Integrand size = 13, antiderivative size = 144 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {46 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{21 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {92 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{21 \left (1+\frac {1}{x}\right )^{3/2} x}+\frac {8 \sqrt {-\frac {1-x}{x}} x (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6311, 6316, 91, 79, 47, 37} \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {2 \sqrt {-\frac {1-x}{x}} (x+1)^{3/2} x^2}{7 \left (\frac {1}{x}+1\right )^{3/2}}+\frac {8 \sqrt {-\frac {1-x}{x}} (x+1)^{3/2} x}{7 \left (\frac {1}{x}+1\right )^{3/2}}+\frac {46 \sqrt {-\frac {1-x}{x}} (x+1)^{3/2}}{21 \left (\frac {1}{x}+1\right )^{3/2}}+\frac {92 \sqrt {-\frac {1-x}{x}} (x+1)^{3/2}}{21 \left (\frac {1}{x}+1\right )^{3/2} x} \]
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rule 91
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {(1+x)^{3/2} \int e^{\coth ^{-1}(x)} \left (1+\frac {1}{x}\right )^{3/2} x^{5/2} \, dx}{\left (1+\frac {1}{x}\right )^{3/2} x^{3/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \text {Subst}\left (\int \frac {(1+x)^2}{\sqrt {1-x} x^{9/2}} \, dx,x,\frac {1}{x}\right )}{\left (1+\frac {1}{x}\right )^{3/2}} \\ & = \frac {2 \sqrt {-\frac {1-x}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}}-\frac {\left (2 \left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \text {Subst}\left (\int \frac {10+\frac {7 x}{2}}{\sqrt {1-x} x^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 \left (1+\frac {1}{x}\right )^{3/2}} \\ & = \frac {8 \sqrt {-\frac {1-x}{x}} x (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}}-\frac {\left (23 \left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{5/2}} \, dx,x,\frac {1}{x}\right )}{7 \left (1+\frac {1}{x}\right )^{3/2}} \\ & = \frac {46 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{21 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {8 \sqrt {-\frac {1-x}{x}} x (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}}-\frac {\left (46 \left (\frac {1}{x}\right )^{3/2} (1+x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx,x,\frac {1}{x}\right )}{21 \left (1+\frac {1}{x}\right )^{3/2}} \\ & = \frac {46 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{21 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {92 \sqrt {-\frac {1-x}{x}} (1+x)^{3/2}}{21 \left (1+\frac {1}{x}\right )^{3/2} x}+\frac {8 \sqrt {-\frac {1-x}{x}} x (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}}+\frac {2 \sqrt {-\frac {1-x}{x}} x^2 (1+x)^{3/2}}{7 \left (1+\frac {1}{x}\right )^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.32 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} \sqrt {1+x} \left (46+23 x+12 x^2+3 x^3\right )}{21 \sqrt {1+\frac {1}{x}}} \]
[In]
[Out]
Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.26
method | result | size |
gosper | \(\frac {2 \left (x -1\right ) \left (3 x^{3}+12 x^{2}+23 x +46\right )}{21 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(37\) |
default | \(\frac {2 \left (x -1\right ) \left (3 x^{3}+12 x^{2}+23 x +46\right )}{21 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(37\) |
risch | \(\frac {2 \left (x -1\right ) \left (3 x^{3}+12 x^{2}+23 x +46\right )}{21 \sqrt {1+x}\, \sqrt {\frac {x -1}{1+x}}}\) | \(37\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.23 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {2}{21} \, {\left (3 \, x^{3} + 12 \, x^{2} + 23 \, x + 46\right )} \sqrt {x + 1} \sqrt {\frac {x - 1}{x + 1}} \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\int \frac {x \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {\frac {x - 1}{x + 1}}}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.19 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\frac {2 \, {\left (3 \, x^{4} + 9 \, x^{3} + 11 \, x^{2} + 23 \, x - 46\right )}}{21 \, \sqrt {x - 1}} \]
[In]
[Out]
Exception generated. \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 4.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.33 \[ \int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx=\sqrt {\frac {x-1}{x+1}}\,\left (\frac {46\,x\,\sqrt {x+1}}{21}+\frac {92\,\sqrt {x+1}}{21}+\frac {8\,x^2\,\sqrt {x+1}}{7}+\frac {2\,x^3\,\sqrt {x+1}}{7}\right ) \]
[In]
[Out]