Optimal. Leaf size=86 \[ \frac{1}{5} x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{20 c}-\frac{3 b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{40 c^3}+\frac{3 b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{40 c^5} \]
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Rubi [A] time = 0.047877, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6284, 266, 51, 63, 208} \[ \frac{1}{5} x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{20 c}-\frac{3 b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{40 c^3}+\frac{3 b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{40 c^5} \]
Antiderivative was successfully verified.
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Rule 6284
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x^4 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{5} x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b \int \frac{x^3}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{5 c}\\ &=\frac{1}{5} x^5 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{10 c}\\ &=\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{40 c^3}\\ &=-\frac{3 b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{40 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{80 c^5}\\ &=-\frac{3 b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{40 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )}{40 c^3}\\ &=-\frac{3 b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{40 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^4}{20 c}+\frac{1}{5} x^5 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{3 b \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{40 c^5}\\ \end{align*}
Mathematica [A] time = 0.0468538, size = 97, normalized size = 1.13 \[ \frac{a x^5}{5}+b \sqrt{\frac{c^2 x^2+1}{c^2 x^2}} \left (\frac{x^4}{20 c}-\frac{3 x^2}{40 c^3}\right )+\frac{3 b \log \left (x \left (\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{40 c^5}+\frac{1}{5} b x^5 \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 108, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{{c}^{5}{x}^{5}a}{5}}+b \left ({\frac{{c}^{5}{x}^{5}{\rm arccsch} \left (cx\right )}{5}}+{\frac{1}{40\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( 2\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}-3\,cx\sqrt{{c}^{2}{x}^{2}+1}+3\,{\it Arcsinh} \left ( cx \right ) \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00616, size = 173, normalized size = 2.01 \begin{align*} \frac{1}{5} \, a x^{5} + \frac{1}{80} \,{\left (16 \, x^{5} \operatorname{arcsch}\left (c x\right ) - \frac{\frac{2 \,{\left (3 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac{3 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac{3 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.3142, size = 443, normalized size = 5.15 \begin{align*} \frac{8 \, a c^{5} x^{5} + 8 \, b c^{5} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 8 \, b c^{5} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 3 \, b \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + 8 \,{\left (b c^{5} x^{5} - b c^{5}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (2 \, b c^{4} x^{4} - 3 \, b c^{2} x^{2}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{40 \, c^{5}} \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 x^{4} \left (a + b \operatorname{acsch}{\left (c x \right )}\right )\, 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{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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