Optimal. Leaf size=110 \[ \frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^6 \sqrt{\frac{1}{c^2 x^2}+1}}{42 c}-\frac{5 b x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{168 c^3}+\frac{5 b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{112 c^5}-\frac{5 b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{112 c^7} \]
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Rubi [A] time = 0.0608451, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^6 \sqrt{\frac{1}{c^2 x^2}+1}}{42 c}-\frac{5 b x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{168 c^3}+\frac{5 b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{112 c^5}-\frac{5 b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{112 c^7} \]
Antiderivative was successfully verified.
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Rule 6284
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x^6 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b \int \frac{x^5}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{7 c}\\ &=\frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{14 c}\\ &=\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^6}{42 c}+\frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{84 c^3}\\ &=-\frac{5 b \sqrt{1+\frac{1}{c^2 x^2}} x^4}{168 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^6}{42 c}+\frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{112 c^5}\\ &=\frac{5 b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{112 c^5}-\frac{5 b \sqrt{1+\frac{1}{c^2 x^2}} x^4}{168 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^6}{42 c}+\frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{224 c^7}\\ &=\frac{5 b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{112 c^5}-\frac{5 b \sqrt{1+\frac{1}{c^2 x^2}} x^4}{168 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^6}{42 c}+\frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )}{112 c^5}\\ &=\frac{5 b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{112 c^5}-\frac{5 b \sqrt{1+\frac{1}{c^2 x^2}} x^4}{168 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^6}{42 c}+\frac{1}{7} x^7 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{5 b \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{112 c^7}\\ \end{align*}
Mathematica [A] time = 0.136911, size = 107, normalized size = 0.97 \[ \frac{a x^7}{7}+b \sqrt{\frac{c^2 x^2+1}{c^2 x^2}} \left (-\frac{5 x^4}{168 c^3}+\frac{5 x^2}{112 c^5}+\frac{x^6}{42 c}\right )-\frac{5 b \log \left (x \left (\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{112 c^7}+\frac{1}{7} b x^7 \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.205, size = 127, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{7}} \left ({\frac{{c}^{7}{x}^{7}a}{7}}+b \left ({\frac{{c}^{7}{x}^{7}{\rm arccsch} \left (cx\right )}{7}}+{\frac{1}{336\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( 8\,{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}-10\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+15\,cx\sqrt{{c}^{2}{x}^{2}+1}-15\,{\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 = 1.03671, size = 213, normalized size = 1.94 \begin{align*} \frac{1}{7} \, a x^{7} + \frac{1}{672} \,{\left (96 \, x^{7} \operatorname{arcsch}\left (c x\right ) + \frac{\frac{2 \,{\left (15 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{6}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - c^{6}} - \frac{15 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} + \frac{15 \, \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.4366, size = 473, normalized size = 4.3 \begin{align*} \frac{48 \, a c^{7} x^{7} + 48 \, b c^{7} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 48 \, b c^{7} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 15 \, b \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + 48 \,{\left (b c^{7} x^{7} - b c^{7}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (8 \, b c^{6} x^{6} - 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{336 \, c^{7}} \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^{6} \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^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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