Optimal. Leaf size=689 \[ \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {\sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}-\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2} \]
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Rubi [A] time = 0.49, antiderivative size = 689, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5062, 80, 50, 63, 331, 299, 1122, 1169, 634, 618, 204, 628} \[ \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {\sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}-\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 204
Rule 299
Rule 331
Rule 618
Rule 628
Rule 634
Rule 1122
Rule 1169
Rule 5062
Rubi steps
\begin {align*} \int e^{\frac {1}{4} i \tan ^{-1}(a x)} x \, dx &=\int \frac {x \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx\\ &=\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \int \frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx}{8 a}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \int \frac {1}{\sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx}{32 a}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-i a x}\right )}{4 a^2}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\operatorname {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{4 a^2}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\operatorname {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 \sqrt {2} a^2}-\frac {\operatorname {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 \sqrt {2} a^2}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {\operatorname {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 \sqrt {2} a^2}+\frac {\operatorname {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 \sqrt {2} a^2}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 \sqrt {2 \left (2-\sqrt {2}\right )} a^2}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 \sqrt {2 \left (2-\sqrt {2}\right )} a^2}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 \sqrt {2 \left (2+\sqrt {2}\right )} a^2}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 \sqrt {2 \left (2+\sqrt {2}\right )} a^2}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 a^2}+\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \operatorname {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \operatorname {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 a^2}+\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 a^2}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^2}-\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^2}-\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^2}-\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^2}\\ &=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}-\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 63, normalized size = 0.09 \[ \frac {(1-i a x)^{7/8} \left (2 \sqrt [8]{2} \, _2F_1\left (-\frac {1}{8},\frac {7}{8};\frac {15}{8};\frac {1}{2} (1-i a x)\right )+7 (1+i a x)^{9/8}\right )}{14 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.90, size = 428, normalized size = 0.62 \[ -\frac {8 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - {\left (4 \, a^{2} x^{2} - i \, a x + 5\right )} \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {1}{4}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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