Optimal. Leaf size=786 \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tan ^{-1}\left (\frac{\sqrt{\frac{1}{c x}+1} \sqrt{c d-\sqrt{-d} \sqrt{e}}}{\sqrt{\frac{1}{c x}-1} \sqrt{c d+\sqrt{-d} \sqrt{e}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{b \tan ^{-1}\left (\frac{\sqrt{\frac{1}{c x}+1} \sqrt{c d+\sqrt{-d} \sqrt{e}}}{\sqrt{\frac{1}{c x}-1} \sqrt{c d-\sqrt{-d} \sqrt{e}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}} \]
[Out]
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Rubi [A] time = 2.83591, antiderivative size = 786, normalized size of antiderivative = 1., number of steps used = 47, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {6293, 5792, 5707, 5802, 93, 205, 5800, 5562, 2190, 2279, 2391} \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tan ^{-1}\left (\frac{\sqrt{\frac{1}{c x}+1} \sqrt{c d-\sqrt{-d} \sqrt{e}}}{\sqrt{\frac{1}{c x}-1} \sqrt{c d+\sqrt{-d} \sqrt{e}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{b \tan ^{-1}\left (\frac{\sqrt{\frac{1}{c x}+1} \sqrt{c d+\sqrt{-d} \sqrt{e}}}{\sqrt{\frac{1}{c x}-1} \sqrt{c d-\sqrt{-d} \sqrt{e}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6293
Rule 5792
Rule 5707
Rule 5802
Rule 93
Rule 205
Rule 5800
Rule 5562
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (a+b \cosh ^{-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{e \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{d \left (e+d x^2\right )^2}+\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{d}+\frac{e \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{2 \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{d}+\frac{e \operatorname{Subst}\left (\int \left (-\frac{d \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}-d x\right )^2}-\frac{d \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{4 e \left (\sqrt{-d} \sqrt{e}+d x\right )^2}-\frac{d \left (a+b \cosh ^{-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}\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}-d x\right )^2} \, dx,x,\frac{1}{x}\right )\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\left (\sqrt{-d} \sqrt{e}+d x\right )^2} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac{1}{x}\right )-\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 d \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{2 d \sqrt{e}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}-\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{2 d \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (\sqrt{-d} \sqrt{e}-d x\right )} \, dx,x,\frac{1}{x}\right )}{4 c d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (\sqrt{-d} \sqrt{e}+d x\right )} \, dx,x,\frac{1}{x}\right )}{4 c d}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{2 d \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{2 d \sqrt{e}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \operatorname{Subst}\left (\int \frac{1}{d+\frac{\sqrt{-d} \sqrt{e}}{c}-\left (-d+\frac{\sqrt{-d} \sqrt{e}}{c}\right ) x^2} \, dx,x,\frac{\sqrt{1+\frac{1}{c x}}}{\sqrt{-1+\frac{1}{c x}}}\right )}{2 c d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-d+\frac{\sqrt{-d} \sqrt{e}}{c}-\left (d+\frac{\sqrt{-d} \sqrt{e}}{c}\right ) x^2} \, dx,x,\frac{\sqrt{1+\frac{1}{c x}}}{\sqrt{-1+\frac{1}{c x}}}\right )}{2 c d}+\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 d \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\frac{1}{x}\right )}{4 d \sqrt{e}}-\frac{\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{sech}^{-1}(c x)\right )}{2 d \sqrt{e}}-\frac{\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{sech}^{-1}(c x)\right )}{2 d \sqrt{e}}-\frac{\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{sech}^{-1}(c x)\right )}{2 d \sqrt{e}}-\frac{\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{sech}^{-1}(c x)\right )}{2 d \sqrt{e}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b \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{sech}^{-1}(c x)\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \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{sech}^{-1}(c x)\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b \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{sech}^{-1}(c x)\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \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{sech}^{-1}(c x)\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}-\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{4 d \sqrt{e}}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{\frac{\sqrt{e}}{c}+\sqrt{-d} \cosh (x)} \, dx,x,\text{sech}^{-1}(c x)\right )}{4 d \sqrt{e}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b \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{sech}^{-1}(c x)}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \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{sech}^{-1}(c x)}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b \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{sech}^{-1}(c x)}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \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{sech}^{-1}(c x)}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\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{sech}^{-1}(c x)\right )}{4 d \sqrt{e}}+\frac{\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{sech}^{-1}(c x)\right )}{4 d \sqrt{e}}+\frac{\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{sech}^{-1}(c x)\right )}{4 d \sqrt{e}}+\frac{\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{sech}^{-1}(c x)\right )}{4 d \sqrt{e}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \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{sech}^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \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{sech}^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \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{sech}^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \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{sech}^{-1}(c x)\right )}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b \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{sech}^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \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{sech}^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \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{sech}^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \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{sech}^{-1}(c x)}\right )}{4 (-d)^{3/2} \sqrt{e}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}-\frac{d}{x}\right )}+\frac{a+b \text{sech}^{-1}(c x)}{4 d \left (\sqrt{-d} \sqrt{e}+\frac{d}{x}\right )}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}+\frac{b \tan ^{-1}\left (\frac{\sqrt{c d+\sqrt{-d} \sqrt{e}} \sqrt{1+\frac{1}{c x}}}{\sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{-1+\frac{1}{c x}}}\right )}{2 d \sqrt{c d-\sqrt{-d} \sqrt{e}} \sqrt{c d+\sqrt{-d} \sqrt{e}}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left (-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}+\sqrt{c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 1.54307, size = 1216, normalized size = 1.55 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.619, size = 1870, normalized size = 2.4 \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 \operatorname{arsech}\left (c x\right ) + a}{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{b \operatorname{arsech}\left (c x\right ) + a}{{\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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