Optimal. Leaf size=73 \[ \frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}+\frac {3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {3 c \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.21, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6157, 6148, 1807, 1809, 844, 216, 266, 63, 208} \[ \frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}+\frac {3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}-\frac {3 c \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1807
Rule 1809
Rule 6148
Rule 6157
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\frac {c \int \frac {e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \frac {(1+a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c \int \frac {-3 a-3 a^2 x-a^3 x^2}{x \sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c \int \frac {3 a^3+3 a^4 x}{x \sqrt {1-a^2 x^2}} \, dx}{a^4}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-(3 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\frac {(3 c) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{a}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \sin ^{-1}(a x)}{a}-\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \sin ^{-1}(a x)}{a}+\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^3}\\ &=\frac {c \sqrt {1-a^2 x^2}}{a}+\frac {c \sqrt {1-a^2 x^2}}{a^2 x}-\frac {3 c \sin ^{-1}(a x)}{a}+\frac {3 c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 56, normalized size = 0.77 \[ \frac {c \left (\sqrt {1-a^2 x^2} (a x+1)+3 a x \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-3 a x \sin ^{-1}(a x)\right )}{a^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 82, normalized size = 1.12 \[ \frac {6 \, a c x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 3 \, a c x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + a c x + \sqrt {-a^{2} x^{2} + 1} {\left (a c x + c\right )}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 129, normalized size = 1.77 \[ -\frac {a^{2} c x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {3 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {3 \, c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{2 \, a^{2} x {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 121, normalized size = 1.66 \[ -\frac {c a \,x^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {c}{a \sqrt {-a^{2} x^{2}+1}}-\frac {c x}{\sqrt {-a^{2} x^{2}+1}}-\frac {3 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {3 c \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{a}+\frac {c}{a^{2} x \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 200, normalized size = 2.74 \[ -a^{3} c {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + 3 \, a^{2} c {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} - \frac {2 \, c x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, c {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} - \frac {{\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c}{a^{2}} + \frac {2 \, c}{\sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 81, normalized size = 1.11 \[ \frac {c\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c\,\sqrt {1-a^2\,x^2}}{a^2\,x}-\frac {c\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.33, size = 150, normalized size = 2.05 \[ - a c \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - 3 c \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - \frac {3 c \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {c \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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