Optimal. Leaf size=48 \[ -a \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )+\frac {b \sin ^{-1}(d x)}{d}-\frac {c \sqrt {1-d^2 x^2}}{d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {1609, 1809, 844, 216, 266, 63, 208} \[ -a \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )+\frac {b \sin ^{-1}(d x)}{d}-\frac {c \sqrt {1-d^2 x^2}}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1609
Rule 1809
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{x \sqrt {1-d x} \sqrt {1+d x}} \, dx &=\int \frac {a+b x+c x^2}{x \sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-d^2 x^2}}{d^2}-\frac {\int \frac {-a d^2-b d^2 x}{x \sqrt {1-d^2 x^2}} \, dx}{d^2}\\ &=-\frac {c \sqrt {1-d^2 x^2}}{d^2}+a \int \frac {1}{x \sqrt {1-d^2 x^2}} \, dx+b \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-d^2 x^2}}{d^2}+\frac {b \sin ^{-1}(d x)}{d}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-d^2 x}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {1-d^2 x^2}}{d^2}+\frac {b \sin ^{-1}(d x)}{d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\frac {1}{d^2}-\frac {x^2}{d^2}} \, dx,x,\sqrt {1-d^2 x^2}\right )}{d^2}\\ &=-\frac {c \sqrt {1-d^2 x^2}}{d^2}+\frac {b \sin ^{-1}(d x)}{d}-a \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 48, normalized size = 1.00 \[ -a \tanh ^{-1}\left (\sqrt {1-d^2 x^2}\right )+\frac {b \sin ^{-1}(d x)}{d}-\frac {c \sqrt {1-d^2 x^2}}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 81, normalized size = 1.69 \[ \frac {a d^{2} \log \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{x}\right ) - 2 \, b d \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right ) - \sqrt {d x + 1} \sqrt {-d x + 1} c}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.03, size = 96, normalized size = 2.00 \[ \frac {\left (-a \,d^{2} \arctanh \left (\frac {1}{\sqrt {-d^{2} x^{2}+1}}\right ) \mathrm {csgn}\relax (d )+b d \arctan \left (\frac {d x \,\mathrm {csgn}\relax (d )}{\sqrt {-\left (d x +1\right ) \left (d x -1\right )}}\right )-\sqrt {-d^{2} x^{2}+1}\, c \,\mathrm {csgn}\relax (d )\right ) \sqrt {-d x +1}\, \sqrt {d x +1}\, \mathrm {csgn}\relax (d )}{\sqrt {-d^{2} x^{2}+1}\, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 57, normalized size = 1.19 \[ -a \log \left (\frac {2 \, \sqrt {-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {b \arcsin \left (d x\right )}{d} - \frac {\sqrt {-d^{2} x^{2} + 1} c}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.92, size = 122, normalized size = 2.54 \[ a\,\left (\ln \left (\frac {{\left (\sqrt {1-d\,x}-1\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}-1\right )-\ln \left (\frac {\sqrt {1-d\,x}-1}{\sqrt {d\,x+1}-1}\right )\right )-\frac {\sqrt {1-d\,x}\,\left (\frac {c}{d^2}+\frac {c\,x}{d}\right )}{\sqrt {d\,x+1}}-\frac {4\,b\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {1-d\,x}-1\right )}{\left (\sqrt {d\,x+1}-1\right )\,\sqrt {d^2}}\right )}{\sqrt {d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 55.20, size = 245, normalized size = 5.10 \[ \frac {i a {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {a {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i b {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} + \frac {b {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {i c {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} - \frac {c {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________