Optimal. Leaf size=188 \[ \frac{a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2}}{7 c^2}+\frac{59 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{35 c}+\frac{472}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\frac{1888 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{105 \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.413827, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {6178, 1635, 795, 657, 649} \[ \frac{a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2}}{7 c^2}+\frac{59 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{35 c}+\frac{472}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\frac{1888 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{105 \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 1635
Rule 795
Rule 657
Rule 649
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (c-\frac{c x}{a}\right )^{7/2}}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{7 a^2}{2}-a x\right ) \left (c-\frac{c x}{a}\right )^{5/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2}}{7 c^2}+\frac{a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\left (59 a^2\right ) \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{5/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{14 c^2}\\ &=\frac{59 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{35 c}+\frac{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2}}{7 c^2}+\frac{a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\left (236 a^2\right ) \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{35 c}\\ &=\frac{472}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\frac{59 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{35 c}+\frac{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2}}{7 c^2}+\frac{a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{1}{105} \left (944 a^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1888 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{105 \sqrt{c-\frac{c}{a x}}}+\frac{472}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\frac{59 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{35 c}+\frac{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2}}{7 c^2}+\frac{a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.123501, size = 78, normalized size = 0.41 \[ \frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (1336 a^4 x^4+668 a^3 x^3-167 a^2 x^2+66 a x-15\right ) \sqrt{c-\frac{c}{a x}}}{105 x^2 \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.115, size = 78, normalized size = 0.4 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 1336\,{x}^{4}{a}^{4}+668\,{x}^{3}{a}^{3}-167\,{a}^{2}{x}^{2}+66\,ax-15 \right ) }{105\,{x}^{3} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70569, size = 173, normalized size = 0.92 \begin{align*} \frac{2 \,{\left (1336 \, a^{4} x^{4} + 668 \, a^{3} x^{3} - 167 \, a^{2} x^{2} + 66 \, a x - 15\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{105 \,{\left (a x^{4} - x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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