Optimal. Leaf size=86 \[ \frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^5 \sqrt{\frac{1}{c^2 x^2}+1}}{30 c}-\frac{2 b x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^3}+\frac{4 b x \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^5} \]
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Rubi [A] time = 0.0441296, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6284, 271, 191} \[ \frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^5 \sqrt{\frac{1}{c^2 x^2}+1}}{30 c}-\frac{2 b x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^3}+\frac{4 b x \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^5} \]
Antiderivative was successfully verified.
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Rule 6284
Rule 271
Rule 191
Rubi steps
\begin{align*} \int x^5 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b \int \frac{x^4}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{6 c}\\ &=\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(2 b) \int \frac{x^2}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{15 c^3}\\ &=-\frac{2 b \sqrt{1+\frac{1}{c^2 x^2}} x^3}{45 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{(4 b) \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{45 c^5}\\ &=\frac{4 b \sqrt{1+\frac{1}{c^2 x^2}} x}{45 c^5}-\frac{2 b \sqrt{1+\frac{1}{c^2 x^2}} x^3}{45 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \text{csch}^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.127229, size = 72, normalized size = 0.84 \[ \frac{a x^6}{6}+b \sqrt{\frac{c^2 x^2+1}{c^2 x^2}} \left (-\frac{2 x^3}{45 c^3}+\frac{4 x}{45 c^5}+\frac{x^5}{30 c}\right )+\frac{1}{6} b x^6 \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 83, normalized size = 1. \begin{align*}{\frac{1}{{c}^{6}} \left ({\frac{{c}^{6}{x}^{6}a}{6}}+b \left ({\frac{{c}^{6}{x}^{6}{\rm arccsch} \left (cx\right )}{6}}+{\frac{ \left ({c}^{2}{x}^{2}+1 \right ) \left ( 3\,{c}^{4}{x}^{4}-4\,{c}^{2}{x}^{2}+8 \right ) }{90\,cx}{\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 = 1.00425, size = 104, normalized size = 1.21 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{90} \,{\left (15 \, x^{6} \operatorname{arcsch}\left (c x\right ) + \frac{3 \, c^{4} x^{5}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 10 \, c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19157, size = 215, normalized size = 2.5 \begin{align*} \frac{15 \, b c^{5} x^{6} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 15 \, a c^{5} x^{6} +{\left (3 \, b c^{4} x^{5} - 4 \, b c^{2} x^{3} + 8 \, b x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{90 \, 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^{5} \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^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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