Integrand size = 27, antiderivative size = 156 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=-\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}+\frac {7 a^4 \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 \sqrt {1-a x} \sqrt {1+a x}} \]
[Out]
Time = 0.40 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6302, 6294, 6264, 100, 156, 12, 94, 214} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}+\frac {7 a^4 x \text {arctanh}\left (\sqrt {1-a x} \sqrt {a x+1}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{8 \sqrt {1-a x} \sqrt {a x+1}}-\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}} \]
[In]
[Out]
Rule 12
Rule 94
Rule 100
Rule 156
Rule 214
Rule 6264
Rule 6294
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx \\ & = -\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^5} \, dx}{\sqrt {1-a x} \sqrt {1+a x}} \\ & = -\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{x^5 \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {8 a-7 a^2 x}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{4 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {21 a^2-16 a^3 x}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{12 \sqrt {1-a x} \sqrt {1+a x}} \\ & = \frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {32 a^3-21 a^4 x}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{24 \sqrt {1-a x} \sqrt {1+a x}} \\ & = -\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {21 a^4}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{24 \sqrt {1-a x} \sqrt {1+a x}} \\ & = -\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}-\frac {\left (7 a^4 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 \sqrt {1-a x} \sqrt {1+a x}} \\ & = -\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}+\frac {\left (7 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \text {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{8 \sqrt {1-a x} \sqrt {1+a x}} \\ & = -\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}+\frac {7 a^4 \sqrt {c-\frac {c}{a^2 x^2}} x \text {arctanh}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 \sqrt {1-a x} \sqrt {1+a x}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.60 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (-6+16 a x-21 a^2 x^2+32 a^3 x^3\right )+21 a^4 x^4 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )\right )}{24 x^3 \sqrt {-1+a^2 x^2}} \]
[In]
[Out]
Time = 0.60 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.02
method | result | size |
risch | \(-\frac {\left (32 a^{5} x^{5}-21 a^{4} x^{4}-16 a^{3} x^{3}+15 a^{2} x^{2}-16 a x +6\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{24 x^{3} \left (a^{2} x^{2}-1\right )}-\frac {7 a^{4} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {c \left (a^{2} x^{2}-1\right )}\, x}{8 \sqrt {-c}\, \left (a^{2} x^{2}-1\right )}\) | \(159\) |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a^{2} \left (-48 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{3} c \,x^{5}+48 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3} x^{3}+48 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{4}-48 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+c x}{\sqrt {c}}\right ) a \,x^{4}+48 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c \,x^{4}-21 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c \,x^{4}-27 \sqrt {-\frac {c}{a^{2}}}\, {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{2} x^{2}-21 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{2} x^{4}+16 a {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} x \sqrt {-\frac {c}{a^{2}}}-6 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{24 x^{3} \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c}\) | \(410\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.38 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\left [\frac {21 \, a^{3} \sqrt {-c} x^{3} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, x^{3}}, -\frac {21 \, a^{3} \sqrt {c} x^{3} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, x^{3}}\right ] \]
[In]
[Out]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{4} \left (a x + 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )} x^{4}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (128) = 256\).
Time = 1.77 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.03 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\frac {1}{12} \, {\left (21 \, a^{2} \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right ) - \frac {21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} a^{2} c \mathrm {sgn}\left (x\right ) + 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} a^{2} c^{2} \mathrm {sgn}\left (x\right ) + 96 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a^{2} c^{3} \mathrm {sgn}\left (x\right ) + 128 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a^{2} c^{4} \mathrm {sgn}\left (x\right ) + 32 \, a c^{\frac {9}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4}}\right )} {\left | a \right |} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x-1\right )}{x^4\,\left (a\,x+1\right )} \,d x \]
[In]
[Out]