Integrand size = 19, antiderivative size = 70 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {400, 211} \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)} \]
[In]
[Out]
Rule 211
Rule 400
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \frac {1}{a+b x^2} \, dx}{b c-a d}-\frac {d \int \frac {1}{c+d x^2} \, dx}{b c-a d} \\ & = \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} (b c-a d)}-\frac {\sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c}}}{b c-a d} \]
[In]
[Out]
Time = 2.78 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right ) \sqrt {a b}}+\frac {d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \sqrt {c d}}\) | \(55\) |
risch | \(\frac {\sqrt {-c d}\, \ln \left (d x +\sqrt {-c d}\right )}{2 c \left (a d -b c \right )}-\frac {\sqrt {-c d}\, \ln \left (d x -\sqrt {-c d}\right )}{2 c \left (a d -b c \right )}+\frac {\sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right )}{2 a \left (a d -b c \right )}-\frac {\sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right )}{2 a \left (a d -b c \right )}\) | \(136\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 292, normalized size of antiderivative = 4.17 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\left [-\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{2 \, {\left (b c - a d\right )}}, -\frac {2 \, \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{2 \, {\left (b c - a d\right )}}, \frac {2 \, \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{2 \, {\left (b c - a d\right )}}, \frac {\sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right )}{b c - a d}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 712 vs. \(2 (60) = 120\).
Time = 2.73 (sec) , antiderivative size = 712, normalized size of antiderivative = 10.17 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\sqrt {- \frac {b}{a}} \log {\left (x + \frac {- \frac {a^{4} c d^{3} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{3} b c^{2} d^{2} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{2} c^{3} d \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} d^{2} \sqrt {- \frac {b}{a}}}{a d - b c} - \frac {a b^{3} c^{4} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{2} c^{2} \sqrt {- \frac {b}{a}}}{a d - b c}}{b d} \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {b}{a}} \log {\left (x + \frac {\frac {a^{4} c d^{3} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{3} b c^{2} d^{2} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{2} c^{3} d \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} d^{2} \sqrt {- \frac {b}{a}}}{a d - b c} + \frac {a b^{3} c^{4} \left (- \frac {b}{a}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{2} c^{2} \sqrt {- \frac {b}{a}}}{a d - b c}}{b d} \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {d}{c}} \log {\left (x + \frac {- \frac {a^{4} c d^{3} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{3} b c^{2} d^{2} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} b^{2} c^{3} d \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} d^{2} \sqrt {- \frac {d}{c}}}{a d - b c} - \frac {a b^{3} c^{4} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b^{2} c^{2} \sqrt {- \frac {d}{c}}}{a d - b c}}{b d} \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {d}{c}} \log {\left (x + \frac {\frac {a^{4} c d^{3} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{3} b c^{2} d^{2} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a^{2} b^{2} c^{3} d \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a^{2} d^{2} \sqrt {- \frac {d}{c}}}{a d - b c} + \frac {a b^{3} c^{4} \left (- \frac {d}{c}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b^{2} c^{2} \sqrt {- \frac {d}{c}}}{a d - b c}}{b d} \right )}}{2 \left (a d - b c\right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} {\left (b c - a d\right )}} - \frac {d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c - a d\right )} \sqrt {c d}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} {\left (b c - a d\right )}} - \frac {d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c - a d\right )} \sqrt {c d}} \]
[In]
[Out]
Time = 0.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.93 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\ln \left (b\,x-\sqrt {-a\,b}\right )\,\sqrt {-a\,b}}{2\,a^2\,d-2\,a\,b\,c}-\frac {\ln \left (d\,x+\sqrt {-c\,d}\right )\,\sqrt {-c\,d}}{2\,\left (b\,c^2-a\,c\,d\right )}-\frac {\ln \left (b\,x+\sqrt {-a\,b}\right )\,\sqrt {-a\,b}}{2\,\left (a^2\,d-a\,b\,c\right )}+\frac {\ln \left (d\,x-\sqrt {-c\,d}\right )\,\sqrt {-c\,d}}{2\,b\,c^2-2\,a\,c\,d} \]
[In]
[Out]