Integrand size = 22, antiderivative size = 196 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {49 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{15 a \sqrt {c-\frac {c}{a x}}}+\frac {31 c^3 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}}{15 a}+\frac {7 c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 a}+c \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2} x-\frac {5 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6317, 6314, 99, 158, 152, 65, 214} \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {x \left (a-\frac {1}{x}\right )^3 \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (80 a-\frac {31}{x}\right ) \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {5 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}{a \left (1-\frac {1}{a x}\right )^{7/2}} \]
[In]
[Out]
Rule 65
Rule 99
Rule 152
Rule 158
Rule 214
Rule 6314
Rule 6317
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-\frac {c}{a x}\right )^{7/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{7/2} \, dx}{\left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {\left (c-\frac {c}{a x}\right )^{7/2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3 \sqrt {1+\frac {x}{a}}}{x^2} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (c-\frac {c}{a x}\right )^{7/2} \text {Subst}\left (\int \frac {\left (-\frac {5}{2 a}-\frac {7 x}{2 a^2}\right ) \left (1-\frac {x}{a}\right )^2}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (2 a \left (c-\frac {c}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {25}{4 a^2}-\frac {31 x}{4 a^3}\right ) \left (1-\frac {x}{a}\right )}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {\left (80 a-\frac {31}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (5 \left (c-\frac {c}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {\left (80 a-\frac {31}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (5 \left (c-\frac {c}{a x}\right )^{7/2}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {\left (80 a-\frac {31}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {5 \left (c-\frac {c}{a x}\right )^{7/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{7/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.52 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} \left (6-28 a x+56 a^2 x^2+15 a^3 x^3\right )-75 a^2 x^2 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{15 a^3 \sqrt {1-\frac {1}{a x}} x^2} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (30 a^{\frac {7}{2}} x^{3} \sqrt {\left (a x +1\right ) x}+112 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}-75 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} x^{3}-56 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+12 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{30 \sqrt {\frac {a x -1}{a x +1}}\, x^{2} a^{\frac {7}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(149\) |
risch | \(\frac {\left (15 a^{4} x^{4}+71 a^{3} x^{3}+28 a^{2} x^{2}-22 a x +6\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {5 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(176\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.12 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\left [\frac {75 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (15 \, a^{4} c^{3} x^{4} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{60 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac {75 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (15 \, a^{4} c^{3} x^{4} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{30 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\right ] \]
[In]
[Out]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
[In]
[Out]