Optimal. Leaf size=146 \[ \frac {(1+a x)^2}{a^5 c \sqrt {1-a^2 x^2}}+\frac {13 \sqrt {1-a^2 x^2}}{3 a^5 c}+\frac {11 x \sqrt {1-a^2 x^2}}{8 a^4 c}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}-\frac {27 \text {ArcSin}(a x)}{8 a^5 c} \]
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Rubi [A]
time = 0.24, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {27 \text {ArcSin}(a x)}{8 a^5 c}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}+\frac {13 \sqrt {1-a^2 x^2}}{3 a^5 c}+\frac {(a x+1)^2}{a^5 c \sqrt {1-a^2 x^2}}+\frac {11 x \sqrt {1-a^2 x^2}}{8 a^4 c}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 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^4}{c-a c x} \, dx &=c \int \frac {x^4 \sqrt {1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=\frac {\int \frac {x^4 (c+a c x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {(1+a x)^2}{a^5 c \sqrt {1-a^2 x^2}}-\frac {\int \frac {(c+a c x) \left (\frac {2}{a^4}+\frac {x}{a^3}+\frac {x^2}{a^2}+\frac {x^3}{a}\right )}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {(1+a x)^2}{a^5 c \sqrt {1-a^2 x^2}}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}+\frac {\int \frac {-\frac {8 c}{a^2}-\frac {12 c x}{a}-11 c x^2-8 a c x^3}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2 c^2}\\ &=\frac {(1+a x)^2}{a^5 c \sqrt {1-a^2 x^2}}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}-\frac {\int \frac {24 c+52 a c x+33 a^2 c x^2}{\sqrt {1-a^2 x^2}} \, dx}{12 a^4 c^2}\\ &=\frac {(1+a x)^2}{a^5 c \sqrt {1-a^2 x^2}}+\frac {11 x \sqrt {1-a^2 x^2}}{8 a^4 c}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}+\frac {\int \frac {-81 a^2 c-104 a^3 c x}{\sqrt {1-a^2 x^2}} \, dx}{24 a^6 c^2}\\ &=\frac {(1+a x)^2}{a^5 c \sqrt {1-a^2 x^2}}+\frac {13 \sqrt {1-a^2 x^2}}{3 a^5 c}+\frac {11 x \sqrt {1-a^2 x^2}}{8 a^4 c}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}-\frac {27 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4 c}\\ &=\frac {(1+a x)^2}{a^5 c \sqrt {1-a^2 x^2}}+\frac {13 \sqrt {1-a^2 x^2}}{3 a^5 c}+\frac {11 x \sqrt {1-a^2 x^2}}{8 a^4 c}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{3 a^3 c}+\frac {x^3 \sqrt {1-a^2 x^2}}{4 a^2 c}-\frac {27 \sin ^{-1}(a x)}{8 a^5 c}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 81, normalized size = 0.55 \begin {gather*} \frac {-\frac {\sqrt {1+a x} \left (-128+47 a x+17 a^2 x^2+10 a^3 x^3+6 a^4 x^4\right )}{\sqrt {1-a x}}+162 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{24 a^5 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 212, normalized size = 1.45
method | result | size |
risch | \(-\frac {\left (6 a^{3} x^{3}+16 a^{2} x^{2}+33 a x +80\right ) \left (a^{2} x^{2}-1\right )}{24 a^{5} \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\frac {27 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{4} \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}}{c}\) | \(130\) |
default | \(-\frac {-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {11 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {11 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}+\frac {-\frac {2 x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {4 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}}{a}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}}{c}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 129, normalized size = 0.88 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{4 \, a^{2} c} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6} c x - a^{5} c} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a^{3} c} + \frac {11 \, \sqrt {-a^{2} x^{2} + 1} x}{8 \, a^{4} c} - \frac {27 \, \arcsin \left (a x\right )}{8 \, a^{5} c} + \frac {10 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 95, normalized size = 0.65 \begin {gather*} \frac {128 \, a x + 162 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (6 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 47 \, a x - 128\right )} \sqrt {-a^{2} x^{2} + 1} - 128}{24 \, {\left (a^{6} c x - a^{5} 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^{4}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{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 [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 = 0.82, size = 163, normalized size = 1.12 \begin {gather*} \frac {10\,\sqrt {1-a^2\,x^2}}{3\,a^5\,c}-\frac {2\,\sqrt {1-a^2\,x^2}}{\sqrt {-a^2}\,\left (a^3\,c\,\sqrt {-a^2}-a^4\,c\,x\,\sqrt {-a^2}\right )}+\frac {11\,x\,\sqrt {1-a^2\,x^2}}{8\,a^4\,c}-\frac {27\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,a^4\,c\,\sqrt {-a^2}}+\frac {x^3\,\sqrt {1-a^2\,x^2}}{4\,a^2\,c}+\frac {2\,x^2\,\sqrt {1-a^2\,x^2}}{3\,a^3\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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