Optimal. Leaf size=128 \[ \frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (1+a x)^2}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2}}{a c^2}-\frac {5 \text {ArcSin}(a x)}{a c^2} \]
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Rubi [A]
time = 0.17, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6266, 6263,
866, 1649, 21, 683, 655, 222} \begin {gather*} \frac {(a x+1)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (a x+1)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (a x+1)^2}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2}}{a c^2}-\frac {5 \text {ArcSin}(a x)}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 222
Rule 655
Rule 683
Rule 866
Rule 1649
Rule 6263
Rule 6266
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=\frac {a^2 \int \frac {e^{3 \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac {a^2 \int \frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^5} \, dx}{c^2}\\ &=\frac {a^2 \int \frac {x^2 (1+a x)^5}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^2 \int \frac {\left (\frac {5}{a^2}+\frac {5 x}{a}\right ) (1+a x)^4}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {(1+a x)^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5 \int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (1+a x)^2}{3 a c^2 \sqrt {1-a^2 x^2}}-\frac {5 \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (1+a x)^2}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2}}{a c^2}-\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (1+a x)^2}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2}}{a c^2}-\frac {5 \sin ^{-1}(a x)}{a c^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 61, normalized size = 0.48 \begin {gather*} \frac {\frac {\sqrt {1-a^2 x^2} \left (-118+279 a x-188 a^2 x^2+15 a^3 x^3\right )}{(-1+a x)^3}-75 \text {ArcSin}(a x)}{15 a c^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. \(348\) vs.
\(2(114)=228\).
time = 0.76, size = 349, normalized size = 2.73
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}-\frac {\left (\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {4 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{6} \left (x -\frac {1}{a}\right )^{3}}+\frac {52 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 a^{5} \left (x -\frac {1}{a}\right )^{2}}+\frac {143 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2}}{c^{2}}\) | \(193\) |
default | \(\frac {a^{2} \left (a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {25 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {12}{a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {8}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {24 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{4}}+\frac {\frac {28}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {28 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{3}}\right )}{c^{2}}\) | \(349\) |
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 = 143, normalized size = 1.12 \begin {gather*} \frac {118 \, a^{3} x^{3} - 354 \, a^{2} x^{2} + 354 \, a x + 150 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{3} x^{3} - 188 \, a^{2} x^{2} + 279 \, a x - 118\right )} \sqrt {-a^{2} x^{2} + 1} - 118}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, 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^{2} \left (\int \frac {x^{2}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \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^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \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^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \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^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 180, normalized size = 1.41 \begin {gather*} -\frac {5 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c^{2} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{2}} - \frac {2 \, {\left (\frac {440 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {670 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {360 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {75 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 103\right )}}{15 \, c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 270, normalized size = 2.11 \begin {gather*} \frac {8\,a^2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^5\,c^2\,x^2-2\,a^4\,c^2\,x+a^3\,c^2\right )}-\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}+\frac {143\,\sqrt {1-a^2\,x^2}}{15\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {4\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}+a^2\,c^2\,x^3\,\sqrt {-a^2}-3\,a\,c^2\,x^2\,\sqrt {-a^2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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