Optimal. Leaf size=580 \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{b \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}(c x)}\right )}{2 e^2}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 e^2}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 e^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (\frac{d}{x^2}+e\right )}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{b e^2}-\frac{\log \left (e^{-2 \text{sech}^{-1}(c x)}+1\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{e^2}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{2 e^{3/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}} \]
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Rubi [A] time = 1.46331, antiderivative size = 562, normalized size of antiderivative = 0.97, number of steps used = 30, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {6303, 5792, 5660, 3718, 2190, 2279, 2391, 5788, 519, 377, 208, 5800, 5562} \[ \frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{b \text{PolyLog}\left (2,-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{b \text{PolyLog}\left (2,\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}-\frac{b \text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^2}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right )}{2 e^2}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}+1\right )}{2 e^2}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1-\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right )}{2 e^2}+\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (\frac{c \sqrt{-d} e^{\text{sech}^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}+1\right )}{2 e^2}-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (\frac{d}{x^2}+e\right )}-\frac{\log \left (e^{2 \text{sech}^{-1}(c x)}+1\right ) \left (a+b \text{sech}^{-1}(c x)\right )}{e^2}+\frac{b \sqrt{\frac{1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}-1}}\right )}{2 e^{3/2} \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1} \sqrt{c^2 d+e}} \]
Warning: Unable to verify antiderivative.
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Rule 6303
Rule 5792
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rule 5788
Rule 519
Rule 377
Rule 208
Rule 5800
Rule 5562
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{x \left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a+b \cosh ^{-1}\left (\frac{x}{c}\right )}{e^2 x}-\frac{d x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e \left (e+d x^2\right )^2}-\frac{d x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e^2 \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 )}{x} \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{d \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{d \operatorname{Subst}\left (\int \frac{x \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac{1}{x}\right )}{e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (e+\frac{d}{x^2}\right )}-\frac{\operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\text{sech}^{-1}(c x)\right )}{e^2}+\frac{d \operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}\left (\frac{x}{c}\right )\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\frac{1}{x}\right )}{e^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x}{c}} \sqrt{1+\frac{x}{c}} \left (e+d x^2\right )} \, dx,x,\frac{1}{x}\right )}{2 c e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b e^2}-\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text{sech}^{-1}(c x)\right )}{e^2}-\frac{\sqrt{-d} \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 e^2}+\frac{\sqrt{-d} \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 e^2}+\frac{\left (b \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+\frac{x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac{1}{x}\right )}{2 c e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (e+\frac{d}{x^2}\right )}+\frac{\left (a+b \text{sech}^{-1}(c x)\right )^2}{2 b e^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^2}+\frac{b \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text{sech}^{-1}(c x)\right )}{e^2}-\frac{\sqrt{-d} \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 e^2}+\frac{\sqrt{-d} \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 e^2}+\frac{\left (b \sqrt{-1+\frac{1}{c^2 x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{e-\left (d+\frac{e}{c^2}\right ) x^2} \, dx,x,\frac{1}{\sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 c e \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (e+\frac{d}{x^2}\right )}+\frac{b \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 e^{3/2} \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^2}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^2}-\frac{\sqrt{-d} \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 e^2}-\frac{\sqrt{-d} \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 e^2}+\frac{\sqrt{-d} \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 e^2}+\frac{\sqrt{-d} \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 e^2}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (e+\frac{d}{x^2}\right )}+\frac{b \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 e^{3/2} \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\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 e^2}+\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 e^2}+\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 e^2}+\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 e^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^2}-\frac{b \text{Li}_2\left (-e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^2}-\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 e^2}-\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 e^2}-\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 e^2}-\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 e^2}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (e+\frac{d}{x^2}\right )}+\frac{b \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 e^{3/2} \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\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 e^2}+\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 e^2}+\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 e^2}+\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 e^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^2}-\frac{b \text{Li}_2\left (-e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^2}-\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 e^2}-\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 e^2}-\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 e^2}-\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 e^2}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{2 e \left (e+\frac{d}{x^2}\right )}+\frac{b \sqrt{-1+\frac{1}{c^2 x^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2 d+e}}{c \sqrt{e} \sqrt{-1+\frac{1}{c^2 x^2}} x}\right )}{2 e^{3/2} \sqrt{c^2 d+e} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}}}+\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 e^2}+\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 e^2}+\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 e^2}+\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 e^2}-\frac{\left (a+b \text{sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text{sech}^{-1}(c x)}\right )}{e^2}+\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 e^2}+\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 e^2}+\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 e^2}+\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 e^2}-\frac{b \text{Li}_2\left (-e^{2 \text{sech}^{-1}(c x)}\right )}{2 e^2}\\ \end{align*}
Mathematica [C] time = 1.21561, size = 1208, normalized size = 2.08 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.461, size = 661, normalized size = 1.1 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{d}{e^{3} x^{2} + d e^{2}} + \frac{\log \left (e x^{2} + d\right )}{e^{2}}\right )} + b \int \frac{x^{3} \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \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^{3} \operatorname{arsech}\left (c x\right ) + a x^{3}}{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{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{3}}{{\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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