Optimal. Leaf size=58 \[ -\frac {c^2 (a x+1) \sqrt {1-a^2 x^2}}{x}+a c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-a c^2 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.10, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6128, 813, 844, 216, 266, 63, 208} \[ -\frac {c^2 (a x+1) \sqrt {1-a^2 x^2}}{x}+a c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-a c^2 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 813
Rule 844
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x^2} \, dx &=c \int \frac {(c-a c x) \sqrt {1-a^2 x^2}}{x^2} \, dx\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{x}-\frac {1}{2} c \int \frac {2 a c+2 a^2 c x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{x}-\left (a c^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (a^2 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{x}-a c^2 \sin ^{-1}(a x)-\frac {1}{2} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{x}-a c^2 \sin ^{-1}(a x)+\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}\\ &=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{x}-a c^2 \sin ^{-1}(a x)+a c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.19, size = 84, normalized size = 1.45 \[ \frac {1}{2} c^2 \left (\frac {2 (a x-1) (a x+1)^2}{x \sqrt {1-a^2 x^2}}+2 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-a \sin ^{-1}(a x)+2 a \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 91, normalized size = 1.57 \[ \frac {2 \, a c^{2} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - a c^{2} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - a c^{2} x - {\left (a c^{2} x + c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 140, normalized size = 2.41 \[ \frac {a^{4} c^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {a^{2} c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {a^{2} c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \sqrt {-a^{2} x^{2} + 1} a c^{2} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{2 \, x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 91, normalized size = 1.57 \[ -c^{2} a \sqrt {-a^{2} x^{2}+1}-\frac {c^{2} a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+c^{2} a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 80, normalized size = 1.38 \[ -a c^{2} \arcsin \left (a x\right ) + a c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \sqrt {-a^{2} x^{2} + 1} a c^{2} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 87, normalized size = 1.50 \[ -a\,c^2\,\sqrt {1-a^2\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{x}-\frac {a^2\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-a\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.60, size = 153, normalized size = 2.64 \[ a^{3} c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - a c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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