Optimal. Leaf size=113 \[ -4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6268, 25, 528,
457, 81, 52, 65, 214} \begin {gather*} -\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^2 \sqrt {c-\frac {c}{a x}}+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 25
Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{x^3 (1+a x)} \, dx\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x^2 (1+a x)} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^3} \, dx}{c}\\ &=\frac {a \text {Subst}\left (\int \frac {x \left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {a^2 \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\left (2 a^2\right ) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right )\\ &=-4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\left (4 a^2 c\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\left (8 a^3\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-4 a^2 \sqrt {c-\frac {c}{a x}}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^2 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 79, normalized size = 0.70 \begin {gather*} -\frac {2 \sqrt {c-\frac {c}{a x}} \left (3-11 a x+38 a^2 x^2\right )}{15 x^2}+4 \sqrt {2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs.
\(2(94)=188\).
time = 0.39, size = 278, normalized size = 2.46
method | result | size |
risch | \(-\frac {2 \left (38 a^{3} x^{3}-49 a^{2} x^{2}+14 a x -3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} \left (a x -1\right )}-\frac {2 a^{2} \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {a c x \left (a x -1\right )}}{\sqrt {c}\, \left (a x -1\right )}\) | \(152\) |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (30 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}\, x^{4}-90 a^{\frac {7}{2}} \sqrt {a \,x^{2}-x}\, \sqrt {\frac {1}{a}}\, x^{4}-30 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{4}+60 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2} \sqrt {\frac {1}{a}}+45 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{4}-45 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{4}-16 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+6 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{15 x^{3} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}}\) | \(278\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 185, normalized size = 1.64 \begin {gather*} \left [\frac {2 \, {\left (15 \, \sqrt {2} a^{2} \sqrt {c} x^{2} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) - {\left (38 \, a^{2} x^{2} - 11 \, a x + 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{15 \, x^{2}}, -\frac {2 \, {\left (30 \, \sqrt {2} a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (38 \, a^{2} x^{2} - 11 \, a x + 3\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{15 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {\sqrt {c - \frac {c}{a x}}}{a x^{4} + x^{3}}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{4} + x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs.
\(2 (94) = 188\).
time = 0.74, size = 278, normalized size = 2.46 \begin {gather*} -\frac {4 \, \sqrt {2} a^{3} c \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (60 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{5} c - 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{4} c^{\frac {3}{2}} {\left | a \right |} + 35 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{5} c^{2} - 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{4} c^{\frac {5}{2}} {\left | a \right |} + 3 \, a^{5} c^{3}\right )}}{15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{2} {\left | a \right |} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.74, size = 96, normalized size = 0.85 \begin {gather*} -4\,a^2\,\sqrt {c-\frac {c}{a\,x}}-\frac {2\,a^2\,{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{3\,c}-\frac {2\,a^2\,{\left (c-\frac {c}{a\,x}\right )}^{5/2}}{5\,c^2}-\sqrt {2}\,a^2\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-\frac {c}{a\,x}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________