Optimal. Leaf size=85 \[ \frac {2 x^3}{3 a^2}+\frac {x^4 \sqrt {\frac {1}{a^2 x^2}+1}}{2 a}-\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{4 a^5}+\frac {x^2 \sqrt {\frac {1}{a^2 x^2}+1}}{4 a^3}+\frac {x^5}{5} \]
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Rubi [A] time = 0.25, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6338, 6742, 266, 47, 51, 63, 208} \[ \frac {x^4 \sqrt {\frac {1}{a^2 x^2}+1}}{2 a}+\frac {2 x^3}{3 a^2}+\frac {x^2 \sqrt {\frac {1}{a^2 x^2}+1}}{4 a^3}-\frac {\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )}{4 a^5}+\frac {x^5}{5} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rule 6338
Rule 6742
Rubi steps
\begin {align*} \int e^{2 \text {csch}^{-1}(a x)} x^4 \, dx &=\int \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2 x^4 \, dx\\ &=\int \left (\frac {2 x^2}{a^2}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}} x^3}{a}+x^4\right ) \, dx\\ &=\frac {2 x^3}{3 a^2}+\frac {x^5}{5}+\frac {2 \int \sqrt {1+\frac {1}{a^2 x^2}} x^3 \, dx}{a}\\ &=\frac {2 x^3}{3 a^2}+\frac {x^5}{5}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x^3} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a^3}\\ &=\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{4 a^3}+\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{8 a^5}\\ &=\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{4 a^3}+\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}+\frac {\operatorname {Subst}\left (\int \frac {1}{-a^2+a^2 x^2} \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}\right )}{4 a^3}\\ &=\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^2}{4 a^3}+\frac {2 x^3}{3 a^2}+\frac {\sqrt {1+\frac {1}{a^2 x^2}} x^4}{2 a}+\frac {x^5}{5}-\frac {\tanh ^{-1}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right )}{4 a^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 0.99 \[ \frac {a^2 x^2 \left (12 a^3 x^3+30 a^2 x^2 \sqrt {\frac {1}{a^2 x^2}+1}+15 \sqrt {\frac {1}{a^2 x^2}+1}+40 a x\right )-15 \log \left (x \left (\sqrt {\frac {1}{a^2 x^2}+1}+1\right )\right )}{60 a^5} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.67, size = 87, normalized size = 1.02 \[ \frac {12 \, a^{5} x^{5} + 40 \, a^{3} x^{3} + 15 \, {\left (2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 15 \, \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{60 \, a^{5}} \]
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 [A] time = 0.04, size = 114, normalized size = 1.34 \[ \frac {x^{5}}{5}+\frac {2 x^{3}}{3 a^{2}}-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, x \left (-2 x \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} a^{4}+x \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right )\right )}{4 a^{5} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 117, normalized size = 1.38 \[ \frac {1}{5} \, x^{5} + \frac {2 \, x^{3}}{3 \, a^{2}} + \frac {\frac {2 \, {\left ({\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \sqrt {\frac {1}{a^{2} x^{2}} + 1}\right )}}{a^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{2} - 2 \, a^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} + a^{4}} - \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right )}{a^{4}} + \frac {\log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{a^{4}}}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.17, size = 73, normalized size = 0.86 \[ \frac {x^5}{5}+\frac {2\,x^3}{3\,a^2}+\frac {x^4\,\sqrt {\frac {1}{a^2\,x^2}+1}}{2\,a}+\frac {x^2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{4\,a^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.72, size = 82, normalized size = 0.96 \[ \frac {x^{5}}{5} + \frac {x^{5}}{2 \sqrt {a^{2} x^{2} + 1}} + \frac {2 x^{3}}{3 a^{2}} + \frac {3 x^{3}}{4 a^{2} \sqrt {a^{2} x^{2} + 1}} + \frac {x}{4 a^{4} \sqrt {a^{2} x^{2} + 1}} - \frac {\operatorname {asinh}{\left (a x \right )}}{4 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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