Integrand size = 22, antiderivative size = 321 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^7 \sqrt {1-\frac {1}{a^2 x^2}} x^6}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 \sqrt {1-\frac {1}{a^2 x^2}} x^5}-\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6332, 6328, 90} \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {c^3 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^7 x^6 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 x^5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
[In]
[Out]
Rule 90
Rule 6328
Rule 6332
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^3 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{7/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\left (c^3 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(-1+a x)^3 (1+a x)^4}{x^7} \, dx}{a^7 \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {\left (c^3 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^7-\frac {1}{x^7}-\frac {a}{x^6}+\frac {3 a^2}{x^5}+\frac {3 a^3}{x^4}-\frac {3 a^4}{x^3}-\frac {3 a^5}{x^2}+\frac {a^6}{x}\right ) \, dx}{a^7 \sqrt {1-\frac {1}{a^2 x^2}}} \\ & = \frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^7 \sqrt {1-\frac {1}{a^2 x^2}} x^6}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 \sqrt {1-\frac {1}{a^2 x^2}} x^5}-\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.29 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (\frac {1}{6 a^7 x^6}+\frac {1}{5 a^6 x^5}-\frac {3}{4 a^5 x^4}-\frac {1}{a^4 x^3}+\frac {3}{2 a^3 x^2}+\frac {3}{a^2 x}+x+\frac {\log (x)}{a}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.35
method | result | size |
default | \(\frac {\left (60 a^{7} x^{7}+60 a^{6} \ln \left (x \right ) x^{6}+180 a^{5} x^{5}+90 a^{4} x^{4}-60 a^{3} x^{3}-45 a^{2} x^{2}+12 a x +10\right ) {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}} x}{60 \left (a x +1\right ) \left (a^{2} x^{2}-1\right )^{3} \sqrt {\frac {a x -1}{a x +1}}}\) | \(112\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.30 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\frac {{\left (60 \, a^{7} c^{3} x^{7} + 60 \, a^{6} c^{3} x^{6} \log \left (x\right ) + 180 \, a^{5} c^{3} x^{5} + 90 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} - 45 \, a^{2} c^{3} x^{2} + 12 \, a c^{3} x + 10 \, c^{3}\right )} \sqrt {a^{2} c}}{60 \, a^{8} x^{6}} \]
[In]
[Out]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
[In]
[Out]
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
[In]
[Out]