Optimal. Leaf size=118 \[ \frac {x \sqrt {c-\frac {c}{a x}}}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6177, 879, 869, 875, 208} \[ \frac {x \sqrt {c-\frac {c}{a x}}}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 869
Rule 875
Rule 879
Rule 6177
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {5 \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}\\ &=\frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {5 \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a c}\\ &=\frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {(5 c) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a^3}\\ &=\frac {5 \sqrt {c-\frac {c}{a x}}}{a c \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {c-\frac {c}{a x}} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 69, normalized size = 0.58 \[ \frac {\sqrt {1-\frac {1}{a x}} \left (5 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {1}{a x}\right )+a x\right )}{a \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.16, size = 303, normalized size = 2.57 \[ \left [\frac {5 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c x - a c\right )}}, \frac {5 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c x - a c\right )}}\right ] \]
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 [A] time = 0.06, size = 149, normalized size = 1.26 \[ -\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+5 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x a -10 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+5 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right )^{2} \sqrt {a}\, c \sqrt {\left (a x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\sqrt {c - \frac {c}{a x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{\sqrt {c-\frac {c}{a\,x}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________