Optimal. Leaf size=74 \[ -\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {3 c \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.21, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6157, 6149, 1807, 1809, 844, 216, 266, 63, 208} \[ -\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {3 c \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1807
Rule 1809
Rule 6149
Rule 6157
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\frac {c \int \frac {e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \frac {(1-a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c \int \frac {3 a-3 a^2 x+a^3 x^2}{x \sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c \int \frac {-3 a^3+3 a^4 x}{x \sqrt {1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-(3 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {(3 c) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \sin ^{-1}(a x)}{a}+\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \sin ^{-1}(a x)}{a}-\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^3}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \sin ^{-1}(a x)}{a}-\frac {3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 57, normalized size = 0.77 \[ -\frac {c \left (\sqrt {1-a^2 x^2} (a x-1)+3 a x \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+3 a x \sin ^{-1}(a x)\right )}{a^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 86, normalized size = 1.16 \[ \frac {6 \, a c x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a c x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - a c x - \sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 130, normalized size = 1.76 \[ -\frac {a^{2} c x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {3 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {3 \, c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{2 \, a^{2} x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 266, normalized size = 3.59 \[ \frac {c \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{a}+\frac {3 c \sqrt {-a^{2} x^{2}+1}}{a}-\frac {3 c \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}+\frac {c \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{a^{2} x}+c x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}+\frac {3 c x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {3 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-\frac {2 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {3 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a}-\frac {9 c \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{2}-\frac {9 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 82, normalized size = 1.11 \[ \frac {c\,\sqrt {1-a^2\,x^2}}{a^2\,x}-\frac {3\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c\,\sqrt {1-a^2\,x^2}}{a}+\frac {c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{a} \]
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 \[ \frac {c \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\right )\, dx + \int \frac {a x \sqrt {- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\, dx + \int \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\, dx + \int \left (- \frac {a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{2} x^{4} + 2 a x^{3} + x^{2}}\right )\, dx\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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