Optimal. Leaf size=373 \[ \frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {475 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4} \]
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Rubi [A]
time = 0.18, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 13, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5170, 99,
158, 152, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {475 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {i (-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac {475 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5170
Rubi steps
\begin {align*} \int e^{\frac {5}{2} i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1+i a x)^{5/4}}{(1-i a x)^{5/4}} \, dx\\ &=-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac {(4 i) \int \frac {x^2 \sqrt [4]{1+i a x} \left (3+\frac {17 i a x}{4}\right )}{\sqrt [4]{1-i a x}} \, dx}{a}\\ &=-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}+\frac {i \int \frac {x \sqrt [4]{1+i a x} \left (-\frac {17 i a}{2}+\frac {113 a^2 x}{8}\right )}{\sqrt [4]{1-i a x}} \, dx}{a^3}\\ &=-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {(475 i) \int \frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx}{64 a^3}\\ &=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {(475 i) \int \frac {1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{128 a^3}\\ &=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{32 a^4}\\ &=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{32 a^4}\\ &=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {475 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 a^4}+\frac {475 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 a^4}\\ &=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 a^4}+\frac {475 \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 a^4}+\frac {475 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}+\frac {475 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}\\ &=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}+\frac {475 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {475 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}\\ &=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {475 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 96, normalized size = 0.26 \begin {gather*} \frac {-\sqrt [4]{1+i a x} (-i+a x)^2 \left (59-5 i a x+6 a^2 x^2\right )+380 \sqrt [4]{2} (1-i a x) \, _2F_1\left (-\frac {5}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-i a x)\right )}{24 a^4 \sqrt [4]{1-i a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}} x^{3}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.88, size = 251, normalized size = 0.67 \begin {gather*} -\frac {96 \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} \log \left (\frac {64}{475} \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} \log \left (-\frac {64}{475} \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 96 \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} \log \left (\frac {64}{475} \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} \log \left (-\frac {64}{475} \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + {\left (48 \, a^{4} x^{4} - 136 i \, a^{3} x^{3} - 226 \, a^{2} x^{2} + 521 i \, a x - 2467\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{192 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x^3\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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