Integrand size = 25, antiderivative size = 115 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {49, 50, 42, 203, 202} \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=-\frac {6 \sqrt [4]{x^2+1} E\left (\left .\frac {\arctan (x)}{2}\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}} \]
[In]
[Out]
Rule 42
Rule 49
Rule 50
Rule 202
Rule 203
Rubi steps \begin{align*} \text {integral}& = \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {3}{5} \int \frac {1}{\sqrt [4]{a-i a x} (a+i a x)^{5/4}} \, dx \\ & = \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {1}{5} (3 a) \int \frac {1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx \\ & = \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {\left (3 a \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac {1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{5 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {\left (3 \sqrt [4]{1+x^2}\right ) \int \frac {1}{\left (1+x^2\right )^{5/4}} \, dx}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ & = \frac {4 i (a-i a x)^{3/4}}{5 a (a+i a x)^{5/4}}-\frac {6 i}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac {6 \sqrt [4]{1+x^2} E\left (\left .\frac {1}{2} \tan ^{-1}(x)\right |2\right )}{5 a \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.61 \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\frac {i \sqrt [4]{1+i x} (a-i a x)^{7/4} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},\frac {9}{4},\frac {11}{4},\frac {1}{2}-\frac {i x}{2}\right )}{7 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {2 \left (3 x^{2}+2 i x +1\right )}{5 \left (x -i\right ) a \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}+\frac {3 x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^{2}\right ) \left (-a^{2} \left (i x -1\right ) \left (i x +1\right )\right )^{\frac {1}{4}}}{5 \left (a^{2}\right )^{\frac {1}{4}} a \left (-a \left (i x -1\right )\right )^{\frac {1}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}}}\) | \(107\) |
[In]
[Out]
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int \frac {\left (- i a \left (x + i\right )\right )^{\frac {3}{4}}}{\left (i a \left (x - i\right )\right )^{\frac {9}{4}}}\, dx \]
[In]
[Out]
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int { \frac {{\left (-i \, a x + a\right )}^{\frac {3}{4}}}{{\left (i \, a x + a\right )}^{\frac {9}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a-i a x)^{3/4}}{(a+i a x)^{9/4}} \, dx=\int \frac {{\left (a-a\,x\,1{}\mathrm {i}\right )}^{3/4}}{{\left (a+a\,x\,1{}\mathrm {i}\right )}^{9/4}} \,d x \]
[In]
[Out]