Integrand size = 25, antiderivative size = 256 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx=-\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}-\frac {3 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {3 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {3 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}-\frac {3 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx=-\frac {3 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {3 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}+\frac {3 i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}}-\frac {3 i \log \left (\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt {2}} \]
[In]
[Out]
Rule 52
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}+\frac {1}{2} (3 a) \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{3/4}} \, dx \\ & = -\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}+6 i \text {Subst}\left (\int \frac {x^2}{\left (2 a-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{a-i a x}\right ) \\ & = -\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}+6 i \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right ) \\ & = -\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}-3 i \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )+3 i \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right ) \\ & = -\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}+\frac {3}{2} i \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )+\frac {3}{2} i \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )+\frac {(3 i) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}+\frac {(3 i) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}} \\ & = -\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}+\frac {3 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}-\frac {3 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}+\frac {(3 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}-\frac {(3 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}} \\ & = -\frac {i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{a}-\frac {3 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {3 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt {2}}+\frac {3 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}-\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}}-\frac {3 i \log \left (1+\frac {\sqrt {a-i a x}}{\sqrt {a+i a x}}+\frac {\sqrt {2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{2 \sqrt {2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.27 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx=\frac {2 i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{7/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{4},\frac {11}{4},\frac {1}{2}-\frac {i x}{2}\right )}{7 a (a+i a x)^{3/4}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.12 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.82
method | result | size |
risch | \(-\frac {i \left (x +i\right ) \left (x -i\right ) a}{\left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}+\frac {\left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \ln \left (-\frac {-\left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) x^{2}+x^{3}-i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {3}{4}}+i \sqrt {-x^{4}+2 i x^{3}+2 i x +1}\, x +2 i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x -2 i x^{2}+\sqrt {-x^{4}+2 i x^{3}+2 i x +1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}}-x}{\left (i x +1\right )^{2}}\right )}{2}-\frac {3 i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \ln \left (-\frac {-i \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}} x +x^{3}-i \sqrt {-x^{4}+2 i x^{3}+2 i x +1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {3}{4}}+i \operatorname {RootOf}\left (\textit {\_Z}^{2}-i\right ) \left (-x^{4}+2 i x^{3}+2 i x +1\right )^{\frac {1}{4}}-2 i x^{2}-\sqrt {-x^{4}+2 i x^{3}+2 i x +1}-x}{\left (i x +1\right )^{2}}\right )}{2}\right ) \left (-\left (i x -1\right ) \left (i x +1\right )^{3}\right )^{\frac {1}{4}} a}{\left (a \left (i x +1\right )\right )^{\frac {3}{4}} \left (-a \left (i x -1\right )\right )^{\frac {1}{4}}}\) | \(465\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.77 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx=\frac {\sqrt {9 i} a \log \left (\frac {\sqrt {9 i} {\left (a x + i \, a\right )} + 3 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{3 \, {\left (x + i\right )}}\right ) - \sqrt {9 i} a \log \left (-\frac {\sqrt {9 i} {\left (a x + i \, a\right )} - 3 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{3 \, {\left (x + i\right )}}\right ) + \sqrt {-9 i} a \log \left (\frac {\sqrt {-9 i} {\left (a x + i \, a\right )} + 3 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{3 \, {\left (x + i\right )}}\right ) - \sqrt {-9 i} a \log \left (-\frac {\sqrt {-9 i} {\left (a x + i \, a\right )} - 3 \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{3 \, {\left (x + i\right )}}\right ) - 2 i \, {\left (i \, a x + a\right )}^{\frac {1}{4}} {\left (-i \, a x + a\right )}^{\frac {3}{4}}}{2 \, a} \]
[In]
[Out]
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx=\int \frac {\left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {3}{4}}}\, dx \]
[In]
[Out]
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {3}{4}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{3/4}} \,d x \]
[In]
[Out]