (\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 \sqrt{-d} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 2.33828, size = 1482, normalized size = 1.94 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.572, size = 1722, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{2} \operatorname{arccsc}\left (c x\right ) + a x^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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