Integrand size = 29, antiderivative size = 102 \[ \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx=-\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}+a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {893, 879, 889, 214} \[ \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx=a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}} \]
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Rule 214
Rule 879
Rule 889
Rule 893
Rubi steps \begin{align*} \text {integral}& = -\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac {1}{2} (a c) \int \frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}} \, dx \\ & = -\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac {1}{2} a \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\left (a^3 c^2\right ) \text {Subst}\left (\int \frac {1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \\ & = -\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}+a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx=\frac {\sqrt {c-a c x} \left ((1+2 a x) \sqrt {1-a^2 x^2}+a x \sqrt {-1+a x} \arctan \left (\frac {\sqrt {-1+a x}}{\sqrt {1-a^2 x^2}}\right )\right )}{x (-1+a x)} \]
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Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\left (-\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a c x +2 a x \sqrt {c \left (a x +1\right )}\, \sqrt {c}+\sqrt {c \left (a x +1\right )}\, \sqrt {c}\right ) \sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x \sqrt {c}}\) | \(95\) |
risch | \(\frac {\left (2 a^{2} x^{2}+3 a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{x \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {a \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(147\) |
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Time = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.13 \[ \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx=\left [\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (2 \, a x + 1\right )}}{2 \, {\left (a x^{2} - x\right )}}, \frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (2 \, a x + 1\right )}}{a x^{2} - x}\right ] \]
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\[ \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx=\int \frac {\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{x^2} \,d x \]
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