Optimal. Leaf size=137 \[ \frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}-\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8-5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}+\frac {3 \sqrt {c} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4} \]
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Rubi [A]
time = 0.29, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6302, 6287,
1823, 847, 794, 223, 209} \begin {gather*} \frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}-\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {3 \sqrt {c} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4}+\frac {3 (8-5 a x) \sqrt {c-a^2 c x^2}}{20 a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 794
Rule 847
Rule 1823
Rule 6287
Rule 6302
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac {x^3 (1-a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x^3 \left (-9 a^2 c+10 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{5 a^2}\\ &=-\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}-\frac {\int \frac {x^2 \left (-30 a^3 c^2+36 a^4 c^2 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{20 a^4 c}\\ &=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}-\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x \left (-72 a^4 c^3+90 a^5 c^3 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{60 a^6 c^2}\\ &=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}-\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8-5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}+\frac {(3 c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{4 a^3}\\ &=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}-\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8-5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}+\frac {(3 c) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^3}\\ &=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}-\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8-5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}+\frac {3 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 96, normalized size = 0.70 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \left (24-15 a x+12 a^2 x^2-10 a^3 x^3+4 a^4 x^4\right )-15 \sqrt {c} \text {ArcTan}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{20 a^4} \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. \(260\) vs.
\(2(113)=226\).
time = 0.20, size = 261, normalized size = 1.91
method | result | size |
risch | \(-\frac {\left (4 a^{4} x^{4}-10 a^{3} x^{3}+12 a^{2} x^{2}-15 a x +24\right ) \left (a^{2} x^{2}-1\right ) c}{20 a^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c}{4 a^{3} \sqrt {a^{2} c}}\) | \(97\) |
default | \(-\frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{5 a^{2} c}-\frac {4 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,a^{4}}-\frac {2 \left (-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 a^{2} c}+\frac {\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}}{4 a^{2}}\right )}{a}-\frac {2 \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{a^{3}}+\frac {2 \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}+\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}}{a^{4}}\) | \(261\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 117, normalized size = 0.85 \begin {gather*} -\frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2}}{5 \, a^{2} c} - \frac {5 \, \sqrt {-a^{2} c x^{2} + c} x}{4 \, a^{3}} + \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{2 \, a^{3} c} + \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{4 \, a^{4}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{4}} - \frac {4 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{5 \, a^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 184, normalized size = 1.34 \begin {gather*} \left [\frac {2 \, {\left (4 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 15 \, a x + 24\right )} \sqrt {-a^{2} c x^{2} + c} + 15 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{40 \, a^{4}}, \frac {{\left (4 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 15 \, a x + 24\right )} \sqrt {-a^{2} c x^{2} + c} - 15 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{20 \, a^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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