Optimal. Leaf size=1224 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 3.9617, antiderivative size = 1224, normalized size of antiderivative = 1., number of steps used = 80, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {5792, 5707, 5802, 96, 93, 208, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right ) c^3}{8 \left (c \sqrt{-d}-\sqrt{e}\right )^{3/2} \left (\sqrt{-d} c+\sqrt{e}\right )^{3/2} e^{5/2}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right ) c^3}{8 \left (c \sqrt{-d}-\sqrt{e}\right )^{3/2} \left (\sqrt{-d} c+\sqrt{e}\right )^{3/2} e^{5/2}}-\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right ) c}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right ) c}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}-\frac{b \sqrt{-d} \sqrt{c x-1} \sqrt{c x+1} c}{16 e^2 \left (d c^2+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b \sqrt{-d} \sqrt{c x-1} \sqrt{c x+1} c}{16 e^2 \left (d c^2+e\right ) \left (\sqrt{e} x+\sqrt{-d}\right )}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{e} x+\sqrt{-d}\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right )}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right )}{16 \sqrt{-d} e^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5792
Rule 5707
Rule 5802
Rule 96
Rule 93
Rule 208
Rule 5800
Rule 5562
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\int \left (\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^3}-\frac{2 d \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^2}+\frac{a+b \cosh ^{-1}(c x)}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx}{e^2}-\frac{(2 d) \int \frac{a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{e^2}+\frac{d^2 \int \frac{a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx}{e^2}\\ &=\frac{\int \left (\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{e^2}-\frac{(2 d) \int \left (-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx}{e^2}+\frac{d^2 \int \left (-\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-e x\right )^3}-\frac{3 e \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+e x\right )^3}-\frac{3 e \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{3 e \left (a+b \cosh ^{-1}(c x)\right )}{8 d^2 \left (-d e-e^2 x^2\right )}\right ) \, dx}{e^2}\\ &=-\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d} e^2}-\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d} e^2}-\frac{3 \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{16 e}-\frac{3 \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{16 e}-\frac{3 \int \frac{a+b \cosh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{8 e}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{2 e}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{2 e}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{e}-\frac{\sqrt{-d} \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^3} \, dx}{8 \sqrt{e}}-\frac{\sqrt{-d} \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^3} \, dx}{8 \sqrt{e}}\\ &=-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{(3 b c) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}-e x\right )} \, dx}{16 e^2}-\frac{(3 b c) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}+e x\right )} \, dx}{16 e^2}-\frac{(b c) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}-e x\right )} \, dx}{2 e^2}+\frac{(b c) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}+e x\right )} \, dx}{2 e^2}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}+\frac{\left (b c \sqrt{-d}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{16 e^{3/2}}-\frac{\left (b c \sqrt{-d}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{16 e^{3/2}}-\frac{3 \int \left (-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{8 e}+\frac{\int \left (-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{e}\\ &=-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{(3 b c) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}+e-\left (c \sqrt{-d} \sqrt{e}-e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{8 e^2}-\frac{(3 b c) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}-e-\left (c \sqrt{-d} \sqrt{e}+e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{8 e^2}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}+e-\left (c \sqrt{-d} \sqrt{e}-e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{e^2}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}-e-\left (c \sqrt{-d} \sqrt{e}+e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{e^2}-\frac{3 \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{16 \sqrt{-d} e^2}-\frac{3 \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{16 \sqrt{-d} e^2}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d} e^2}+\frac{\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d} e^2}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}+\frac{\left (b c^3 d\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}-e x\right )} \, dx}{16 e^2 \left (c^2 d+e\right )}-\frac{\left (b c^3 d\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}+e x\right )} \, dx}{16 e^2 \left (c^2 d+e\right )}\\ &=-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^2}-\frac{3 \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^2}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}+\frac{\left (b c^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}+e-\left (c \sqrt{-d} \sqrt{e}-e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{8 e^2 \left (c^2 d+e\right )}-\frac{\left (b c^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}-e-\left (c \sqrt{-d} \sqrt{e}+e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{8 e^2 \left (c^2 d+e\right )}\\ &=-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{b c^3 d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2} \left (c^2 d+e\right )}+\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c^3 d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2} \left (c^2 d+e\right )}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^2}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^2}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^2}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^2}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}+\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^2}\\ &=-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{b c^3 d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2} \left (c^2 d+e\right )}+\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c^3 d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2} \left (c^2 d+e\right )}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^{5/2}}+\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^{5/2}}-\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^{5/2}}+\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-d} e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d} e^{5/2}}\\ &=-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{b c^3 d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2} \left (c^2 d+e\right )}+\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c^3 d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2} \left (c^2 d+e\right )}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 \sqrt{-d} e^{5/2}}+\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 \sqrt{-d} e^{5/2}}-\frac{b \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 \sqrt{-d} e^{5/2}}\\ &=-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-d} \sqrt{-1+c x} \sqrt{1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{16 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{b c^3 d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2} \left (c^2 d+e\right )}+\frac{5 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c^3 d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2} \left (c^2 d+e\right )}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 \sqrt{-d} e^{5/2}}\\ \end{align*}
Mathematica [C] time = 6.90056, size = 1185, normalized size = 0.97 \[ -\frac{5 a x}{8 e^2 \left (e x^2+d\right )}+\frac{a d x}{4 e^2 \left (e x^2+d\right )^2}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{d} e^{5/2}}+b \left (-\frac{5 \left (\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{c \log \left (\frac{2 e \left (\sqrt{d} x c^2+i \sqrt{e}-i \sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c \sqrt{-d c^2-e} \left (i \sqrt{e} x+\sqrt{d}\right )}\right )}{\sqrt{-d c^2-e}}\right )}{16 e^{5/2}}+\frac{5 \left (-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \log \left (\frac{2 e \left (-i \sqrt{d} x c^2-\sqrt{e}+\sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c \sqrt{-d c^2-e} \left (\sqrt{e} x+i \sqrt{d}\right )}\right )}{\sqrt{-d c^2-e}}\right )}{16 e^{5/2}}+\frac{i \sqrt{d} \left (\frac{\sqrt{d} \left (\log \left (\frac{e \sqrt{d c^2+e} \left (-\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c^3 \left (d+i \sqrt{e} x \sqrt{d}\right )}\right )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left (d c^2+e\right ) \left (\sqrt{e} x-i \sqrt{d}\right )}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x-i \sqrt{d}\right )^2}\right )}{16 e^2}-\frac{i \sqrt{d} \left (-\frac{\sqrt{d} \left (\log \left (\frac{e \sqrt{d c^2+e} \left (\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c^3 \left (d-i \sqrt{d} \sqrt{e} x\right )}\right )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left (d c^2+e\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x+i \sqrt{d}\right )^2}\right )}{16 e^2}+\frac{3 i \left (\cosh ^{-1}(c x) \left (2 \left (\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i c \sqrt{d}-\sqrt{-d c^2-e}}+1\right )+\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i \sqrt{d} c+\sqrt{-d c^2-e}}+1\right )\right )-\cosh ^{-1}(c x)\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )\right )}{32 \sqrt{d} e^{5/2}}-\frac{3 i \left (\cosh ^{-1}(c x) \left (2 \left (\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d c^2-e}-i c \sqrt{d}}+1\right )+\log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )\right )-\cosh ^{-1}(c x)\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )\right )}{32 \sqrt{d} e^{5/2}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.02, size = 3125, normalized size = 2.6 \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 x^{4} \operatorname{arcosh}\left (c x\right ) + a x^{4}}{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{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{4}}{{\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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