Integrand size = 27, antiderivative size = 137 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8+5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {3 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4} \]
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6302, 6286, 1823, 847, 794, 223, 209} \[ \int e^{2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}-\frac {3 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4}+\frac {3 (5 a x+8) \sqrt {c-a^2 c x^2}}{20 a^4} \]
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Rule 209
Rule 223
Rule 794
Rule 847
Rule 1823
Rule 6286
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx \\ & = -\left (c \int \frac {x^3 (1+a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right ) \\ & = \frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x^3 \left (-9 a^2 c-10 a^3 c x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{5 a^2} \\ & = \frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}-\frac {\int \frac {x^2 \left (30 a^3 c^2+36 a^4 c^2 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{20 a^4 c} \\ & = \frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {\int \frac {x \left (-72 a^4 c^3-90 a^5 c^3 x\right )}{\sqrt {c-a^2 c x^2}} \, dx}{60 a^6 c^2} \\ & = \frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8+5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {(3 c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{4 a^3} \\ & = \frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8+5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {(3 c) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^3} \\ & = \frac {3 x^2 \sqrt {c-a^2 c x^2}}{5 a^2}+\frac {x^3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{5} x^4 \sqrt {c-a^2 c x^2}+\frac {3 (8+5 a x) \sqrt {c-a^2 c x^2}}{20 a^4}-\frac {3 \sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^4} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.70 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (24+15 a x+12 a^2 x^2+10 a^3 x^3+4 a^4 x^4\right )+15 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{20 a^4} \]
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Time = 0.66 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.71
method | result | size |
risch | \(-\frac {\left (4 a^{4} x^{4}+10 a^{3} x^{3}+12 a^{2} x^{2}+15 a x +24\right ) \left (a^{2} x^{2}-1\right ) c}{20 a^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c}{4 a^{3} \sqrt {a^{2} c}}\) | \(97\) |
default | \(-\frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{5 a^{2} c}-\frac {4 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{5 c \,a^{4}}+\frac {x \sqrt {-a^{2} c \,x^{2}+c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}}{a^{3}}+\frac {-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 a^{2} c}+\frac {\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}}{2 a^{2}}}{a}+\frac {2 \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}-\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}}{a^{4}}\) | \(270\) |
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Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.34 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\left [\frac {2 \, {\left (4 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 15 \, a x + 24\right )} \sqrt {-a^{2} c x^{2} + c} + 15 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{40 \, a^{4}}, \frac {{\left (4 \, a^{4} x^{4} + 10 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 15 \, a x + 24\right )} \sqrt {-a^{2} c x^{2} + c} + 15 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{20 \, a^{4}}\right ] \]
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\[ \int e^{2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int e^{2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=-\frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2}}{5 \, a^{2} c} + \frac {5 \, \sqrt {-a^{2} c x^{2} + c} x}{4 \, a^{3}} - \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{2 \, a^{3} c} - \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{4 \, a^{4}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{4}} - \frac {4 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{5 \, a^{4} c} \]
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Exception generated. \[ \int e^{2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{2 \coth ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx=\int \frac {x^3\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
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