Optimal. Leaf size=95 \[ \frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c}+\frac {3 \text {ArcSin}(a x)}{a^3 c} \]
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Rubi [A]
time = 0.22, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6283, 1649, 21,
683, 655, 222} \begin {gather*} \frac {3 \text {ArcSin}(a x)}{a^3 c}+\frac {(a x+1)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (a x+1)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 222
Rule 655
Rule 683
Rule 1649
Rule 6283
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^2 (1+a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {\left (\frac {3}{a^2}+\frac {3 x}{a}\right ) (1+a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^2}{a^3 c \sqrt {1-a^2 x^2}}+\frac {3 \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c}+\frac {3 \sin ^{-1}(a x)}{a^3 c}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 78, normalized size = 0.82 \begin {gather*} \frac {14-5 a x-16 a^2 x^2+3 a^3 x^3+9 (-1+a x) \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a^3 c (-1+a x) \sqrt {1-a^2 x^2}} \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. \(206\) vs.
\(2(87)=174\).
time = 0.08, size = 207, normalized size = 2.18
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a^{3} \sqrt {-a^{2} x^{2}+1}\, c}+\frac {\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {13 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a^{4} \left (x -\frac {1}{a}\right )}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}}{c}\) | \(147\) |
default | \(-\frac {a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {7 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {4}{a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {4}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}+\frac {4 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}}{a^{3}}}{c}\) | \(207\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 364 vs.
\(2 (87) = 174\).
time = 0.54, size = 364, normalized size = 3.83 \begin {gather*} \frac {1}{6} \, {\left (\frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{8} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4}} - \frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{8} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4}} + \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac {6 \, x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2} c} - \frac {26 \, x}{\sqrt {-a^{2} x^{2} + 1} a^{3} c} + \frac {18 \, \arcsin \left (a x\right )}{a^{4} c} - \frac {36}{\sqrt {-a^{2} x^{2} + 1} a^{4} c}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 103, normalized size = 1.08 \begin {gather*} -\frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{5} c x^{2} - 2 \, a^{4} c x + a^{3} c\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*} \frac {\int \frac {x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \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 [B]
time = 0.94, size = 133, normalized size = 1.40 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^5\,x^2-2\,c\,a^4\,x+c\,a^3\right )}+\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\left (a\,c\,\sqrt {-a^2}-a^2\,c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3\,c}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,c\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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