Optimal. Leaf size=78 \[ \frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6302, 6286,
1821, 821, 272, 65, 214} \begin {gather*} \frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rule 6286
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx\\ &=-\left (c \int \frac {(1+a x)^2}{x^3 \sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {1}{2} \int \frac {-4 a c-3 a^2 c x}{x^2 \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {1}{2} \left (3 a^2 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {1}{4} \left (3 a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 76, normalized size = 0.97 \begin {gather*} \frac {1}{2} \left (\frac {(1+4 a x) \sqrt {c-a^2 c x^2}}{x^2}-3 a^2 \sqrt {c} \log (x)+3 a^2 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(238\) vs.
\(2(64)=128\).
time = 0.20, size = 239, normalized size = 3.06
method | result | size |
risch | \(-\frac {\left (4 a^{3} x^{3}+a^{2} x^{2}-4 a x -1\right ) c}{2 x^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 a^{2} \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{2}\) | \(78\) |
default | \(-2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )+\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )}{2}+2 a^{2} \left (\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}-\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )\) | \(239\) |
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 = 148, normalized size = 1.90 \begin {gather*} \left [\frac {3 \, a^{2} \sqrt {c} x^{2} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x + 1\right )}}{4 \, x^{2}}, \frac {3 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x + 1\right )}}{2 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{x^{3} \left (a x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 200 vs.
\(2 (64) = 128\).
time = 0.41, size = 200, normalized size = 2.56 \begin {gather*} -\frac {3 \, a^{2} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {{\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{2} c - 4 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a \sqrt {-c} c {\left | a \right |} + {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{2} c^{2} + 4 \, a \sqrt {-c} c^{2} {\left | a \right |}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{x^3\,\left (a\,x-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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