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 = 2.34, antiderivative size = 395, normalized size of antiderivative = 1.00, 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 12
Rule 43
Rule 50
Rule 63
Rule 208
Rule 266
Rule 783
Rule 848
Rule 6310
Rule 6721
Rule 6742
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*}
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Mathematica [A] time = 0.31, size = 214, normalized size = 0.54 \[ -\frac {105 a \sqrt {1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right )+840 b \log \left (c^2 x^3+x\right )+105 b \sqrt {1-c^4 x^4} \left (3 c^8 x^8+4 c^4 x^4+8\right ) \text {csch}^{-1}(c x)-840 b \log \left (c^2 x^2+c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {1-c^4 x^4}+1\right )+\frac {b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {1-c^4 x^4} \left (35 c^8 x^8-5 c^6 x^6+78 c^4 x^4-36 c^2 x^2+768\right )}{c^2 x^2+1}}{3150 c^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 382, normalized size = 0.97 \[ -\frac {105 \, {\left (3 \, b c^{10} x^{10} + 3 \, b c^{8} x^{8} + 4 \, b c^{6} x^{6} + 4 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (35 \, b c^{9} x^{9} - 5 \, b c^{7} x^{7} + 78 \, b c^{5} x^{5} - 36 \, b c^{3} x^{3} + 768 \, b c x\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - 420 \, {\left (b c^{2} x^{2} + b\right )} \log \left (\frac {c^{2} x^{2} + \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + 420 \, {\left (b c^{2} x^{2} + b\right )} \log \left (-\frac {c^{2} x^{2} - \sqrt {-c^{4} x^{4} + 1} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c^{2} x^{2} + 1}\right ) + 105 \, {\left (3 \, a c^{10} x^{10} + 3 \, a c^{8} x^{8} + 4 \, a c^{6} x^{6} + 4 \, a c^{4} x^{4} + 8 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{3150 \, {\left (c^{14} x^{2} + c^{12}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {x^{11} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{30} \, a {\left (\frac {3 \, {\left (-c^{4} x^{4} + 1\right )}^{\frac {5}{2}}}{c^{12}} - \frac {10 \, {\left (-c^{4} x^{4} + 1\right )}^{\frac {3}{2}}}{c^{12}} + \frac {15 \, \sqrt {-c^{4} x^{4} + 1}}{c^{12}}\right )} + \frac {1}{30} \, b {\left (\frac {{\left (3 \, c^{12} x^{12} + c^{8} x^{8} + 4 \, c^{4} x^{4} - 8\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {c^{2} x^{2} + 1} \sqrt {c x + 1} \sqrt {-c x + 1} c^{12}} - 30 \, \int {\left (x^{11} \log \relax (c) + x^{11} \log \relax (x)\right )} e^{\left (-\frac {1}{2} \, \log \left (c^{2} x^{2} + 1\right ) - \frac {1}{2} \, \log \left (c x + 1\right ) - \frac {1}{2} \, \log \left (-c x + 1\right )\right )}\,{d x} - 30 \, \int \frac {3 \, c^{10} x^{11} - 3 \, c^{8} x^{9} + 4 \, c^{6} x^{7} - 4 \, c^{4} x^{5} + 8 \, c^{2} x^{3} - 8 \, x}{30 \, {\left (\sqrt {c^{2} x^{2} + 1} \sqrt {c x + 1} \sqrt {-c x + 1} c^{10} + \sqrt {c x + 1} \sqrt {-c x + 1} c^{10}\right )}}\,{d x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{11}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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