Optimal. Leaf size=96 \[ \frac {a^2 x^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a c^2}+\frac {\text {ArcSin}(a x)}{a c^2} \]
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Rubi [A]
time = 0.11, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6292, 6283,
833, 655, 222} \begin {gather*} -\frac {x (4 a x+3)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a c^2}+\frac {a^2 x^3 (a x+1)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {\text {ArcSin}(a x)}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 655
Rule 833
Rule 6283
Rule 6292
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx &=\frac {a^4 \int \frac {e^{\tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^2} \, dx}{c^2}\\ &=\frac {a^4 \int \frac {x^4 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {a^2 x^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 \int \frac {x^2 (3+4 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac {a^2 x^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {3+8 a x}{\sqrt {1-a^2 x^2}} \, dx}{3 c^2}\\ &=\frac {a^2 x^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a c^2}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {a^2 x^3 (1+a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a c^2}+\frac {\sin ^{-1}(a x)}{a c^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 78, normalized size = 0.81 \begin {gather*} \frac {8-5 a x-7 a^2 x^2+3 a^3 x^3+3 (-1+a x) \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a c^2 (-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. \(215\) vs.
\(2(84)=168\).
time = 0.41, size = 216, normalized size = 2.25
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{6 a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {19 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 a^{6} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{6} \left (x +\frac {1}{a}\right )}\right ) a^{4}}{c^{2}}\) | \(184\) |
default | \(\frac {a^{4} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{2 a^{6}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{6} \left (x +\frac {1}{a}\right )}+\frac {7 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{6} \left (x -\frac {1}{a}\right )}\right )}{c^{2}}\) | \(216\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 142, normalized size = 1.48 \begin {gather*} -\frac {8 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 8 \, a x + 6 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} + 8}{3 \, {\left (a^{4} c^{2} x^{3} - a^{3} c^{2} x^{2} - a^{2} c^{2} x + a c^{2}\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 {a^{4} \int \frac {x^{4}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 188, normalized size = 1.96 \begin {gather*} \frac {a\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}+\frac {\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}+\frac {c^2\,\sqrt {-a^2}}{a}\right )}-\frac {19\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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