Optimal. Leaf size=1134 \[ \text{result too large to display} \]
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Rubi [A] time = 4.04886, antiderivative size = 1134, normalized size of antiderivative = 1., number of steps used = 81, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5231, 4733, 4667, 4743, 731, 725, 206, 4741, 4519, 2190, 2279, 2391} \[ -\frac{b \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{b \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )}-\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )}+\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\frac{d}{x}+\sqrt{-d} \sqrt{e}\right )^2}+\frac{5 b \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt{d c^2+e}}-\frac{b e \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (d c^2+e\right )^{3/2}}+\frac{5 b \tanh ^{-1}\left (\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt{d c^2+e}}-\frac{b e \tanh ^{-1}\left (\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (d c^2+e\right )^{3/2}}-\frac{3 \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{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left (2,-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{PolyLog}\left (2,\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 5231
Rule 4733
Rule 4667
Rule 4743
Rule 731
Rule 725
Rule 206
Rule 4741
Rule 4519
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x^4 \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{e^2 \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^3}-\frac{2 e \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}+\frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{(2 e) \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 )}{d^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac{1}{x}\right )}{d^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{(2 e) \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 )}{d^2}-\frac{e^2 \operatorname{Subst}\left (\int \left (-\frac{d^3 \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt{-d} \sqrt{e}-d x\right )^3}-\frac{3 d \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{16 e^2 \left (\sqrt{-d} \sqrt{e}-d x\right )^2}-\frac{d^3 \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt{-d} \sqrt{e}+d x\right )^3}-\frac{3 d \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{16 e^2 \left (\sqrt{-d} \sqrt{e}+d x\right )^2}-\frac{3 d \left (a+b \sin ^{-1}\left (\frac{x}{c}\right )\right )}{8 e^2 \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )}{d^2}\\ &=\frac{3 \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 )}{16 d}+\frac{3 \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 )}{16 d}+\frac{3 \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 )}{8 d}-\frac{\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 )}{2 d}-\frac{\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 )}{2 d}-\frac{\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 )}{d}-\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 )}{2 d^2 \sqrt{e}}-\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 )}{2 d^2 \sqrt{e}}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}-d x\right )^3} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{-d}}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}+d x\right )^3} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{-d}}\\ &=\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}+\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{(3 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 )}{16 c d^2}+\frac{(3 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 )}{16 c d^2}+\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 )}{2 c d^2}-\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 )}{2 c d^2}+\frac{3 \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 )}{8 d}-\frac{\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 )}{d}-\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 )}{2 d^2 \sqrt{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 )}{2 d^2 \sqrt{e}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-d} \sqrt{e}-d x\right )^2 \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{16 c (-d)^{3/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-d} \sqrt{e}+d x\right )^2 \sqrt{1-\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{16 c (-d)^{3/2}}\\ &=-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}+\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{(3 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 )}{16 c d^2}-\frac{(3 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 )}{16 c d^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 )}{2 c d^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 )}{2 c d^2}-\frac{3 \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 )}{16 d^2 \sqrt{e}}-\frac{3 \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 )}{16 d^2 \sqrt{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 )}{2 d^2 \sqrt{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 )}{2 d^2 \sqrt{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 )}{2 d^2 \sqrt{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 )}{2 d^2 \sqrt{e}}+\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 )}{2 d^2 \sqrt{e}}+\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 )}{2 d^2 \sqrt{e}}-\frac{(b e) \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 )}{16 c d^2 \left (c^2 d+e\right )}+\frac{(b e) \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 )}{16 c d^2 \left (c^2 d+e\right )}\\ &=-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}+\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{5 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 )}{16 d^{5/2} \sqrt{c^2 d+e}}+\frac{5 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 )}{16 d^{5/2} \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 )}{2 (-d)^{5/2} \sqrt{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 )}{2 (-d)^{5/2} \sqrt{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 )}{2 (-d)^{5/2} \sqrt{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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 d^2 \sqrt{e}}-\frac{3 \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 )}{16 d^2 \sqrt{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 )}{2 d^2 \sqrt{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 )}{2 d^2 \sqrt{e}}+\frac{(b e) \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 )}{16 c d^2 \left (c^2 d+e\right )}-\frac{(b e) \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 )}{16 c d^2 \left (c^2 d+e\right )}\\ &=-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}+\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \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 )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac{5 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 )}{16 d^{5/2} \sqrt{c^2 d+e}}-\frac{b e \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 )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac{5 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 )}{16 d^{5/2} \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 )}{2 (-d)^{5/2} \sqrt{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 )}{2 (-d)^{5/2} \sqrt{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 )}{2 (-d)^{5/2} \sqrt{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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 d^2 \sqrt{e}}-\frac{3 \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 )}{16 d^2 \sqrt{e}}-\frac{3 \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 )}{16 d^2 \sqrt{e}}-\frac{3 \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 )}{16 d^2 \sqrt{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 )}{2 d^2 \sqrt{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 )}{2 d^2 \sqrt{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 )}{2 d^2 \sqrt{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 )}{2 d^2 \sqrt{e}}\\ &=-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}+\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \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 )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac{5 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 )}{16 d^{5/2} \sqrt{c^2 d+e}}-\frac{b e \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 )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac{5 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 )}{16 d^{5/2} \sqrt{c^2 d+e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}+\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}\\ &=-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}+\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \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 )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac{5 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 )}{16 d^{5/2} \sqrt{c^2 d+e}}-\frac{b e \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 )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac{5 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 )}{16 d^{5/2} \sqrt{c^2 d+e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}+\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 )}{2 (-d)^{5/2} \sqrt{e}}-\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 )}{2 (-d)^{5/2} \sqrt{e}}\\ &=-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{b c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )^2}-\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{\sqrt{e} \left (a+b \csc ^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )^2}+\frac{5 \left (a+b \csc ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{b e \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 )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac{5 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 )}{16 d^{5/2} \sqrt{c^2 d+e}}-\frac{b e \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 )}{16 d^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac{5 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 )}{16 d^{5/2} \sqrt{c^2 d+e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 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 )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 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 )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 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 )}{16 (-d)^{5/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 6.06086, size = 2060, normalized size = 1.82 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.95, size = 3166, normalized size = 2.8 \begin{align*} \text{output 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 \operatorname{arccsc}\left (c x\right ) + a}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, 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{b \operatorname{arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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