Optimal. Leaf size=58 \[ \frac {c \sqrt {1-a^2 x^2} (a x+1)}{a^2 x}-\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}+\frac {c \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.11, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6157, 6149, 813, 844, 216, 266, 63, 208} \[ \frac {c \sqrt {1-a^2 x^2} (a x+1)}{a^2 x}-\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}+\frac {c \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 813
Rule 844
Rule 6149
Rule 6157
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\frac {c \int \frac {e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2} \, dx}{a^2}\\ &=\frac {c (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c \int \frac {2 a+2 a^2 x}{x \sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=\frac {c (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}+c \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {c \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=\frac {c (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c \sin ^{-1}(a x)}{a}+\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {c (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c \sin ^{-1}(a x)}{a}-\frac {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 (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c \sin ^{-1}(a x)}{a}-\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 0.95 \[ \frac {c \left (\sqrt {1-a^2 x^2} (a x+1)-a x \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+a x \sin ^{-1}(a x)\right )}{a^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 85, normalized size = 1.47 \[ -\frac {2 \, a c x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - a c x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - a c x - \sqrt {-a^{2} x^{2} + 1} {\left (a c x + c\right )}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 128, normalized size = 2.21 \[ -\frac {a^{2} c x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} + \frac {c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{2 \, a^{2} x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 100, normalized size = 1.72 \[ \frac {c \sqrt {-a^{2} x^{2}+1}}{a}-\frac {c \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}+\frac {c \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{a^{2} x}+c x \sqrt {-a^{2} x^{2}+1}+\frac {c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}} \]
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 \[ c {\left (\frac {\arcsin \left (a x\right )}{a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a}\right )} - c \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1}}{a^{3} x^{3} + a^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 76, normalized size = 1.31 \[ \frac {c\,\sqrt {1-a^2\,x^2}}{a}-\frac {c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{a}+\frac {c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c\,\sqrt {1-a^2\,x^2}}{a^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.91, size = 177, normalized size = 3.05 \[ \frac {c \left (\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {c \left (\begin {cases} - \frac {i a^{2} x}{\sqrt {a^{2} x^{2} - 1}} + i a \operatorname {acosh}{\left (a x \right )} + \frac {i}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} x}{\sqrt {- a^{2} x^{2} + 1}} - a \operatorname {asin}{\left (a x \right )} - \frac {1}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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