Optimal. Leaf size=65 \[ -\frac {c^2 (a x+1) \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {c^2 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.12, antiderivative size = 65, 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 = {6131, 6128, 813, 844, 216, 266, 63, 208} \[ -\frac {c^2 (a x+1) \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {c^2 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 813
Rule 844
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{\tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2} \, dx}{a^2}\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^2 \int \frac {2 a+2 a^2 x}{x \sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}-c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {c^2 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^2 \sin ^{-1}(a x)}{a}-\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^2 \sin ^{-1}(a x)}{a}+\frac {c^2 \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^2 (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^2 \sin ^{-1}(a x)}{a}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 78, normalized size = 1.20 \[ -\frac {c^2 \left (a x \sqrt {1-a^2 x^2}+\sqrt {1-a^2 x^2}-a x \log \left (\sqrt {1-a^2 x^2}+1\right )+a x \log (a x)+a x \sin ^{-1}(a x)\right )}{a^2 x} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 94, normalized size = 1.45 \[ \frac {2 \, a c^{2} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - a c^{2} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - a c^{2} x - {\left (a c^{2} x + c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 139, normalized size = 2.14 \[ \frac {a^{2} c^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{2 \, a^{2} x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 95, normalized size = 1.46 \[ -\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{a}-\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {c^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}-\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 89, normalized size = 1.37 \[ -\frac {c^{2} \arcsin \left (a x\right )}{a} + \frac {c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 91, normalized size = 1.40 \[ -\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a^2\,x}-\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.26, size = 151, normalized size = 2.32 \[ a c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - c^{2} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - \frac {c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} + \frac {c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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