Optimal. Leaf size=99 \[ \frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x}{2 a}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{2 a}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x \text {ArcSin}(a x)}{2 a \sqrt {1-a x} \sqrt {1+a x}} \]
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Rubi [A]
time = 0.15, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6294, 6264, 52,
41, 222} \begin {gather*} \frac {3 x \text {ArcSin}(a x) \sqrt {c-\frac {c}{a^2 x^2}}}{2 a \sqrt {1-a x} \sqrt {a x+1}}+\frac {x (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}+\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}}}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 52
Rule 222
Rule 6264
Rule 6294
Rubi steps
\begin {align*} \int 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 e^{-2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x} \, 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}}{\sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{2 a}+\frac {\left (3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1-a x}}{\sqrt {1+a x}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x}{2 a}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{2 a}+\frac {\left (3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x}{2 a}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{2 a}+\frac {\left (3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x}{2 a}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x (1-a x)}{2 a}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}} x \sin ^{-1}(a x)}{2 a \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 100, normalized size = 1.01 \begin {gather*} \frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {1+a x} \left (4-5 a x+a^2 x^2\right )-6 \sqrt {1-a x} \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.05, size = 147, normalized size = 1.48
method | result | size |
risch | \(-\frac {\left (a x -4\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 a}-\frac {3 \ln \left (\frac {c \,a^{2} x}{\sqrt {c \,a^{2}}}+\sqrt {a^{2} c \,x^{2}-c}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 \sqrt {c \,a^{2}}\, \left (a^{2} x^{2}-1\right )}\) | \(114\) |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (-x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}+\sqrt {c}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-4 \sqrt {c}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right )+4 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a \right )}{2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}}\) | \(147\) |
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 = 0.45, size = 188, normalized size = 1.90 \begin {gather*} \left [-\frac {2 \, {\left (a^{2} x^{2} - 4 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \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 )}{4 \, a^{2}}, -\frac {{\left (a^{2} x^{2} - 4 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \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 )}{2 \, a^{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 {x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\right )\, dx - \int \frac {a x^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 106, normalized size = 1.07 \begin {gather*} -\frac {1}{4} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left (\frac {x \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {4 \, \mathrm {sgn}\left (x\right )}{a^{3}}\right )} - \frac {6 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{2} {\left | a \right |}} + \frac {{\left (3 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 8 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\left (x\right )}{a^{3} {\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 {x\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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