Integrand size = 21, antiderivative size = 49 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} \sqrt {b c-a d}} \]
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Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {385, 211} \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} \sqrt {b c-a d}} \]
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Rule 211
Rule 385
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right ) \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} \sqrt {b c-a d}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {\arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a} \sqrt {b c-a d}} \]
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Time = 2.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\) | \(41\) |
default | \(-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(306\) |
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (39) = 78\).
Time = 0.31 (sec) , antiderivative size = 241, normalized size of antiderivative = 4.92 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left [-\frac {\sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, {\left (a b c - a^{2} d\right )}}, \frac {\arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{2 \, \sqrt {a b c - a^{2} d}}\right ] \]
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\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right ) \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}}} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left \{\begin {array}{cl} \frac {\mathrm {atan}\left (\frac {x\,\sqrt {b\,c-a\,d}}{\sqrt {a}\,\sqrt {d\,x^2+c}}\right )}{\sqrt {-a\,\left (a\,d-b\,c\right )}} & \text {\ if\ \ }0<b\,c-a\,d\\ \frac {\ln \left (\frac {\sqrt {a\,\left (d\,x^2+c\right )}+x\,\sqrt {a\,d-b\,c}}{\sqrt {a\,\left (d\,x^2+c\right )}-x\,\sqrt {a\,d-b\,c}}\right )}{2\,\sqrt {a\,\left (a\,d-b\,c\right )}} & \text {\ if\ \ }b\,c-a\,d<0\\ \int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}} \,d x & \text {\ if\ \ }b\,c-a\,d\notin \mathbb {R}\vee a\,d=b\,c \end {array}\right . \]
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