Optimal. Leaf size=159 \[ \frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^3}{a^5 c^2 \sqrt {1-a^2 x^2}}-\frac {5 \sqrt {1-a^2 x^2}}{2 a^5 c^2}-\frac {(5+a x) \sqrt {1-a^2 x^2}}{6 a^5 c^2}-\frac {(5+a x)^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}+\frac {17 \text {ArcSin}(a x)}{2 a^5 c^2} \]
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Rubi [A]
time = 0.36, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6263, 866,
1649, 1639, 1668, 21, 757, 655, 222} \begin {gather*} \frac {17 \text {ArcSin}(a x)}{2 a^5 c^2}-\frac {2 (a x+1)^3}{a^5 c^2 \sqrt {1-a^2 x^2}}+\frac {(a x+1)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {(a x+5)^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {(a x+5) \sqrt {1-a^2 x^2}}{6 a^5 c^2}-\frac {5 \sqrt {1-a^2 x^2}}{2 a^5 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 222
Rule 655
Rule 757
Rule 866
Rule 1639
Rule 1649
Rule 1668
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^4}{(c-a c x)^2} \, dx &=c \int \frac {x^4 \sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx\\ &=\frac {\int \frac {x^4 (c+a c x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^5}\\ &=\frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(c+a c x)^2 \left (\frac {3}{a^4}+\frac {3 x}{a^3}+\frac {3 x^2}{a^2}+\frac {3 x^3}{a}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^4}\\ &=\frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(c+a c x)^3 \left (\frac {3}{a^4 c}+\frac {3 x^2}{a^2 c}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^4}\\ &=\frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^3}{a^5 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {\left (\frac {15}{a^4 c}+\frac {3 x}{a^3 c}\right ) (c+a c x)^2}{\sqrt {1-a^2 x^2}} \, dx}{3 c^3}\\ &=\frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^3}{a^5 c^2 \sqrt {1-a^2 x^2}}-\frac {(5+a x)^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {a^4 \int \frac {\left (-\frac {45}{a^4}-\frac {9 x}{a^3}\right ) \left (\frac {15}{a^4 c}+\frac {3 x}{a^3 c}\right )}{\sqrt {1-a^2 x^2}} \, dx}{81 c}\\ &=\frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^3}{a^5 c^2 \sqrt {1-a^2 x^2}}-\frac {(5+a x)^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}+\frac {a^4 \int \frac {\left (-\frac {45}{a^4}-\frac {9 x}{a^3}\right )^2}{\sqrt {1-a^2 x^2}} \, dx}{243 c^2}\\ &=\frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^3}{a^5 c^2 \sqrt {1-a^2 x^2}}-\frac {(5+a x) \sqrt {1-a^2 x^2}}{6 a^5 c^2}-\frac {(5+a x)^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {a^2 \int \frac {-\frac {4131}{a^6}-\frac {1215 x}{a^5}}{\sqrt {1-a^2 x^2}} \, dx}{486 c^2}\\ &=\frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^3}{a^5 c^2 \sqrt {1-a^2 x^2}}-\frac {5 \sqrt {1-a^2 x^2}}{2 a^5 c^2}-\frac {(5+a x) \sqrt {1-a^2 x^2}}{6 a^5 c^2}-\frac {(5+a x)^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}+\frac {17 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^4 c^2}\\ &=\frac {(1+a x)^3}{3 a^5 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^3}{a^5 c^2 \sqrt {1-a^2 x^2}}-\frac {5 \sqrt {1-a^2 x^2}}{2 a^5 c^2}-\frac {(5+a x) \sqrt {1-a^2 x^2}}{6 a^5 c^2}-\frac {(5+a x)^2 \sqrt {1-a^2 x^2}}{3 a^5 c^2}+\frac {17 \sin ^{-1}(a x)}{2 a^5 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 80, normalized size = 0.50 \begin {gather*} -\frac {\frac {\sqrt {1+a x} \left (80-109 a x+18 a^2 x^2+5 a^3 x^3+2 a^4 x^4\right )}{(1-a x)^{3/2}}+102 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{6 a^5 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.04, size = 275, normalized size = 1.73
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}+9 a x +34\right ) \left (a^{2} x^{2}-1\right )}{6 a^{5} \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\frac {17 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{4} \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {25 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{6} \left (x -\frac {1}{a}\right )}}{c^{2}}\) | \(162\) |
default | \(\frac {\frac {-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}}{a}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}-\frac {5 \sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{6}}+\frac {9 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}}{c^{2}}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 153, normalized size = 0.96 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{7} c^{2} x^{2} - 2 \, a^{6} c^{2} x + a^{5} c^{2}\right )}} + \frac {25 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{6} c^{2} x - a^{5} c^{2}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a^{3} c^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{4} c^{2}} + \frac {17 \, \arcsin \left (a x\right )}{2 \, a^{5} c^{2}} - \frac {17 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 125, normalized size = 0.79 \begin {gather*} -\frac {80 \, a^{2} x^{2} - 160 \, a x + 102 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{4} x^{4} + 5 \, a^{3} x^{3} + 18 \, a^{2} x^{2} - 109 \, a x + 80\right )} \sqrt {-a^{2} x^{2} + 1} + 80}{6 \, {\left (a^{7} c^{2} x^{2} - 2 \, a^{6} c^{2} x + a^{5} 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 {\int \frac {x^{4}}{a^{2} x^{2} \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 x^{5}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 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(-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 = 0.81, size = 189, normalized size = 1.19 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{3\,\left (a^7\,c^2\,x^2-2\,a^6\,c^2\,x+a^5\,c^2\right )}+\frac {25\,\sqrt {1-a^2\,x^2}}{3\,\left (a^3\,c^2\,\sqrt {-a^2}-a^4\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {17\,\sqrt {1-a^2\,x^2}}{3\,a^5\,c^2}-\frac {3\,x\,\sqrt {1-a^2\,x^2}}{2\,a^4\,c^2}+\frac {17\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^4\,c^2\,\sqrt {-a^2}}-\frac {x^2\,\sqrt {1-a^2\,x^2}}{3\,a^3\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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