Optimal. Leaf size=401 \[ -\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}-\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{9/2}}{90 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{7/2}}{70 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{13 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{150 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{7 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{90 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{15 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{15 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}} \]
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Rubi [A] time = 2.52723, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {266, 43, 5247, 12, 6721, 6742, 848, 50, 63, 208, 783} \[ -\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}-\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{9/2}}{90 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{3 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{7/2}}{70 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{13 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{150 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{7 b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{90 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{15 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{15 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5247
Rule 12
Rule 6721
Rule 6742
Rule 848
Rule 50
Rule 63
Rule 208
Rule 783
Rubi steps
\begin{align*} \int \frac{x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt{1-c^4 x^4}} \, dx &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac{b \int \frac{\sqrt{1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{30 c^{12} \sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac{b \int \frac{\sqrt{1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{\sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{30 c^{13}}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{1-c^4 x^4} \left (-8-4 c^4 x^4-3 c^8 x^8\right )}{x \sqrt{1-c^2 x^2}} \, dx}{30 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2} \left (8+4 c^4 x^2+3 c^8 x^4\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{60 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8 \sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}}+\frac{4 c^4 x \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}}+\frac{3 c^8 x^3 \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}}\right ) \, dx,x,x^2\right )}{60 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{15 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{15 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^3 \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{20 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+c^2 x}}{x} \, dx,x,x^2\right )}{15 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x \sqrt{1+c^2 x} \, dx,x,x^2\right )}{15 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sqrt{1+c^2 x} \, dx,x,x^2\right )}{20 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{15 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{15 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{1+c^2 x}}{c^2}+\frac{\left (1+c^2 x\right )^{3/2}}{c^2}\right ) \, dx,x,x^2\right )}{15 c^9 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{1+c^2 x}}{c^6}+\frac{3 \left (1+c^2 x\right )^{3/2}}{c^6}-\frac{3 \left (1+c^2 x\right )^{5/2}}{c^6}+\frac{\left (1+c^2 x\right )^{7/2}}{c^6}\right ) \, dx,x,x^2\right )}{20 c^5 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{15 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{7 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{13 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{3 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}-\frac{\left (4 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{15 c^{15} \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{15 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{7 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{90 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{13 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{150 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{3 b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{7/2}}{70 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{9/2}}{90 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{10 c^{12}}+\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{15 c^{13} \sqrt{1-\frac{1}{c^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.290755, size = 194, normalized size = 0.48 \[ -\frac{105 a \sqrt{1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right )+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^4 x^4} \left (35 c^8 x^8+5 c^6 x^6+78 c^4 x^4+36 c^2 x^2+768\right )}{c^2 x^2-1}+840 b \tan ^{-1}\left (\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right )+105 b \sqrt{1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right ) \csc ^{-1}(c x)}{3150 c^{12}} \]
Antiderivative was successfully verified.
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Maple [F] time = 22.796, size = 0, normalized size = 0. \begin{align*} \int{{x}^{11} \left ( a+b{\rm arccsc} \left (cx\right ) \right ){\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{30} \, a{\left (\frac{3 \,{\left (-c^{4} x^{4} + 1\right )}^{\frac{5}{2}}}{c^{12}} - \frac{10 \,{\left (-c^{4} x^{4} + 1\right )}^{\frac{3}{2}}}{c^{12}} + \frac{15 \, \sqrt{-c^{4} x^{4} + 1}}{c^{12}}\right )} + \frac{{\left (c^{12} \int \frac{{\left (3 \, c^{10} x^{11} + 3 \, c^{8} x^{9} + 4 \, c^{6} x^{7} + 4 \, c^{4} x^{5} + 8 \, c^{2} x^{3} + 8 \, x\right )} e^{\left (-\frac{1}{2} \, \log \left (c^{2} x^{2} + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )}}{{\left (c x + 1\right )}{\left (c x - 1\right )} \sqrt{-c x + 1} c^{10} + \sqrt{-c x + 1} c^{10}}\,{d x} -{\left (3 \, c^{8} x^{8} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 4 \, c^{4} x^{4} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 8 \, \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{c x + 1} \sqrt{-c x + 1}\right )} b}{30 \, c^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{11}}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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