Optimal. Leaf size=72 \[ \frac {x^2}{2 c^3}+\frac {x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{3 c^2}-\frac {\log \left (c^2 x^2+1\right )}{2 c^5}-\frac {2 x \sqrt {\frac {1}{c^2 x^2}+1}}{3 c^4} \]
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Rubi [A] time = 0.10, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6342, 271, 191, 266, 43} \[ \frac {x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{3 c^2}+\frac {x^2}{2 c^3}-\frac {2 x \sqrt {\frac {1}{c^2 x^2}+1}}{3 c^4}-\frac {\log \left (c^2 x^2+1\right )}{2 c^5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 191
Rule 266
Rule 271
Rule 6342
Rubi steps
\begin {align*} \int \frac {e^{\text {csch}^{-1}(c x)} x^4}{1+c^2 x^2} \, dx &=\frac {\int \frac {x^2}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{c^2}+\frac {\int \frac {x^3}{1+c^2 x^2} \, dx}{c}\\ &=\frac {\sqrt {1+\frac {1}{c^2 x^2}} x^3}{3 c^2}-\frac {2 \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{3 c^4}+\frac {\operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {2 \sqrt {1+\frac {1}{c^2 x^2}} x}{3 c^4}+\frac {\sqrt {1+\frac {1}{c^2 x^2}} x^3}{3 c^2}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{2 c}\\ &=-\frac {2 \sqrt {1+\frac {1}{c^2 x^2}} x}{3 c^4}+\frac {x^2}{2 c^3}+\frac {\sqrt {1+\frac {1}{c^2 x^2}} x^3}{3 c^2}-\frac {\log \left (1+c^2 x^2\right )}{2 c^5}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 64, normalized size = 0.89 \[ \frac {c x \left (2 c^2 x^2 \sqrt {\frac {1}{c^2 x^2}+1}-4 \sqrt {\frac {1}{c^2 x^2}+1}+3 c x\right )-3 \log \left (c^2 x^2+1\right )}{6 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 58, normalized size = 0.81 \[ \frac {3 \, c^{2} x^{2} + 2 \, {\left (c^{3} x^{3} - 2 \, c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - 3 \, \log \left (c^{2} x^{2} + 1\right )}{6 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 85, normalized size = 1.18 \[ -\frac {\log \left (c^{2} x^{2} + 1\right )}{2 \, c^{5}} + \frac {2 \, {\left | c \right |} \mathrm {sgn}\relax (x)}{3 \, c^{6}} + \frac {2 \, {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{12} {\left | c \right |} \mathrm {sgn}\relax (x) - 6 \, \sqrt {c^{2} x^{2} + 1} c^{12} {\left | c \right |} \mathrm {sgn}\relax (x) + 3 \, {\left (c^{2} x^{2} + 1\right )} c^{13}}{6 \, c^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 120, normalized size = 1.67 \[ \frac {\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \left (\left (\frac {c^{2} x^{2}+1}{c^{2}}\right )^{\frac {3}{2}} c^{2}-3 \sqrt {-\frac {\left (-c^{2} x +\sqrt {-c^{2}}\right ) \left (c^{2} x +\sqrt {-c^{2}}\right )}{c^{4}}}\right )}{3 c^{4} \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}}+\frac {x^{2}}{2 c^{3}}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 49, normalized size = 0.68 \[ \frac {x^{2}}{2 \, c^{3}} + \frac {\sqrt {c^{2} x^{2} + 1} {\left (c^{2} x^{2} - 2\right )}}{3 \, c^{5}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{2 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.35, size = 61, normalized size = 0.85 \[ \frac {x^3\,\sqrt {\frac {1}{c^2\,x^2}+1}}{3\,c^2}-\frac {2\,x\,\sqrt {\frac {1}{c^2\,x^2}+1}}{3\,c^4}-\frac {\ln \left (c^2\,x^2+1\right )-c^2\,x^2}{2\,c^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {c x^{4} \sqrt {1 + \frac {1}{c^{2} x^{2}}}}{c^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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