Optimal. Leaf size=142 \[ \frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{c x+1}}}+\frac{5 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{112 c^7} \]
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Rubi [A] time = 0.0609944, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6283, 100, 12, 90, 41, 216} \[ \frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{c x+1}}}-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{c x+1}}}+\frac{5 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{112 c^7} \]
Antiderivative was successfully verified.
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Rule 6283
Rule 100
Rule 12
Rule 90
Rule 41
Rule 216
Rubi steps
\begin{align*} \int x^6 \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{7} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^6}{\sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{5 x^4}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{42 c^2}\\ &=-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^4}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{42 c^2}\\ &=-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{3 x^2}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{168 c^4}\\ &=-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{56 c^4}\\ &=-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{1+c x}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c x} \sqrt{1+c x}} \, dx}{112 c^6}\\ &=-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{1+c x}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{\left (5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{112 c^6}\\ &=-\frac{5 b x \sqrt{1-c x}}{112 c^6 \sqrt{\frac{1}{1+c x}}}-\frac{5 b x^3 \sqrt{1-c x}}{168 c^4 \sqrt{\frac{1}{1+c x}}}-\frac{b x^5 \sqrt{1-c x}}{42 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{7} x^7 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{5 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{112 c^7}\\ \end{align*}
Mathematica [C] time = 0.187581, size = 143, normalized size = 1.01 \[ \frac{a x^7}{7}+b \sqrt{\frac{1-c x}{c x+1}} \left (-\frac{x^5}{42 c^2}-\frac{5 x^4}{168 c^3}-\frac{5 x^3}{168 c^4}-\frac{5 x^2}{112 c^5}-\frac{5 x}{112 c^6}-\frac{x^6}{42 c}\right )+\frac{5 i b \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )}{112 c^7}+\frac{1}{7} b x^7 \text{sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.217, size = 138, normalized size = 1. \begin{align*}{\frac{1}{{c}^{7}} \left ({\frac{{c}^{7}{x}^{7}a}{7}}+b \left ({\frac{{c}^{7}{x}^{7}{\rm arcsech} \left (cx\right )}{7}}+{\frac{cx}{336}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \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\,\arcsin \left ( cx \right ) \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49201, size = 182, normalized size = 1.28 \begin{align*} \frac{1}{7} \, a x^{7} + \frac{1}{336} \,{\left (48 \, x^{7} \operatorname{arsech}\left (c x\right ) - \frac{\frac{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}}{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 \, \arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )}{c^{6}}}{c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33069, size = 406, normalized size = 2.86 \begin{align*} \frac{48 \, a c^{7} x^{7} - 48 \, b c^{7} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 30 \, b \arctan \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{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{asech}{\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{arsech}\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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