Optimal. Leaf size=117 \[ \sqrt {c-\frac {c}{a^2 x^2}}-\frac {a x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt {1-a x} \sqrt {a x+1}}+\frac {2 a x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{\sqrt {1-a x} \sqrt {a x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.55, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6167, 6159, 6129, 98, 157, 41, 216, 92, 208} \[ \sqrt {c-\frac {c}{a^2 x^2}}-\frac {a x \sqrt {c-\frac {c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt {1-a x} \sqrt {a x+1}}+\frac {2 a x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{\sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 92
Rule 98
Rule 157
Rule 208
Rule 216
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x} \, dx\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^2} \, 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^2 \sqrt {1-a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\sqrt {c-\frac {c}{a^2 x^2}}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {-2 a-a^2 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\sqrt {c-\frac {c}{a^2 x^2}}-\frac {\left (2 a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {\left (a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\sqrt {c-\frac {c}{a^2 x^2}}-\frac {\left (a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (2 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\sqrt {c-\frac {c}{a^2 x^2}}-\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x \sin ^{-1}(a x)}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{\sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 82, normalized size = 0.70 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {a^2 x^2-1}+a x \log \left (\sqrt {a^2 x^2-1}+a x\right )-2 a x \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )\right )}{\sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 252, normalized size = 2.15 \[ \left [-\sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}, -2 \, \sqrt {c} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \frac {1}{2} \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 127, normalized size = 1.09 \[ {\left (\frac {4 \, \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x)}{a} - \frac {\sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{{\left | a \right |}} + \frac {2 \, c^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 306, normalized size = 2.62 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (-\sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{2} a^{3} c +a^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}+\sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x a -2 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x a -2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x \,a^{2} c +2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} \sqrt {-\frac {c}{a^{2}}}\, c x +2 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x \,c^{2}\right )}{a \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x - 1\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{x\,\left (a\,x-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x \left (a x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________