Optimal. Leaf size=92 \[ x \sqrt {c-\frac {c}{a x}}-\frac {5 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6167, 6133, 25, 514, 375, 98, 156, 63, 208} \[ x \sqrt {c-\frac {c}{a x}}-\frac {5 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 25
Rule 63
Rule 98
Rule 156
Rule 208
Rule 375
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\\ &=-\int \frac {\sqrt {c-\frac {c}{a x}} (1-a x)}{1+a x} \, dx\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x}{1+a x} \, dx}{c}\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{a+\frac {1}{x}} \, dx}{c}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\sqrt {c-\frac {c}{a x}} x+\frac {\operatorname {Subst}\left (\int \frac {\frac {5 c^2}{2}-\frac {3 c^2 x}{2 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\sqrt {c-\frac {c}{a x}} x+\frac {(5 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}-\frac {(4 c) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\sqrt {c-\frac {c}{a x}} x-5 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )+8 \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=\sqrt {c-\frac {c}{a x}} x-\frac {5 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 92, normalized size = 1.00 \[ x \sqrt {c-\frac {c}{a x}}-\frac {5 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 219, normalized size = 2.38 \[ \left [\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} + 4 \, \sqrt {2} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 5 \, \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a}, \frac {a x \sqrt {\frac {a c x - c}{a x}} - 4 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + 5 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 190, normalized size = 2.07 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-4 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+4 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}+6 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{2 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________