Optimal. Leaf size=118 \[ -\sqrt {c-\frac {c}{a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x \text {ArcSin}(a x)}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}} \]
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Rubi [A]
time = 0.28, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6294, 6264,
100, 163, 41, 222, 94, 214} \begin {gather*} \frac {a x \text {ArcSin}(a x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a x} \sqrt {a x+1}}-\sqrt {c-\frac {c}{a^2 x^2}}+\frac {2 a x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{\sqrt {1-a x} \sqrt {a x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 94
Rule 100
Rule 163
Rule 214
Rule 222
Rule 6264
Rule 6294
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^2} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{x^2 \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\sqrt {c-\frac {c}{a^2 x^2}}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {2 a-a^2 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\sqrt {c-\frac {c}{a^2 x^2}}-\frac {\left (2 a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\sqrt {c-\frac {c}{a^2 x^2}}+\frac {\left (a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (2 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\sqrt {c-\frac {c}{a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x \sin ^{-1}(a x)}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 83, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2}+2 a x \text {ArcTan}\left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+a x \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{\sqrt {-1+a^2 x^2}} \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. \(304\) vs.
\(2(100)=200\).
time = 0.06, size = 305, normalized size = 2.58
method | result | size |
risch | \(-\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}-\frac {\left (\frac {a^{2} \ln \left (\frac {c \,a^{2} x}{\sqrt {c \,a^{2}}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {c \,a^{2}}}+\frac {2 a \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{\sqrt {-c}}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{a^{2} x^{2}-1}\) | \(148\) |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (-\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{3} c \,x^{2}+a^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}-2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c x \,a^{2} \sqrt {-\frac {c}{a^{2}}}+2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c x -2 c^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a x +c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {-\frac {c}{a^{2}}}\, a x -2 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{2} x \right )}{a \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c \sqrt {-\frac {c}{a^{2}}}}\) | \(305\) |
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 = 0.40, size = 255, normalized size = 2.16 \begin {gather*} \left [\sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}, -2 \, \sqrt {c} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \frac {1}{2} \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - \sqrt {\frac {a^{2} c x^{2} - c}{a^{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 \left (- \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{2} + x}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{2} + x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 126, normalized size = 1.07 \begin {gather*} {\left (\frac {4 \, \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a} + \frac {\sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{{\left | a \right |}} - \frac {2 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )} {\left | a \right |}}\right )} {\left | a \right |} \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-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )}{x\,{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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