Integrand size = 26, antiderivative size = 48 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c-d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {455, 65, 223, 212} \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c-d x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Rule 65
Rule 212
Rule 223
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a-b x} \sqrt {c-d x}} \, dx,x,x^2\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a-b x^2}\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a-b x^2}}{\sqrt {c-d x^2}}\right )}{b} \\ & = -\frac {\tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-b x^2}}{\sqrt {b} \sqrt {c-d x^2}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c-d x^2}}{\sqrt {d} \sqrt {a-b x^2}}\right )}{\sqrt {b} \sqrt {d}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(36)=72\).
Time = 3.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.98
method | result | size |
default | \(\frac {\ln \left (-\frac {-2 b d \,x^{2}+a d +b c -2 \sqrt {\left (-b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}{2 \sqrt {b d}\, \sqrt {\left (-b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}}\) | \(95\) |
elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \ln \left (\frac {-\frac {1}{2} a d -\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (-a d -b c \right ) x^{2}+a c}\right )}{2 \sqrt {-b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}\, \sqrt {b d}}\) | \(95\) |
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (36) = 72\).
Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 4.23 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx=\left [\frac {\sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} - b c - a d\right )} \sqrt {-b x^{2} + a} \sqrt {-d x^{2} + c} \sqrt {b d}\right )}{4 \, b d}, -\frac {\sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} - b c - a d\right )} \sqrt {-b x^{2} + a} \sqrt {-d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d - {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right )}{2 \, b d}\right ] \]
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\[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx=\int \frac {x}{\sqrt {a - b x^{2}} \sqrt {c - d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx=\frac {b \log \left ({\left | -\sqrt {-b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c - {\left (b x^{2} - a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} {\left | b \right |}} \]
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Time = 5.43 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int \frac {x}{\sqrt {a-b x^2} \sqrt {c-d x^2}} \, dx=\frac {2\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {c-d\,x^2}-\sqrt {c}\right )}{\sqrt {-b\,d}\,\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}\right )}{\sqrt {-b\,d}} \]
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