Optimal. Leaf size=66 \[ -a c \sqrt {1-a^2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{x}-3 a c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+3 a c \sin ^{-1}(a x) \]
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Rubi [A] time = 0.19, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6148, 1807, 1809, 844, 216, 266, 63, 208} \[ -a c \sqrt {1-a^2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{x}-3 a c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+3 a c \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1807
Rule 1809
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^2} \, dx &=c \int \frac {(1+a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{x}-c \int \frac {-3 a-3 a^2 x-a^3 x^2}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-a c \sqrt {1-a^2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{x}+\frac {c \int \frac {3 a^3+3 a^4 x}{x \sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-a c \sqrt {1-a^2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{x}+(3 a c) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx+\left (3 a^2 c\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-a c \sqrt {1-a^2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{x}+3 a c \sin ^{-1}(a x)+\frac {1}{2} (3 a c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-a c \sqrt {1-a^2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{x}+3 a c \sin ^{-1}(a x)-\frac {(3 c) \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}\\ &=-a c \sqrt {1-a^2 x^2}-\frac {c \sqrt {1-a^2 x^2}}{x}+3 a c \sin ^{-1}(a x)-3 a c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 52, normalized size = 0.79 \[ c \left (-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+3 a \sin ^{-1}(a x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 80, normalized size = 1.21 \[ -\frac {6 \, a c x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 3 \, a c x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + a c x + \sqrt {-a^{2} x^{2} + 1} {\left (a c x + c\right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 131, normalized size = 1.98 \[ \frac {a^{4} c x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} + \frac {3 \, a^{2} c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {3 \, a^{2} c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \sqrt {-a^{2} x^{2} + 1} a c - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{2 \, x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 122, normalized size = 1.85 \[ \frac {c \,a^{3} x^{2}}{\sqrt {-a^{2} x^{2}+1}}-\frac {c a}{\sqrt {-a^{2} x^{2}+1}}+\frac {c \,a^{2} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {3 c \,a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-3 c a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {c}{x \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 111, normalized size = 1.68 \[ \frac {a^{3} c x^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {a^{2} c x}{\sqrt {-a^{2} x^{2} + 1}} + 3 \, a c \arcsin \left (a x\right ) - 3 \, a c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {a c}{\sqrt {-a^{2} x^{2} + 1}} - \frac {c}{\sqrt {-a^{2} x^{2} + 1} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 79, normalized size = 1.20 \[ \frac {3\,a^2\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-a\,c\,\sqrt {1-a^2\,x^2}-\frac {c\,\sqrt {1-a^2\,x^2}}{x}+a\,c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.98, size = 150, normalized size = 2.27 \[ a^{3} c \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) + 3 a c \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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