Optimal. Leaf size=266 \[ \frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]
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Rubi [A] time = 0.139989, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {47, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 47
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)^{3/4}}{(a+i a x)^{7/4}} \, dx &=\frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\int \frac{1}{\sqrt [4]{a-i a x} (a+i a x)^{3/4}} \, dx\\ &=\frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac{(4 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 )}{a}\\ &=\frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac{(4 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 )}{a}\\ &=\frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}+\frac{(2 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 )}{a}-\frac{(2 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 )}{a}\\ &=\frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac{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 )}{a}-\frac{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 )}{a}-\frac{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 )}{\sqrt{2} a}-\frac{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 )}{\sqrt{2} a}\\ &=\frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}-\frac{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 )}{\sqrt{2} a}+\frac{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 )}{\sqrt{2} a}-\frac{\left (i \sqrt{2}\right ) \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 )}{a}+\frac{\left (i \sqrt{2}\right ) \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 )}{a}\\ &=\frac{4 i (a-i a x)^{3/4}}{3 a (a+i a x)^{3/4}}+\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{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 )}{\sqrt{2} a}+\frac{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 )}{\sqrt{2} a}\\ \end{align*}
Mathematica [C] time = 0.0223752, size = 70, normalized size = 0.26 \[ \frac{i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{7/4} \, _2F_1\left (\frac{7}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2}-\frac{i x}{2}\right )}{7 a^2 (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a-iax \right ) ^{{\frac{3}{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{3}{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 = 1.67501, size = 763, normalized size = 2.87 \begin{align*} -\frac{3 \,{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x + i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{2 \, x + 2 i}\right ) - 3 \,{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x + i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{2 \, x + 2 i}\right ) + 3 \,{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x + i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{2 \, x + 2 i}\right ) - 3 \,{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x + i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{2 \, x + 2 i}\right ) - 8 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{6 \,{\left (a^{2} x - i \, a^{2}\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{3}{4}}}{\left (a \left (i x + 1\right )\right )^{\frac{7}{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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