Optimal. Leaf size=117 \[ -\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{7 x^2 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {8 a^3 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {8 a^3 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{35 \sqrt {c-\frac {c}{a x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6178, 871, 795, 649} \[ \frac {8 a^3 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{7 x^2 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {8 a^3 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{35 \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 649
Rule 795
Rule 871
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2} x^2}+\frac {1}{7} (4 a c) \operatorname {Subst}\left (\int \frac {x \sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {8 a^3 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{35 \sqrt {c-\frac {c}{a x}}}-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2} x^2}+\frac {1}{35} \left (4 a^2 c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 a^3 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {8 a^3 c \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{35 \sqrt {c-\frac {c}{a x}}}-\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{7 \left (c-\frac {c}{a x}\right )^{3/2} x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 66, normalized size = 0.56 \[ -\frac {2 a \sqrt {1-\frac {1}{a^2 x^2}} \left (8 a^3 x^3-4 a^2 x^2+3 a x+15\right ) \sqrt {c-\frac {c}{a x}}}{105 x^2 (a x-1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 77, normalized size = 0.66 \[ -\frac {2 \, {\left (8 \, a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} + 18 \, a x + 15\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a x^{4} - x^{3}\right )}} \]
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 = 0.04, size = 55, normalized size = 0.47 \[ -\frac {2 \left (a x +1\right ) \left (8 a^{2} x^{2}-12 a x +15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{105 x^{3} \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c - \frac {c}{a x}}}{x^{4} \sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.41, size = 100, normalized size = 0.85 \[ -\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (8\,a^3\,x^3+12\,a^2\,x^2+11\,a\,x+29\right )\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3}-\frac {88\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3\,\left (a\,x-1\right )} \]
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]
________________________________________________________________________________________