Optimal. Leaf size=58 \[ \frac {c (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c \text {ArcSin}(a x)}{a}-\frac {c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a} \]
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Rubi [A]
time = 0.07, 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 = {6292, 6284,
827, 858, 222, 272, 65, 214} \begin {gather*} \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 \text {ArcSin}(a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 827
Rule 858
Rule 6284
Rule 6292
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 \text {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 \text {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 = 0.03, size = 55, normalized size = 0.95 \begin {gather*} \frac {c \left ((1+a x) \sqrt {1-a^2 x^2}+a x \text {ArcSin}(a x)-a x \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right )}{a^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 101, normalized size = 1.74
method | result | size |
risch | \(-\frac {\left (a^{2} x^{2}-1\right ) c}{x \sqrt {-a^{2} x^{2}+1}\, a^{2}}+\frac {\left (a \sqrt {-a^{2} x^{2}+1}+\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) c}{a^{2}}\) | \(97\) |
default | \(\frac {c \left (a \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}+2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{a^{2}}\) | \(101\) |
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 = 0.41, size = 85, normalized size = 1.47 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.98, size = 177, normalized size = 3.05 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (54) = 108\).
time = 0.42, size = 128, normalized size = 2.21 \begin {gather*} -\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}\left (a\right )}{{\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 |}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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