Optimal. Leaf size=114 \[ \frac{1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x^6 \sqrt{1-\frac{1}{c^2 x^2}}}{42 c}+\frac{5 b x^4 \sqrt{1-\frac{1}{c^2 x^2}}}{168 c^3}+\frac{5 b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{112 c^5}+\frac{5 b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{112 c^7} \]
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Rubi [A] time = 0.0614145, antiderivative size = 114, 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 = {5221, 266, 51, 63, 208} \[ \frac{1}{7} x^7 \left (a+b \csc ^{-1}(c x)\right )+\frac{b x^6 \sqrt{1-\frac{1}{c^2 x^2}}}{42 c}+\frac{5 b x^4 \sqrt{1-\frac{1}{c^2 x^2}}}{168 c^3}+\frac{5 b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{112 c^5}+\frac{5 b \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{112 c^7} \]
Antiderivative was successfully verified.
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Rule 5221
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int x^6 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac{1}{7} x^7 \left (a+b \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 = 0.142633, size = 107, normalized size = 0.94 \[ \frac{a x^7}{7}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left (\frac{5 x^4}{168 c^3}+\frac{5 x^2}{112 c^5}+\frac{x^6}{42 c}\right )+\frac{5 b \log \left (x \left (\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{112 c^7}+\frac{1}{7} b x^7 \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 177, normalized size = 1.6 \begin{align*}{\frac{{x}^{7}a}{7}}+{\frac{b{x}^{7}{\rm arccsc} \left (cx\right )}{7}}+{\frac{b{x}^{6}}{42\,c}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{b{x}^{4}}{168\,{c}^{3}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,b{x}^{2}}{336\,{c}^{5}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{5\,b}{112\,{c}^{7}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}+{\frac{5\,b}{112\,{c}^{8}x}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952431, size = 217, normalized size = 1.9 \begin{align*} \frac{1}{7} \, a x^{7} + \frac{1}{672} \,{\left (96 \, x^{7} \operatorname{arccsc}\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 [A] time = 2.70702, size = 275, normalized size = 2.41 \begin{align*} \frac{48 \, a c^{7} x^{7} - 96 \, b c^{7} \arctan \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) + 48 \,{\left (b c^{7} x^{7} - b c^{7}\right )} \operatorname{arccsc}\left (c x\right ) - 15 \, b \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} \sqrt{c^{2} x^{2} - 1}}{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{acsc}{\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{arccsc}\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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