Optimal. Leaf size=59 \[ \frac {1}{2} c^2 (2-a x) \sqrt {1-a^2 x^2}-c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} c^2 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.10, antiderivative size = 59, 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, 815, 844, 216, 266, 63, 208} \[ \frac {1}{2} c^2 (2-a x) \sqrt {1-a^2 x^2}-c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} c^2 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 815
Rule 844
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x} \, dx &=c \int \frac {(c-a c x) \sqrt {1-a^2 x^2}}{x} \, dx\\ &=\frac {1}{2} c^2 (2-a x) \sqrt {1-a^2 x^2}-\frac {c \int \frac {-2 a^2 c+a^3 c x}{x \sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=\frac {1}{2} c^2 (2-a x) \sqrt {1-a^2 x^2}+c^2 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\frac {1}{2} \left (a c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{2} c^2 (2-a x) \sqrt {1-a^2 x^2}-\frac {1}{2} c^2 \sin ^{-1}(a x)+\frac {1}{2} c^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} c^2 (2-a x) \sqrt {1-a^2 x^2}-\frac {1}{2} 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^2}\\ &=\frac {1}{2} c^2 (2-a x) \sqrt {1-a^2 x^2}-\frac {1}{2} c^2 \sin ^{-1}(a x)-c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [B] time = 0.08, size = 125, normalized size = 2.12 \[ \frac {c^2 \left (a^3 x^3-2 a^2 x^2+\sqrt {1-a^2 x^2} \sin ^{-1}(a x)+4 \sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-2 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-a x+2\right )}{2 \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 76, normalized size = 1.29 \[ c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac {1}{2} \, {\left (a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.72, size = 84, normalized size = 1.42 \[ -\frac {a c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} - \frac {a c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {1}{2} \, {\left (a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 86, normalized size = 1.46 \[ -\frac {c^{2} a x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {c^{2} a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+c^{2} \sqrt {-a^{2} x^{2}+1}-c^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 76, normalized size = 1.29 \[ -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x - \frac {1}{2} \, c^{2} \arcsin \left (a x\right ) - c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \sqrt {-a^{2} x^{2} + 1} c^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 77, normalized size = 1.31 \[ c^2\,\sqrt {1-a^2\,x^2}-c^2\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {a\,c^2\,x\,\sqrt {1-a^2\,x^2}}{2}-\frac {a\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.93, size = 201, normalized size = 3.41 \[ a^{3} c^{2} \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) + c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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