Optimal. Leaf size=58 \[ -\frac {2}{5 a^2 x^5}-\frac {2}{5} a^3 \left (\frac {1}{a^2 x^2}+1\right )^{5/2}+\frac {2}{3} a^3 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {1}{3 x^3} \]
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Rubi [A] time = 0.23, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6338, 6742, 266, 43} \[ -\frac {2}{5} a^3 \left (\frac {1}{a^2 x^2}+1\right )^{5/2}+\frac {2}{3} a^3 \left (\frac {1}{a^2 x^2}+1\right )^{3/2}-\frac {2}{5 a^2 x^5}-\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 6338
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^4} \, dx &=\int \frac {\left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2}{x^4} \, dx\\ &=\int \left (\frac {2}{a^2 x^6}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a x^5}+\frac {1}{x^4}\right ) \, dx\\ &=-\frac {2}{5 a^2 x^5}-\frac {1}{3 x^3}+\frac {2 \int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x^5} \, dx}{a}\\ &=-\frac {2}{5 a^2 x^5}-\frac {1}{3 x^3}-\frac {\operatorname {Subst}\left (\int x \sqrt {1+\frac {x}{a^2}} \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=-\frac {2}{5 a^2 x^5}-\frac {1}{3 x^3}-\frac {\operatorname {Subst}\left (\int \left (-a^2 \sqrt {1+\frac {x}{a^2}}+a^2 \left (1+\frac {x}{a^2}\right )^{3/2}\right ) \, dx,x,\frac {1}{x^2}\right )}{a}\\ &=\frac {2}{3} a^3 \left (1+\frac {1}{a^2 x^2}\right )^{3/2}-\frac {2}{5} a^3 \left (1+\frac {1}{a^2 x^2}\right )^{5/2}-\frac {2}{5 a^2 x^5}-\frac {1}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 54, normalized size = 0.93 \[ -\frac {5 a^2 x^2+2 a x \sqrt {\frac {1}{a^2 x^2}+1} \left (-2 a^4 x^4+a^2 x^2+3\right )+6}{15 a^2 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 67, normalized size = 1.16 \[ \frac {4 \, a^{5} x^{5} - 5 \, a^{2} x^{2} + 2 \, {\left (2 \, a^{5} x^{5} - a^{3} x^{3} - 3 \, a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - 6}{15 \, a^{2} x^{5}} \]
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 = 0.05, size = 73, normalized size = 1.26 \[ \frac {-\frac {a^{2}}{3 x^{3}}-\frac {1}{5 x^{5}}}{a^{2}}+\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} x^{2}+1\right ) \left (2 a^{2} x^{2}-3\right )}{15 a \,x^{4}}-\frac {1}{5 x^{5} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 52, normalized size = 0.90 \[ -\frac {2 \, {\left (3 \, a^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 5 \, a^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}}\right )}}{15 \, a} - \frac {1}{3 \, x^{3}} - \frac {2}{5 \, a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 67, normalized size = 1.16 \[ \frac {4\,a^3\,\sqrt {\frac {1}{a^2\,x^2}+1}}{15}-\frac {\frac {2\,a\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{15}+\frac {1}{3}}{x^3}-\frac {\frac {2}{5\,a^2}+\frac {2\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{5\,a}}{x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.07, size = 76, normalized size = 1.31 \[ \frac {4 a^{2} \sqrt {a^{2} x^{2} + 1}}{15 x} - \frac {2 \sqrt {a^{2} x^{2} + 1}}{15 x^{3}} - \frac {1}{3 x^{3}} - \frac {2 \sqrt {a^{2} x^{2} + 1}}{5 a^{2} x^{5}} - \frac {2}{5 a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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