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 = 0.41, antiderivative size = 188, normalized size of antiderivative = 1.00, 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 649
Rule 657
Rule 795
Rule 1635
Rule 6178
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*}
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Mathematica [A] time = 0.13, 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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fricas [A] time = 0.82, size = 77, normalized size = 0.41 \[ \frac {2 \, {\left (1336 \, a^{4} x^{4} + 668 \, a^{3} x^{3} - 167 \, a^{2} x^{2} + 66 \, a x - 15\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a x^{4} - x^{3}\right )}} \]
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 = 78, normalized size = 0.41 \[ \frac {2 \left (a x +1\right ) \left (1336 x^{4} a^{4}+668 x^{3} a^{3}-167 a^{2} x^{2}+66 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{105 x^{3} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 100, normalized size = 0.53 \[ \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (1336\,a^3\,x^3+2004\,a^2\,x^2+1837\,a\,x+1903\right )\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3}+\frac {3776\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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