Optimal. Leaf size=114 \[ \frac {(1+a x)^2}{a^4 c \sqrt {1-a^2 x^2}}+\frac {11 \sqrt {1-a^2 x^2}}{3 a^4 c}+\frac {x \sqrt {1-a^2 x^2}}{a^3 c}+\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2 c}-\frac {3 \text {ArcSin}(a x)}{a^4 c} \]
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Rubi [A]
time = 0.20, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6263, 866,
1649, 1829, 655, 222} \begin {gather*} -\frac {3 \text {ArcSin}(a x)}{a^4 c}+\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2 c}+\frac {11 \sqrt {1-a^2 x^2}}{3 a^4 c}+\frac {(a x+1)^2}{a^4 c \sqrt {1-a^2 x^2}}+\frac {x \sqrt {1-a^2 x^2}}{a^3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 655
Rule 866
Rule 1649
Rule 1829
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{c-a c x} \, dx &=c \int \frac {x^3 \sqrt {1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=\frac {\int \frac {x^3 (c+a c x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {(1+a x)^2}{a^4 c \sqrt {1-a^2 x^2}}-\frac {\int \frac {(c+a c x) \left (\frac {2}{a^3}+\frac {x}{a^2}+\frac {x^2}{a}\right )}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {(1+a x)^2}{a^4 c \sqrt {1-a^2 x^2}}+\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2 c}+\frac {\int \frac {-\frac {6 c}{a}-11 c x-6 a c x^2}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2 c^2}\\ &=\frac {(1+a x)^2}{a^4 c \sqrt {1-a^2 x^2}}+\frac {x \sqrt {1-a^2 x^2}}{a^3 c}+\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2 c}-\frac {\int \frac {18 a c+22 a^2 c x}{\sqrt {1-a^2 x^2}} \, dx}{6 a^4 c^2}\\ &=\frac {(1+a x)^2}{a^4 c \sqrt {1-a^2 x^2}}+\frac {11 \sqrt {1-a^2 x^2}}{3 a^4 c}+\frac {x \sqrt {1-a^2 x^2}}{a^3 c}+\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2 c}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 c}\\ &=\frac {(1+a x)^2}{a^4 c \sqrt {1-a^2 x^2}}+\frac {11 \sqrt {1-a^2 x^2}}{3 a^4 c}+\frac {x \sqrt {1-a^2 x^2}}{a^3 c}+\frac {x^2 \sqrt {1-a^2 x^2}}{3 a^2 c}-\frac {3 \sin ^{-1}(a x)}{a^4 c}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 72, normalized size = 0.63 \begin {gather*} \frac {-\frac {\sqrt {1+a x} \left (-14+5 a x+2 a^2 x^2+a^3 x^3\right )}{\sqrt {1-a x}}+18 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{3 a^4 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 169, normalized size = 1.48
method | result | size |
risch | \(-\frac {\left (a^{2} x^{2}+3 a x +8\right ) \left (a^{2} x^{2}-1\right )}{3 a^{4} \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{5} \left (x -\frac {1}{a}\right )}}{c}\) | \(121\) |
default | \(-\frac {-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}+\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}}{a}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{5} \left (x -\frac {1}{a}\right )}}{c}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 105, normalized size = 0.92 \begin {gather*} -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{5} c x - a^{4} c} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a^{2} c} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{3} c} - \frac {3 \, \arcsin \left (a x\right )}{a^{4} c} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 86, normalized size = 0.75 \begin {gather*} \frac {14 \, a x + 18 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a^{3} x^{3} + 2 \, a^{2} x^{2} + 5 \, a x - 14\right )} \sqrt {-a^{2} x^{2} + 1} - 14}{3 \, {\left (a^{5} c x - a^{4} 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^{3}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{a x \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 [A]
time = 0.42, size = 101, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} {\left (x {\left (\frac {x}{a^{2} c} + \frac {3}{a^{3} c}\right )} + \frac {8}{a^{4} c}\right )} - \frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{a^{3} c {\left | a \right |}} + \frac {4}{a^{3} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\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 = 158, normalized size = 1.39 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{3\,c\,{\left (-a^2\right )}^{3/2}}-\frac {2}{a^2\,c\,\sqrt {-a^2}}+\frac {a^2\,x^2}{3\,c\,{\left (-a^2\right )}^{3/2}}+\frac {x\,\sqrt {-a^2}}{a^3\,c}\right )}{\sqrt {-a^2}}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^3\,c\,\sqrt {-a^2}}+\frac {2\,\sqrt {1-a^2\,x^2}}{a^3\,c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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