Optimal. Leaf size=395 \[ -\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{csch}^{-1}(c x)\right )}{2 c^{12}}-\frac{b \sqrt{c^2 x^2+1} \left (1-c^2 x^2\right )^{9/2}}{90 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{3 b \sqrt{c^2 x^2+1} \left (1-c^2 x^2\right )^{7/2}}{70 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{13 b \sqrt{c^2 x^2+1} \left (1-c^2 x^2\right )^{5/2}}{150 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{7 b \sqrt{c^2 x^2+1} \left (1-c^2 x^2\right )^{3/2}}{90 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{4 b \sqrt{c^2 x^2+1} \sqrt{1-c^2 x^2}}{15 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{4 b \sqrt{c^2 x^2+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{15 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}} \]
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Rubi [A] time = 2.3444, antiderivative size = 395, 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, 6310, 12, 6721, 6742, 848, 50, 63, 208, 783} \[ -\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{10 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{csch}^{-1}(c x)\right )}{2 c^{12}}-\frac{b \sqrt{c^2 x^2+1} \left (1-c^2 x^2\right )^{9/2}}{90 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{3 b \sqrt{c^2 x^2+1} \left (1-c^2 x^2\right )^{7/2}}{70 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{13 b \sqrt{c^2 x^2+1} \left (1-c^2 x^2\right )^{5/2}}{150 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{7 b \sqrt{c^2 x^2+1} \left (1-c^2 x^2\right )^{3/2}}{90 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{4 b \sqrt{c^2 x^2+1} \sqrt{1-c^2 x^2}}{15 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{4 b \sqrt{c^2 x^2+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{15 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 6310
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 \text{csch}^{-1}(c x)\right )}{\sqrt{1-c^4 x^4}} \, dx &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \left (1-c^2 x^2\right )^{3/2} \sqrt{1+c^2 x^2}}{90 c^{13} \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{13 b \left (1-c^2 x^2\right )^{5/2} \sqrt{1+c^2 x^2}}{150 c^{13} \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{3 b \left (1-c^2 x^2\right )^{7/2} \sqrt{1+c^2 x^2}}{70 c^{13} \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{b \left (1-c^2 x^2\right )^{9/2} \sqrt{1+c^2 x^2}}{90 c^{13} \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 \left (1-c^2 x^2\right )^{3/2} \sqrt{1+c^2 x^2}}{90 c^{13} \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{13 b \left (1-c^2 x^2\right )^{5/2} \sqrt{1+c^2 x^2}}{150 c^{13} \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{3 b \left (1-c^2 x^2\right )^{7/2} \sqrt{1+c^2 x^2}}{70 c^{13} \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{b \left (1-c^2 x^2\right )^{9/2} \sqrt{1+c^2 x^2}}{90 c^{13} \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{csch}^{-1}(c x)\right )}{2 c^{12}}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^{12}}-\frac{\left (1-c^4 x^4\right )^{5/2} \left (a+b \text{csch}^{-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 = 0.309092, size = 214, normalized size = 0.54 \[ -\frac{105 a \sqrt{1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right )+\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{1-c^4 x^4} \left (35 c^8 x^8-5 c^6 x^6+78 c^4 x^4-36 c^2 x^2+768\right )}{c^2 x^2+1}+840 b \log \left (c^2 x^3+x\right )-840 b \log \left (c^2 x^2+c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{1-c^4 x^4}+1\right )+105 b \sqrt{1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right ) \text{csch}^{-1}(c x)}{3150 c^{12}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.483, size = 0, normalized size = 0. \begin{align*} \int{{x}^{11} \left ( a+b{\rm arccsch} \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{1}{30} \, b{\left (\frac{{\left (3 \, c^{12} x^{12} + c^{8} x^{8} + 4 \, c^{4} x^{4} - 8\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{\sqrt{c^{2} x^{2} + 1} \sqrt{c x + 1} \sqrt{-c x + 1} c^{12}} - 30 \, \int{\left (x^{11} \log \left (c\right ) + x^{11} \log \left (x\right )\right )} e^{\left (-\frac{1}{2} \, \log \left (c^{2} x^{2} + 1\right ) - \frac{1}{2} \, \log \left (c x + 1\right ) - \frac{1}{2} \, \log \left (-c x + 1\right )\right )}\,{d x} - 30 \, \int \frac{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}{30 \,{\left (\sqrt{c^{2} x^{2} + 1} \sqrt{c x + 1} \sqrt{-c x + 1} c^{10} + \sqrt{c x + 1} \sqrt{-c x + 1} c^{10}\right )}}\,{d x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.68003, size = 846, normalized size = 2.14 \begin{align*} -\frac{105 \,{\left (3 \, b c^{10} x^{10} + 3 \, b c^{8} x^{8} + 4 \, b c^{6} x^{6} + 4 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt{-c^{4} x^{4} + 1} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (35 \, b c^{9} x^{9} - 5 \, b c^{7} x^{7} + 78 \, b c^{5} x^{5} - 36 \, b c^{3} x^{3} + 768 \, b c x\right )} \sqrt{-c^{4} x^{4} + 1} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 420 \,{\left (b c^{2} x^{2} + b\right )} \log \left (\frac{c^{2} x^{2} + \sqrt{-c^{4} x^{4} + 1} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + 420 \,{\left (b c^{2} x^{2} + b\right )} \log \left (-\frac{c^{2} x^{2} - \sqrt{-c^{4} x^{4} + 1} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + 105 \,{\left (3 \, a c^{10} x^{10} + 3 \, a c^{8} x^{8} + 4 \, a c^{6} x^{6} + 4 \, a c^{4} x^{4} + 8 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt{-c^{4} x^{4} + 1}}{3150 \,{\left (c^{14} x^{2} + c^{12}\right )}} \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{arcsch}\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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