Integrand size = 23, antiderivative size = 162 \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx=\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {418, 1233, 1232} \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx=\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}} \]
[In]
[Out]
Rule 418
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c}+\frac {\int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c} \\ & = \frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{2 c \sqrt {a-b x^4}}+\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{2 c \sqrt {a-b x^4}} \\ & = \frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx=-\frac {5 a c x \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{\sqrt {a-b x^4} \left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.13
method | result | size |
default | \(-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d}\) | \(183\) |
elliptic | \(-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d}\) | \(183\) |
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx=- \int \frac {1}{- c \sqrt {a - b x^{4}} + d x^{4} \sqrt {a - b x^{4}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{\sqrt {-b x^{4} + a} {\left (d x^{4} - c\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{\sqrt {-b x^{4} + a} {\left (d x^{4} - c\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx=\int \frac {1}{\sqrt {a-b\,x^4}\,\left (c-d\,x^4\right )} \,d x \]
[In]
[Out]