Optimal. Leaf size=58 \[ -\frac{2}{5} a^3 \left (\frac{1}{a^2 x^2}+1\right )^{5/2}+\frac{2}{3} a^3 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{2}{5 a^2 x^5}-\frac{1}{3 x^3} \]
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Rubi [A] time = 0.233002, antiderivative size = 58, normalized size of antiderivative = 1., 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 6338
Rule 6742
Rule 266
Rule 43
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*}
Mathematica [A] time = 0.0536828, 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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Maple [A] time = 0.184, size = 73, normalized size = 1.3 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{1}{5\,{x}^{5}}}-{\frac{{a}^{2}}{3\,{x}^{3}}} \right ) }+{\frac{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ) \left ( 2\,{a}^{2}{x}^{2}-3 \right ) }{15\,{x}^{4}a}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}-{\frac{1}{5\,{x}^{5}{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00086, size = 70, normalized size = 1.21 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.68057, size = 146, normalized size = 2.52 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.76731, size = 76, normalized size = 1.31 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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