Optimal. Leaf size=98 \[ -\frac {x (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}-\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}+\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{2 a \sqrt {1-a x} \sqrt {a x+1}} \]
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Rubi [A] time = 0.21, antiderivative size = 98, 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 = {6159, 6129, 50, 41, 216} \[ -\frac {x (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}-\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}}}{2 a}+\frac {3 x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{2 a \sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 216
Rule 6129
Rule 6159
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 = 0.06, size = 77, normalized size = 0.79 \[ -\frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 188, normalized size = 1.92 \[ \left [-\frac {2 \, {\left (a^{2} x^{2} + 4 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{4 \, a^{2}}, -\frac {{\left (a^{2} x^{2} + 4 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 3 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{2 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 106, normalized size = 1.08 \[ -\frac {1}{4} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left (\frac {x \mathrm {sgn}\relax (x)}{a^{2}} + \frac {4 \, \mathrm {sgn}\relax (x)}{a^{3}}\right )} - \frac {6 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a^{2} {\left | a \right |}} + \frac {{\left (3 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) - 8 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\relax (x)}{a^{3} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 147, normalized size = 1.50 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (-x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}+\sqrt {c}\, \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )-4 \sqrt {c}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right )-4 \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, a \right )}{2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}} x}{a^{2} x^{2} - 1}\,{d x} \]
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 \[ \int -\frac {x\,\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
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 \[ - \int \frac {x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x - 1}\, dx - \int \frac {a x^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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