Integrand size = 29, antiderivative size = 76 \[ \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {x \sqrt {-1+c^2 x^2}}{2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {\sqrt {-1+c^2 x^2} \text {arctanh}(c x)}{2 c d \sqrt {d-c^2 d x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {23, 205, 214} \[ \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\text {arctanh}(c x) \sqrt {d-c^2 d x^2}}{2 c d^2 \sqrt {c^2 x^2-1}}+\frac {x \sqrt {d-c^2 d x^2}}{2 d^2 \left (1-c^2 x^2\right ) \sqrt {c^2 x^2-1}} \]
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Rule 23
Rule 205
Rule 214
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d-c^2 d x^2} \int \frac {1}{\left (d-c^2 d x^2\right )^2} \, dx}{\sqrt {-1+c^2 x^2}} \\ & = \frac {x \sqrt {d-c^2 d x^2}}{2 d^2 \left (1-c^2 x^2\right ) \sqrt {-1+c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \int \frac {1}{d-c^2 d x^2} \, dx}{2 d \sqrt {-1+c^2 x^2}} \\ & = \frac {x \sqrt {d-c^2 d x^2}}{2 d^2 \left (1-c^2 x^2\right ) \sqrt {-1+c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{2 c d^2 \sqrt {-1+c^2 x^2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {c x+\left (1-c^2 x^2\right ) \text {arctanh}(c x)}{2 c d \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}} \]
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Time = 2.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.24
method | result | size |
default | \(-\frac {\sqrt {-\left (c^{2} x^{2}-1\right ) d}\, \left (\ln \left (c x -1\right ) c^{2} x^{2}-\ln \left (c x +1\right ) c^{2} x^{2}+2 c x -\ln \left (c x -1\right )+\ln \left (c x +1\right )\right )}{4 \sqrt {c^{2} x^{2}-1}\, d^{2} \left (c x -1\right ) c \left (c x +1\right )}\) | \(94\) |
risch | \(\frac {x}{2 d \sqrt {c^{2} x^{2}-1}\, \sqrt {-\left (c^{2} x^{2}-1\right ) d}}-\frac {\sqrt {c^{2} x^{2}-1}\, \ln \left (-c x -1\right )}{4 d \sqrt {-\left (c^{2} x^{2}-1\right ) d}\, c}+\frac {\sqrt {c^{2} x^{2}-1}\, \ln \left (c x -1\right )}{4 d \sqrt {-\left (c^{2} x^{2}-1\right ) d}\, c}\) | \(112\) |
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none
Time = 0.30 (sec) , antiderivative size = 314, normalized size of antiderivative = 4.13 \[ \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx=\left [-\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c x + {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right )}{8 \, {\left (c^{5} d^{2} x^{4} - 2 \, c^{3} d^{2} x^{2} + c d^{2}\right )}}, -\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c x - {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right )}{4 \, {\left (c^{5} d^{2} x^{4} - 2 \, c^{3} d^{2} x^{2} + c d^{2}\right )}}\right ] \]
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\[ \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {\left (c x - 1\right ) \left (c x + 1\right )} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} \sqrt {c^{2} x^{2} - 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} \sqrt {c^{2} x^{2} - 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d-c^2\,d\,x^2\right )}^{3/2}\,\sqrt {c^2\,x^2-1}} \,d x \]
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