Integrand size = 29, antiderivative size = 33 \[ \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{x}+\frac {b \text {arccosh}(c x)}{c} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {465, 54} \[ \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {a \sqrt {c x-1} \sqrt {c x+1}}{x}+\frac {b \text {arccosh}(c x)}{c} \]
[In]
[Out]
Rule 54
Rule 465
Rubi steps \begin{align*} \text {integral}& = \frac {a \sqrt {-1+c x} \sqrt {1+c x}}{x}+b \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {a \sqrt {-1+c x} \sqrt {1+c x}}{x}+\frac {b \cosh ^{-1}(c x)}{c} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{x}+\frac {2 b \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{c} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.33
method | result | size |
default | \(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\operatorname {csgn}\left (c \right ) c \sqrt {c^{2} x^{2}-1}\, a +\ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right )+c x \right ) \operatorname {csgn}\left (c \right )\right ) b x \right ) \operatorname {csgn}\left (c \right )}{\sqrt {c^{2} x^{2}-1}\, x c}\) | \(77\) |
risch | \(\frac {a \sqrt {c x -1}\, \sqrt {c x +1}}{x}+\frac {b \ln \left (\frac {c^{2} x}{\sqrt {c^{2}}}+\sqrt {c^{2} x^{2}-1}\right ) \sqrt {\left (c x -1\right ) \left (c x +1\right )}}{\sqrt {c^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}}\) | \(78\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {a c^{2} x + \sqrt {c x + 1} \sqrt {c x - 1} a c - b x \log \left (-c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{c x} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 15.32 (sec) , antiderivative size = 148, normalized size of antiderivative = 4.48 \[ \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx=- \frac {a c {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i a c {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {b {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c} - \frac {i b {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {b \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c} + \frac {\sqrt {c^{2} x^{2} - 1} a}{x} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {\frac {16 \, a c^{2}}{{\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 4} - b \log \left ({\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4}\right )}{2 \, c} \]
[In]
[Out]
Time = 6.85 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.85 \[ \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {a\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{x}-\frac {4\,b\,\mathrm {atan}\left (\frac {c\,\left (\sqrt {c\,x-1}-\mathrm {i}\right )}{\left (\sqrt {c\,x+1}-1\right )\,\sqrt {-c^2}}\right )}{\sqrt {-c^2}} \]
[In]
[Out]