Optimal. Leaf size=839 \[ \frac{x \cosh ^{-1}(c x) b}{e^2}+\frac{c d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right ) b}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}-\frac{c d \tanh ^{-1}\left (\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right ) b}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}-\frac{3 \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac{3 \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac{3 \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac{3 \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} b}{c e^2}+\frac{a x}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{e} x+\sqrt{-d}\right )}+\frac{3 \sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right )}{4 e^{5/2}} \]
[Out]
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Rubi [A] time = 2.18778, antiderivative size = 839, normalized size of antiderivative = 1., number of steps used = 49, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5792, 5654, 74, 5707, 5802, 93, 208, 5800, 5562, 2190, 2279, 2391} \[ \frac{x \cosh ^{-1}(c x) b}{e^2}+\frac{c d \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right ) b}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}-\frac{c d \tanh ^{-1}\left (\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right ) b}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}-\frac{3 \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac{3 \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac{3 \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac{3 \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} b}{c e^2}+\frac{a x}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{e} x+\sqrt{-d}\right )}+\frac{3 \sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right )}{4 e^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5792
Rule 5654
Rule 74
Rule 5707
Rule 5802
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 )^2} \, dx &=\int \left (\frac{a+b \cosh ^{-1}(c x)}{e^2}+\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )^2}-\frac{2 d \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{e^2}-\frac{(2 d) \int \frac{a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx}{e^2}+\frac{d^2 \int \frac{a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{e^2}\\ &=\frac{a x}{e^2}+\frac{b \int \cosh ^{-1}(c x) \, dx}{e^2}-\frac{(2 d) \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{d^2 \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{a x}{e^2}+\frac{b x \cosh ^{-1}(c x)}{e^2}-\frac{(b c) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e^2}-\frac{\sqrt{-d} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{e^2}-\frac{\sqrt{-d} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{e^2}-\frac{d \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 e}-\frac{d \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 e}-\frac{d \int \frac{a+b \cosh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 e}\\ &=\frac{a x}{e^2}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}+\frac{b x \cosh ^{-1}(c x)}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e^2}-\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e^2}+\frac{(b c d) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}-e x\right )} \, dx}{4 e^2}-\frac{(b c d) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}+e x\right )} \, dx}{4 e^2}-\frac{d \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}{2 e}\\ &=\frac{a x}{e^2}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}+\frac{b x \cosh ^{-1}(c x)}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{\sqrt{-d} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 e^2}+\frac{\sqrt{-d} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 e^2}-\frac{\sqrt{-d} \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 )}{e^2}-\frac{\sqrt{-d} \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 )}{e^2}-\frac{\sqrt{-d} \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 )}{e^2}-\frac{\sqrt{-d} \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 )}{e^2}+\frac{(b c d) \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 )}{2 e^2}-\frac{(b c d) \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 )}{2 e^2}\\ &=\frac{a x}{e^2}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}+\frac{b x \cosh ^{-1}(c x)}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\sqrt{-d} \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 )}{e^{5/2}}+\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{4 e^2}+\frac{\sqrt{-d} \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{4 e^2}\\ &=\frac{a x}{e^2}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}+\frac{b x \cosh ^{-1}(c x)}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\sqrt{-d} \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 )}{e^{5/2}}+\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\sqrt{-d} \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 )}{e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{e^{5/2}}+\frac{\sqrt{-d} \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 )}{4 e^2}+\frac{\sqrt{-d} \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 )}{4 e^2}+\frac{\sqrt{-d} \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 )}{4 e^2}+\frac{\sqrt{-d} \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 )}{4 e^2}\\ &=\frac{a x}{e^2}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}+\frac{b x \cosh ^{-1}(c x)}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{e^{5/2}}+\frac{b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{e^{5/2}}-\frac{b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{e^{5/2}}+\frac{b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}\\ &=\frac{a x}{e^2}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}+\frac{b x \cosh ^{-1}(c x)}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{e^{5/2}}+\frac{b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{e^{5/2}}-\frac{b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{e^{5/2}}+\frac{b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}+\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}-\frac{\left (b \sqrt{-d}\right ) \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 )}{4 e^{5/2}}\\ &=\frac{a x}{e^2}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}+\frac{b x \cosh ^{-1}(c x)}{e^2}-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}-\frac{b c 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 )}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}+\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 \sqrt{-d} \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 )}{4 e^{5/2}}-\frac{3 b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{4 e^{5/2}}+\frac{3 b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{4 e^{5/2}}-\frac{3 b \sqrt{-d} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{4 e^{5/2}}+\frac{3 b \sqrt{-d} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{4 e^{5/2}}\\ \end{align*}
Mathematica [C] time = 2.22501, size = 776, normalized size = 0.92 \[ \frac{b \left (-3 i \sqrt{d} \left (2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right )+\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{-\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right )+\log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right )\right )\right )\right )+3 i \sqrt{d} \left (2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right )+\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-i c \sqrt{d}}\right )+\log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right )\right )\right )\right )+2 d \left (\frac{c \log \left (\frac{2 e \left (-i \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}+c^2 \sqrt{d} x+i \sqrt{e}\right )}{c \sqrt{c^2 (-d)-e} \left (\sqrt{d}+i \sqrt{e} x\right )}\right )}{\sqrt{c^2 (-d)-e}}+\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}\right )+2 d \left (\frac{c \log \left (\frac{2 e \left (\sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}-i c^2 \sqrt{d} x-\sqrt{e}\right )}{c \sqrt{c^2 (-d)-e} \left (\sqrt{e} x+i \sqrt{d}\right )}\right )}{\sqrt{c^2 (-d)-e}}+\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right )+\frac{8 \sqrt{e} \left (c x \cosh ^{-1}(c x)-\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right )}{c}\right )+\frac{4 a d \sqrt{e} x}{d+e x^2}-12 a \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )+8 a \sqrt{e} x}{8 e^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.26, size = 1749, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{4} \operatorname{arcosh}\left (c x\right ) + a x^{4}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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