Optimal. Leaf size=758 \[ \frac{3 b \sqrt{e} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{4 (-d)^{5/2}}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{a}{d^2 x}-\frac{b e \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right )}{4 d^{5/2} \sqrt{c^2 d-e}}-\frac{b e \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right )}{4 d^{5/2} \sqrt{c^2 d-e}}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{d^2}-\frac{b \text{csch}^{-1}(c x)}{d^2 x} \]
[Out]
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Rubi [A] time = 2.24642, antiderivative size = 758, normalized size of antiderivative = 1., number of steps used = 50, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {6304, 5791, 5653, 261, 5706, 5801, 725, 206, 5799, 5561, 2190, 2279, 2391} \[ \frac{3 b \sqrt{e} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right )}{4 (-d)^{5/2}}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{a}{d^2 x}-\frac{b e \tanh ^{-1}\left (\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right )}{4 d^{5/2} \sqrt{c^2 d-e}}-\frac{b e \tanh ^{-1}\left (\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right )}{4 d^{5/2} \sqrt{c^2 d-e}}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{d^2}-\frac{b \text{csch}^{-1}(c x)}{d^2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6304
Rule 5791
Rule 5653
Rule 261
Rule 5706
Rule 5801
Rule 725
Rule 206
Rule 5799
Rule 5561
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^2} \, dx &=-\operatorname{Subst}\left (\int \frac{x^4 \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{d^2}+\frac{e^2 \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}-\frac{2 e \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right ) \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{d^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{d^2}\\ &=-\frac{a}{d^2 x}-\frac{b \operatorname{Subst}\left (\int \sinh ^{-1}\left (\frac{x}{c}\right ) \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{(2 e) \operatorname{Subst}\left (\int \left (\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \sinh ^{-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{e^2 \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}-d x\right )^2}-\frac{d \left (a+b \sinh ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}+d x\right )^2}-\frac{d \left (a+b \sinh ^{-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{a}{d^2 x}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}+\frac{b \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{c d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{d^2}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}-d x\right )^2} \, dx,x,\frac{1}{x}\right )}{4 d}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}+d x\right )^2} \, dx,x,\frac{1}{x}\right )}{4 d}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac{1}{x}\right )}{2 d}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{d^2}-\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 )}{4 c d^2}+\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 )}{4 c d^2}+\frac{e \operatorname{Subst}\left (\int \left (-\frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}-\frac{a+b \sinh ^{-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 d}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{a+b \sinh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{d^2}+\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{d^2}+\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 )}{4 c d^2}-\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 )}{4 c d^2}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 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 )}{4 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 )}{4 d^{5/2} \sqrt{c^2 d-e}}-\frac{\sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{\sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{(-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{(-d)^{5/2}}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{(a+b x) \cosh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \sinh (x)} \, dx,x,\text{csch}^{-1}(c x)\right )}{4 d^2}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 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 )}{4 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 )}{4 d^{5/2} \sqrt{c^2 d-e}}-\frac{\sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{\sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{(-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{(-d)^{5/2}}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}-\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{4 d^2}-\frac{\sqrt{e} \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}+\sqrt{-d} e^x} \, dx,x,\text{csch}^{-1}(c x)\right )}{4 d^2}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 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 )}{4 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 )}{4 d^{5/2} \sqrt{c^2 d-e}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{-d} e^x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{4 (-d)^{5/2}}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 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 )}{4 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 )}{4 d^{5/2} \sqrt{c^2 d-e}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{(-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{4 (-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}-\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{4 (-d)^{5/2}}-\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{4 (-d)^{5/2}}+\frac{\left (b \sqrt{e}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-d} x}{\frac{\sqrt{e}}{c}+\frac{\sqrt{-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{4 (-d)^{5/2}}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{d^2}-\frac{a}{d^2 x}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}+\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 d^2 \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}-\frac{e \left (a+b \text{csch}^{-1}(c x)\right )}{4 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 )}{4 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 )}{4 d^{5/2} \sqrt{c^2 d-e}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left (a+b \text{csch}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{-c^2 d+e}}\right )}{4 (-d)^{5/2}}\\ \end{align*}
Mathematica [C] time = 2.87637, size = 1487, normalized size = 1.96 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 19.474, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{{x}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}\, dx \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{arcsch}\left (c x\right ) + a}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{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{b \operatorname{arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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