Optimal. Leaf size=287 \[ \frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i (a+i a x)^{3/4} \sqrt [4]{a-i a x}}{a}+\frac{5 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}-\frac{5 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{5 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{5 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.177951, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {47, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i (a+i a x)^{3/4} \sqrt [4]{a-i a x}}{a}+\frac{5 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}-\frac{5 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{5 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{5 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(a-i a x)^{5/4}}{(a+i a x)^{5/4}} \, dx &=\frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}-5 \int \frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}} \, dx\\ &=\frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac{1}{2} (5 a) \int \frac{1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx\\ &=\frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-10 i \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2 a-x^4}} \, dx,x,\sqrt [4]{a-i a x}\right )\\ &=\frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-10 i \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )\\ &=\frac{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-5 i \operatorname{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 )-5 i \operatorname{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{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}-\frac{5}{2} i \operatorname{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{5}{2} i \operatorname{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{(5 i) \operatorname{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{(5 i) \operatorname{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{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+\frac{5 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{5 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{(5 i) \operatorname{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{(5 i) \operatorname{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{4 i (a-i a x)^{5/4}}{a \sqrt [4]{a+i a x}}+\frac{5 i \sqrt [4]{a-i a x} (a+i a x)^{3/4}}{a}+\frac{5 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{5 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}+\frac{5 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{5 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*}
Mathematica [C] time = 0.0270854, size = 70, normalized size = 0.24 \[ \frac{i 2^{3/4} \sqrt [4]{1+i x} (a-i a x)^{9/4} \, _2F_1\left (\frac{5}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2}-\frac{i x}{2}\right )}{9 a^2 \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a-iax \right ) ^{{\frac{5}{4}}} \left ( a+iax \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-i \, a x + a\right )}^{\frac{5}{4}}}{{\left (i \, a x + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81101, size = 686, normalized size = 2.39 \begin{align*} -\frac{\sqrt{25 i}{\left (a x - i \, a\right )} \log \left (\frac{\sqrt{25 i}{\left (a x - i \, a\right )} + 5 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{5 \, x - 5 i}\right ) - \sqrt{25 i}{\left (a x - i \, a\right )} \log \left (-\frac{\sqrt{25 i}{\left (a x - i \, a\right )} - 5 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{5 \, x - 5 i}\right ) + \sqrt{-25 i}{\left (a x - i \, a\right )} \log \left (\frac{\sqrt{-25 i}{\left (a x - i \, a\right )} + 5 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{5 \, x - 5 i}\right ) - \sqrt{-25 i}{\left (a x - i \, a\right )} \log \left (-\frac{\sqrt{-25 i}{\left (a x - i \, a\right )} - 5 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{5 \, x - 5 i}\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, x - 9\right )}}{2 \,{\left (a x - i \, a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- a \left (i x - 1\right )\right )^{\frac{5}{4}}}{\left (a \left (i x + 1\right )\right )^{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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