Integrand size = 25, antiderivative size = 133 \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx=-\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}-\frac {4 i}{7 a^3 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}+\frac {32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}} \]
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Time = 0.02 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {47, 37} \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx=\frac {32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}}+\frac {16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}-\frac {4 i}{7 a^3 (a+i a x)^{5/4} (a-i a x)^{3/4}}-\frac {2 i}{7 a^2 (a+i a x)^{5/4} (a-i a x)^{7/4}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}+\frac {6 \int \frac {1}{(a-i a x)^{7/4} (a+i a x)^{9/4}} \, dx}{7 a} \\ & = -\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}-\frac {4 i}{7 a^3 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {8 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{9/4}} \, dx}{7 a^2} \\ & = -\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}-\frac {4 i}{7 a^3 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}+\frac {16 \int \frac {1}{(a-i a x)^{3/4} (a+i a x)^{5/4}} \, dx}{35 a^3} \\ & = -\frac {2 i}{7 a^2 (a-i a x)^{7/4} (a+i a x)^{5/4}}-\frac {4 i}{7 a^3 (a-i a x)^{3/4} (a+i a x)^{5/4}}+\frac {16 i \sqrt [4]{a-i a x}}{35 a^4 (a+i a x)^{5/4}}+\frac {32 i \sqrt [4]{a-i a x}}{35 a^5 \sqrt [4]{a+i a x}} \\ \end{align*}
Time = 7.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx=\frac {2 \left (9 i+22 x+8 i x^2+16 x^3\right )}{35 a^4 (a-i a x)^{3/4} \sqrt [4]{a+i a x} \left (1+x^2\right )} \]
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Time = 0.40 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.36
method | result | size |
gosper | \(-\frac {2 \left (x +i\right ) \left (-x +i\right ) \left (16 x^{3}+8 i x^{2}+22 x +9 i\right )}{35 \left (-i a x +a \right )^{\frac {11}{4}} \left (i a x +a \right )^{\frac {9}{4}}}\) | \(48\) |
risch | \(\frac {\frac {32}{35} x^{3}+\frac {16}{35} i x^{2}+\frac {44}{35} x +\frac {18}{35} i}{a^{4} \left (-a \left (i x -1\right )\right )^{\frac {3}{4}} \left (a \left (i x +1\right )\right )^{\frac {1}{4}} \left (x -i\right ) \left (x +i\right )}\) | \(56\) |
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Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx=\frac {2 \, {\left (16 \, x^{3} + 8 i \, x^{2} + 22 \, x + 9 i\right )} {\left (i \, a x + a\right )}^{\frac {3}{4}} {\left (-i \, a x + a\right )}^{\frac {1}{4}}}{35 \, {\left (a^{6} x^{4} + 2 \, a^{6} x^{2} + a^{6}\right )}} \]
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Timed out. \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx=\int { \frac {1}{{\left (i \, a x + a\right )}^{\frac {9}{4}} {\left (-i \, a x + a\right )}^{\frac {11}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.73 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.42 \[ \int \frac {1}{(a-i a x)^{11/4} (a+i a x)^{9/4}} \, dx=\frac {2\,{\left (-a\,\left (-1+x\,1{}\mathrm {i}\right )\right )}^{1/4}\,\left (x^4\,16{}\mathrm {i}+8\,x^3+x^2\,30{}\mathrm {i}+13\,x+9{}\mathrm {i}\right )}{35\,a^5\,{\left (x^2+1\right )}^2\,{\left (a\,\left (1+x\,1{}\mathrm {i}\right )\right )}^{1/4}} \]
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