Optimal. Leaf size=67 \[ -\frac {c^2 (1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}+\frac {1}{2} a^2 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+a^2 c^2 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.10, antiderivative size = 67, 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, 811, 844, 216, 266, 63, 208} \[ -\frac {c^2 (1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}+\frac {1}{2} a^2 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+a^2 c^2 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 811
Rule 844
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)^2}{x^3} \, dx &=c \int \frac {(c-a c x) \sqrt {1-a^2 x^2}}{x^3} \, dx\\ &=-\frac {c^2 (1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}-\frac {1}{4} c \int \frac {2 a^2 c-4 a^3 c x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c^2 (1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}-\frac {1}{2} \left (a^2 c^2\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx+\left (a^3 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c^2 (1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}+a^2 c^2 \sin ^{-1}(a x)-\frac {1}{4} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {c^2 (1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}+a^2 c^2 \sin ^{-1}(a x)+\frac {1}{2} 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 )\\ &=-\frac {c^2 (1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}+a^2 c^2 \sin ^{-1}(a x)+\frac {1}{2} a^2 c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [B] time = 0.11, size = 147, normalized size = 2.19 \[ -\frac {c^2 \left (4 a^3 x^3-2 a^2 x^2+a^2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+10 a^2 x^2 \sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-2 a^2 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-4 a x+2\right )}{4 x^2 \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.01, size = 95, normalized size = 1.42 \[ -\frac {4 \, a^{2} c^{2} x^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + a^{2} c^{2} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (2 \, a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 192, normalized size = 2.87 \[ \frac {a^{3} c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {a^{3} c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {{\left (a^{3} c^{2} - \frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{2}}{x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} + \frac {\frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{2} {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 95, normalized size = 1.42 \[ \frac {c^{2} a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {c^{2} a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {c^{2} a \sqrt {-a^{2} x^{2}+1}}{x}-\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 86, normalized size = 1.28 \[ a^{2} c^{2} \arcsin \left (a x\right ) + \frac {1}{2} \, a^{2} c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{2}}{x} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 90, normalized size = 1.34 \[ \frac {a\,c^2\,\sqrt {1-a^2\,x^2}}{x}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{2\,x^2}+\frac {a^3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {a^2\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.71, size = 226, normalized size = 3.37 \[ a^{3} c^{2} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a}{2 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{2 a x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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