Optimal. Leaf size=149 \[ -\frac {104 a^3 c \sqrt {1-\frac {1}{a^2 x^2}}}{105 \sqrt {c-\frac {c}{a x}}}-\frac {104}{105} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {2 c \sqrt {1-\frac {1}{a^2 x^2}}}{7 \sqrt {c-\frac {c}{a x}} x^3}-\frac {26 a c \sqrt {1-\frac {1}{a^2 x^2}}}{35 \sqrt {c-\frac {c}{a x}} x^2} \]
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Rubi [A]
time = 0.22, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6313, 895, 885,
809, 663} \begin {gather*} -\frac {26 a c \sqrt {1-\frac {1}{a^2 x^2}}}{35 x^2 \sqrt {c-\frac {c}{a x}}}+\frac {2 c \sqrt {1-\frac {1}{a^2 x^2}}}{7 x^3 \sqrt {c-\frac {c}{a x}}}-\frac {104}{105} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}-\frac {104 a^3 c \sqrt {1-\frac {1}{a^2 x^2}}}{105 \sqrt {c-\frac {c}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 809
Rule 885
Rule 895
Rule 6313
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^2 \left (c-\frac {c x}{a}\right )^{3/2}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {2 c \sqrt {1-\frac {1}{a^2 x^2}}}{7 \sqrt {c-\frac {c}{a x}} x^3}-\frac {13}{7} \text {Subst}\left (\int \frac {x^2 \sqrt {c-\frac {c x}{a}}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 c \sqrt {1-\frac {1}{a^2 x^2}}}{7 \sqrt {c-\frac {c}{a x}} x^3}-\frac {26 a c \sqrt {1-\frac {1}{a^2 x^2}}}{35 \sqrt {c-\frac {c}{a x}} x^2}+\frac {1}{35} (52 a) \text {Subst}\left (\int \frac {x \sqrt {c-\frac {c x}{a}}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {104}{105} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {2 c \sqrt {1-\frac {1}{a^2 x^2}}}{7 \sqrt {c-\frac {c}{a x}} x^3}-\frac {26 a c \sqrt {1-\frac {1}{a^2 x^2}}}{35 \sqrt {c-\frac {c}{a x}} x^2}-\frac {1}{105} \left (52 a^2\right ) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {104 a^3 c \sqrt {1-\frac {1}{a^2 x^2}}}{105 \sqrt {c-\frac {c}{a x}}}-\frac {104}{105} a^3 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+\frac {2 c \sqrt {1-\frac {1}{a^2 x^2}}}{7 \sqrt {c-\frac {c}{a x}} x^3}-\frac {26 a c \sqrt {1-\frac {1}{a^2 x^2}}}{35 \sqrt {c-\frac {c}{a x}} x^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 66, normalized size = 0.44 \begin {gather*} -\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} \left (-15+39 a x-52 a^2 x^2+104 a^3 x^3\right )}{105 x^2 (-1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 70, normalized size = 0.47
method | result | size |
gosper | \(-\frac {2 \left (a x +1\right ) \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{105 \left (a x -1\right ) x^{3}}\) | \(70\) |
default | \(-\frac {2 \left (a x +1\right ) \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a x -1}{a x +1}}}{105 \left (a x -1\right ) x^{3}}\) | \(70\) |
risch | \(-\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (104 a^{4} x^{4}+52 a^{3} x^{3}-13 a^{2} x^{2}+24 a x -15\right )}{105 \left (a x -1\right ) x^{3}}\) | \(73\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 77, normalized size = 0.52 \begin {gather*} -\frac {2 \, {\left (104 \, a^{4} x^{4} + 52 \, a^{3} x^{3} - 13 \, a^{2} x^{2} + 24 \, 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.41, size = 100, normalized size = 0.67 \begin {gather*} -\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (104\,a^3\,x^3+156\,a^2\,x^2+143\,a\,x+167\right )\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3}-\frac {304\,\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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