Optimal. Leaf size=66 \[ -\frac {1}{2} a c x \sqrt {1-a^2 x^2}-3 c \sqrt {1-a^2 x^2}-c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {7}{2} c \sin ^{-1}(a x) \]
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Rubi [A] time = 0.20, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6148, 1809, 844, 216, 266, 63, 208} \[ -\frac {1}{2} a c x \sqrt {1-a^2 x^2}-3 c \sqrt {1-a^2 x^2}-c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {7}{2} c \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
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} \, dx &=c \int \frac {(1+a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {1}{2} a c x \sqrt {1-a^2 x^2}-\frac {c \int \frac {-2 a^2-7 a^3 x-6 a^4 x^2}{x \sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=-3 c \sqrt {1-a^2 x^2}-\frac {1}{2} a c x \sqrt {1-a^2 x^2}+\frac {c \int \frac {2 a^4+7 a^5 x}{x \sqrt {1-a^2 x^2}} \, dx}{2 a^4}\\ &=-3 c \sqrt {1-a^2 x^2}-\frac {1}{2} a c x \sqrt {1-a^2 x^2}+c \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx+\frac {1}{2} (7 a c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-3 c \sqrt {1-a^2 x^2}-\frac {1}{2} a c x \sqrt {1-a^2 x^2}+\frac {7}{2} c \sin ^{-1}(a x)+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-3 c \sqrt {1-a^2 x^2}-\frac {1}{2} a c x \sqrt {1-a^2 x^2}+\frac {7}{2} c \sin ^{-1}(a x)-\frac {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^2}\\ &=-3 c \sqrt {1-a^2 x^2}-\frac {1}{2} a c x \sqrt {1-a^2 x^2}+\frac {7}{2} c \sin ^{-1}(a x)-c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 0.74 \[ -\frac {1}{2} c \left (\sqrt {1-a^2 x^2} (a x+6)+2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-7 \sin ^{-1}(a x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 69, normalized size = 1.05 \[ -7 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 6 \, c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 76, normalized size = 1.15 \[ \frac {7 \, a c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} - \frac {a c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 6 \, c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 121, normalized size = 1.83 \[ \frac {c \,a^{3} x^{3}}{2 \sqrt {-a^{2} x^{2}+1}}-\frac {c a x}{2 \sqrt {-a^{2} x^{2}+1}}+\frac {7 c a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+\frac {3 c \,a^{2} x^{2}}{\sqrt {-a^{2} x^{2}+1}}-\frac {3 c}{\sqrt {-a^{2} x^{2}+1}}-c \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 111, normalized size = 1.68 \[ \frac {a^{3} c x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c x^{2}}{\sqrt {-a^{2} x^{2} + 1}} - \frac {a c x}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7}{2} \, c \arcsin \left (a x\right ) - c \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, c}{\sqrt {-a^{2} x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 70, normalized size = 1.06 \[ \frac {7\,a\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {a\,c\,x\,\sqrt {1-a^2\,x^2}}{2}-3\,c\,\sqrt {1-a^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.68, size = 197, normalized size = 2.98 \[ a^{3} c \left (\begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) + 3 a c \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) + c \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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