Optimal. Leaf size=110 \[ \frac {(a x+1)^2}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 (1-a x) (a x+1)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2 \sqrt {1-a x} \sqrt {a x+1} \sin ^{-1}(a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rubi [A] time = 0.24, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6159, 6129, 78, 50, 41, 216} \[ \frac {(a x+1)^2}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {2 (1-a x) (a x+1)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2 \sqrt {1-a x} \sqrt {a x+1} \sin ^{-1}(a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 78
Rule 216
Rule 6129
Rule 6159
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx &=\frac {\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} x}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {x \sqrt {1+a x}}{(1-a x)^{3/2}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {\sqrt {1+a x}}{\sqrt {1-a x}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}+\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}+\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\\ &=\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}+\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {2 \sqrt {1-a x} \sqrt {1+a x} \sin ^{-1}(a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 69, normalized size = 0.63 \[ \frac {-a^2 x^2-2 \sqrt {a^2 x^2-1} \log \left (\sqrt {a^2 x^2-1}+a x\right )+2 a x+3}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.55, size = 216, normalized size = 1.96 \[ \left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x - a c}, \frac {2 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x - a c}\right ] \]
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 \[ \int -\frac {{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 178, normalized size = 1.62 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \left (\sqrt {c}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x \,a^{2}+2 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x a c -2 a \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, \sqrt {c}-\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a \sqrt {c}-2 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) c \right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,c^{\frac {3}{2}} a \left (a x -1\right )} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}\,{d x} \]
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 \[ \int -\frac {{\left (a\,x+1\right )}^2}{\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {a x}{a x \sqrt {c - \frac {c}{a^{2} x^{2}}} - \sqrt {c - \frac {c}{a^{2} x^{2}}}}\, dx - \int \frac {1}{a x \sqrt {c - \frac {c}{a^{2} x^{2}}} - \sqrt {c - \frac {c}{a^{2} x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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