Optimal. Leaf size=291 \[ \frac{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i \sqrt [4]{a+i a x} (a-i a x)^{3/4}}{3 a}-\frac{7 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{7 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{7 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{7 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.180831, antiderivative size = 291, 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, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i \sqrt [4]{a+i a x} (a-i a x)^{3/4}}{3 a}-\frac{7 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{7 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{7 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{7 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(a-i a x)^{7/4}}{(a+i a x)^{7/4}} \, dx &=\frac{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}-\frac{7}{3} \int \frac{(a-i a x)^{3/4}}{(a+i a x)^{3/4}} \, dx\\ &=\frac{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{3 a}-\frac{1}{2} (7 a) \int \frac{1}{\sqrt [4]{a-i a x} (a+i a x)^{3/4}} \, dx\\ &=\frac{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{3 a}-14 i \operatorname{Subst}\left (\int \frac{x^2}{\left (2 a-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{a-i a x}\right )\\ &=\frac{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{3 a}-14 i \operatorname{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{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{3 a}+7 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 )-7 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)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{3 a}-\frac{7}{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{7}{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{(7 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{(7 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)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{3 a}-\frac{7 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{7 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{(7 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{(7 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)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i (a-i a x)^{3/4} \sqrt [4]{a+i a x}}{3 a}+\frac{7 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{7 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{7 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{7 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 = 0.0273962, size = 70, normalized size = 0.24 \[ \frac{i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{11/4} \, _2F_1\left (\frac{7}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2}-\frac{i x}{2}\right )}{11 a^2 (a+i a x)^{3/4}} \]
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{7}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{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{7}{4}}}{{\left (i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70361, size = 701, normalized size = 2.41 \begin{align*} -\frac{3 \, \sqrt{49 i}{\left (a x - i \, a\right )} \log \left (\frac{\sqrt{49 i}{\left (a x + i \, a\right )} + 7 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{7 \, x + 7 i}\right ) - 3 \, \sqrt{49 i}{\left (a x - i \, a\right )} \log \left (-\frac{\sqrt{49 i}{\left (a x + i \, a\right )} - 7 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{7 \, x + 7 i}\right ) + 3 \, \sqrt{-49 i}{\left (a x - i \, a\right )} \log \left (\frac{\sqrt{-49 i}{\left (a x + i \, a\right )} + 7 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{7 \, x + 7 i}\right ) - 3 \, \sqrt{-49 i}{\left (a x - i \, a\right )} \log \left (-\frac{\sqrt{-49 i}{\left (a x + i \, a\right )} - 7 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{7 \, x + 7 i}\right ) + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}{\left (-3 i \, x - 11\right )}}{6 \,{\left (a x - i \, a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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