Optimal. Leaf size=66 \[ -3 c \sqrt {1-a^2 x^2}-\frac {1}{2} a c x \sqrt {1-a^2 x^2}+\frac {7}{2} c \text {ArcSin}(a x)-c \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.15, 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 = {6283, 1823,
858, 222, 272, 65, 214} \begin {gather*} -\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 \text {ArcSin}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1823
Rule 6283
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 \text {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 \text {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 = 0.03, size = 49, normalized size = 0.74 \begin {gather*} -\frac {1}{2} c \left ((6+a x) \sqrt {1-a^2 x^2}-7 \text {ArcSin}(a x)+2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs.
\(2(56)=112\).
time = 0.07, size = 222, normalized size = 3.36
method | result | size |
default | \(-c \left (a^{5} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+3 a^{4} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+2 a^{3} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )-\frac {3}{\sqrt {-a^{2} x^{2}+1}}-\frac {3 a x}{\sqrt {-a^{2} x^{2}+1}}+\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(222\) |
meijerg | \(-\frac {2 c \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}+\frac {c \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }}-\frac {a c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {2 a c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}+\frac {3 a c x}{\sqrt {-a^{2} x^{2}+1}}\) | \(288\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 111, normalized size = 1.68 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 69, normalized size = 1.05 \begin {gather*} -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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 8.06, size = 197, normalized size = 2.98 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 76, normalized size = 1.15 \begin {gather*} \frac {7 \, a c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{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 )} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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